28:null 29:["$","title",null,{"children":"Commentary on even-degree trigonometric interpolation | Jacob Rus | Observable"}] 2e:T179a,Gauss_1865 = md` * * * ### Gauss (1865)
“Theoria Interpolationis Methodo Nova Tractata” [§10–11](https://archive.org/details/werkecarlf03gausrich/page/n292/mode/2up) This was likely written circa 1805, but was only published posthumously.

${md` 10. Sit ${$`X`} functio arcus indeterminati ${$`x`} huius formae ${$$`\begin{alignedat}{4} \alpha &{}+ \alpha'& \cos{x} +{}& \alpha''& \cos{2x} +{}& \alpha'''& \cos{3x} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{x} +{}& \beta'' & \sin{2x} +{}& \beta''' & \sin{3x} +{} \mathrm{etc.} \end{alignedat}`} quae non excurrat in infinitum, sed cum ${$`\cos{mx}`} et ${$`\sin{mx}`} abrumpatur, ita ut multitudo coëfficientium (incognitorum) sit ${$`2m + 1`}. Pro totidem valoribus diversis ipsius ${$`x`}, puta ${$`a`}, ${$`b`}, ${$`c`}, ${$`d`} etc. dati sint valores respondentes functionis ${$`X`} puta ${$`A`}, ${$`B`}, ${$`C`}, ${$`D`} . . . (Ceterum valores ipsius ${$`x`}, quorum diiferentia est peripheria integra sive eius multiplum, manifesto hic pro diversis haberi nequeunt). Ex his datis quaeritur formula pro valore ${$`T`}, quem functio ${$`X`} pro quocunque alio valore ipsius ${$`x`}, puta ${$`t`} nanciscitur. Habentur itaque ${$nb`2m + 2`} aequationes ${$$`\begin{alignedat}{4} A = \alpha &{}+ \alpha'& \cos{a} +{}& \alpha''& \cos{2a} +{}& \alpha'''& \cos{3a} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{a} +{}& \beta'' & \sin{2a} +{}& \beta''' & \sin{3a} +{} \mathrm{etc.} \\[.3em] B = \alpha &{}+ \alpha'& \cos{b} +{}& \alpha''& \cos{2b} +{}& \alpha'''& \cos{3b} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{b} +{}& \beta'' & \sin{2b} +{}& \beta''' & \sin{3b} +{} \mathrm{etc.} \\[.3em] C = \alpha &{}+ \alpha'& \cos{c} +{}& \alpha''& \cos{2c} +{}& \alpha'''& \cos{3c} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{c} +{}& \beta'' & \sin{2c} +{}& \beta''' & \sin{3c} +{} \mathrm{etc.} \\[.3em] D = \alpha &{}+ \alpha'& \cos{d} +{}& \alpha''& \cos{2d} +{}& \alpha'''& \cos{3d} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{d} +{}& \beta'' & \sin{2d} +{}& \beta''' & \sin{3d} +{} \mathrm{etc.} \\ \quad \mathrm{etc.} \\ T = \alpha &{}+ \alpha'&{} \cos{t} +{}& \alpha''&{} \cos{2t} +{}& \alpha'''&{} \cos{3t} +{} \mathrm{etc.} \\ &{}+ \beta' &{} \sin{t} +{}& \beta'' &{} \sin{2t} +{}& \beta''' &{} \sin{3t} +{} \mathrm{etc.} \end{alignedat}`} Multiplicando has aequationes resp. per ${$$`\begin{aligned} &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm \sin\!\tfrac{1}{2}(a - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm \sin\!\tfrac{1}{2}(b - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm \sin\!\tfrac{1}{2}(c - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm \sin\!\tfrac{1}{2}(d - t)}} \\[1.2em] &\:\mathrm{etc.}\\ &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} \end{aligned}`} addendoque producta prodeat ${$$`\begin{aligned} &\frac{\small{A}}{\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm \sin\!\tfrac{1}{2}(a - t)}} \\[.8em] {}+{}&\frac{\small{B}}{\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm \sin\!\tfrac{1}{2}(b - t)}} \\[.8em] {}+{}&\frac{\small{C}}{\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm \sin\!\tfrac{1}{2}(c - t)}} \\[.8em] {}+{}&\frac{\small{D}}{\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm \sin\!\tfrac{1}{2}(d - t)}} \\[1.2em] {}+{}&\:\mathrm{etc.}\\ {}+{}&\frac{\small{T}}{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} = W \end{aligned}`} Tunc ex art. 2, ubi ${$`m`} idem erat, quod hic est ${$`2m + 2`}, adeoque casus secundus locum habet, fit ${$`W = 0`}; quamobrem multiplicando per ${$$` \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm `} prodit ${$$`\begin{aligned} T = \quad&\frac{\small{ \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm }} A \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm }} B \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm }} C \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm }} D \\[1.2em] {}+{}&\:\mathrm{etc.}\\ \end{aligned}`} Quum haec formula indefinite pro valore quocunque ipsius ${$`t`} locum habeat, manifestum est, si producta sinuum in numeratoribus in cosinus sinusque arcuum multiplicium evolvantur, id quod provenit cum ${$$`\begin{alignedat}{4} \alpha &{}+ \alpha'& \cos{x} +{}& \alpha''& \cos{2x} +{}& \alpha'''& \cos{3x} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{x} +{}& \beta'' & \sin{2x} +{}& \beta''' & \sin{3x} +{} \mathrm{etc.} \end{alignedat}`} *identicum* esse debere, unde coëfficientes ${$`\alpha, \alpha', \beta', \alpha'', \beta''`} etc. innotescent. Ceterum formula pro ${$`T`}, ut hie exhibita est, ita est comparata, ut sponte et sine calculo pateat, substitutis pro ${$`t`} resp. ${$`a`}, ${$`b`}, ${$`c`}, ${$`d`} etc. valoribus propositis ${$`A`}, ${$`B`}, ${$`C`}, ${$`D`} etc. probe satisfieri. `}

` 2f:T2013,md`

${md` 11. Si multitudo valorum datorum ${$`A`}, ${$`B`}, ${$`C`}, ${$`D`} etc. unitate minor esset, quam supposuimus, puta ${$`= 2m`}, hi non sufficient ad determinationem ${$`2m+1`} coëfficientium incognitorum , nisi inter eos relatio aliqua cognita fuerit. In hoc itaque casu expressio generalis pro ${$`T`} aliter exhiberi nequit, nisi ita, ut constantem incognitam implicet. Ratiocinia art praec. hie adhiberi non possunt, quoniam summatio ${$`W = 0`}, ex art. 2 petita suppositioni innixa est, multitudinem quantitatum ${$`A`}, ${$`B`} . . . ${$`T`} esse parem. Formula quidem in art. praec. pro ${$`T`} inventa valoribus ${$`A`}, ${$`B`} etc. generaliter satisfacit: sive herum multitudo par sit sive impar: sed levi attentione perspicitur, ex evolutione productorum e sinubus prodire, in nostro casu, expressionem propositae non similem, sed huius formae ${$$`\begin{alignedat}{4} &{}\mathfrak{A}& \cos{\tfrac{1}{2}t} +{}& \mathfrak{A}'& \cos{\tfrac{3}{2}t} +{}& \mathfrak{A}''& \cos{\tfrac{5}{2}t} +{} \mathrm{etc.} \\[.4em] {}+{}& \mathfrak{B}& \sin{\tfrac{1}{2}t} +{}& \mathfrak{B}'& \sin{\tfrac{3}{2}t} +{}& \mathfrak{B}''& \sin{\tfrac{5}{2}t} +{} \mathrm{etc.} \end{alignedat}`} Quamobrem pro hoc casu viam aliam ingrodi oportebit. Multiplicatur ${$`X`} per ${$`\cos\bigl(\tfrac{1}{2}x+k\bigr)`}, proditque, si termini ultimi in ${$`X`} statuuntur ${$`\alpha^m \cos{mx} + \beta^m \sin{mx}`} ${$$`\begin{alignedat}{4} & \alpha& \cos\bigl(\tfrac{1}{2}x + k\bigr) &{}+ \tfrac{1}{2}\alpha' & \cos\bigl(\tfrac{3}{2}x + k\bigr) &{}+ \tfrac{1}{2}\alpha'' \cos\bigl(\tfrac{5}{2}x + k\bigr) + \dots \\[.3em] &&&&&{}+ \tfrac{1}{2}\alpha^m \cos\bigl(\bigl(m - \tfrac{1}{2}\bigr)x + k\bigr) \\[.3em] {}+\tfrac{1}{2} & \alpha' & \cos\bigl(\tfrac{1}{2}x - k\bigr) &{}+ \tfrac{1}{2}\alpha'' & \cos\bigl(\tfrac{3}{2}x - k\bigr) &{}+ \tfrac{1}{2}\alpha''' \cos\bigl(\tfrac{5}{2}x - k\bigr) + \dots \\[.3em] &&&& &{}+ \tfrac{1}{2}\alpha^m \cos\bigl(\bigl(m - \tfrac{1}{2}\bigr)x - k\bigr) \\[.3em] && &{}+ \tfrac{1}{2}\beta' & \sin\bigl(\tfrac{3}{2}x + k\bigr) &{}+ \tfrac{1}{2}\beta'' \sin\bigl(\tfrac{5}{2}x + k\bigr) + \dots \\[.3em] &&&&&{}+ \tfrac{1}{2}\beta^m \sin\bigl(\bigl(m - \tfrac{1}{2}\bigr)x + k\bigr) \\[.3em] {}+\tfrac{1}{2} & \beta' & \sin\bigl(\tfrac{1}{2}x - k\bigr) &{}+ \tfrac{1}{2}\beta'' & \sin\bigl(\tfrac{3}{2}x - k\bigr) &{}+ \tfrac{1}{2}\beta''' \sin\bigl(\tfrac{5}{2}x - k\bigr) + \dots \\[.3em] &&&& &{}+ \tfrac{1}{2}\beta^m \sin\bigl(\bigl(m - \tfrac{1}{2}\bigr)x - k\bigr) \end{alignedat}`} Iam si in hac expressione pro ${$`x`} successive substituitur ${$`a`}, ${$`b`}, ${$`c`}, ${$`d`} . . . . ${$`t`}, quod inde provenit resp. per ${$$`\begin{aligned} &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm \sin\!\tfrac{1}{2}(a - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm \sin\!\tfrac{1}{2}(b - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm \sin\!\tfrac{1}{2}(c - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm \sin\!\tfrac{1}{2}(d - t)}} \\[1.2em] &\:\mathrm{etc.}\\ &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} \end{aligned}`} multiplicatur, productorumque aggregatum ${$`= W`} ponitur: ex iis, quae in art. 2, (in casu priori, ubi ${$`m`} idem erat, quod hic est ${$`2m + 1`}) tradidimus, facile sequitur, haberi ${$$`\begin{alignedat}{2} W = \quad&\tfrac{1}{2}(2i)^{2m}\alpha^m \cos &\bigl( \tfrac{1}{2}(a + b + c + d + \dots + t) + k \bigr) \\[.3em] {}+{}&\tfrac{1}{2}(2i)^{2m}\beta^m \sin &\bigl( \tfrac{1}{2}(a + b + c + d + \dots + t) + k \bigr) \end{alignedat}`} Ex altera vero parte erit ${$$`\begin{aligned} W = \quad&\frac{\small{A\cos\bigl(\tfrac{1}{2}a+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm \sin\!\tfrac{1}{2}(a - t)}} \\[.8em] {}+{}&\frac{\small{B\cos\bigl(\tfrac{1}{2}b+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm \sin\!\tfrac{1}{2}(b - t)}} \\[.8em] {}+{}&\frac{\small{C\cos\bigl(\tfrac{1}{2}c+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm \sin\!\tfrac{1}{2}(c - t)}} \\[.8em] {}+{}&\frac{\small{D\cos\bigl(\tfrac{1}{2}d+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm \sin\!\tfrac{1}{2}(d - t)}} \\[1.2em] {}+{}&\:\mathrm{etc.}\\ {}+{}&\frac{\small{T\cos\bigl(\tfrac{1}{2}t+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} \end{aligned}`} Quodsi iam quantitatem ${$`k`}, quae prorsus arbitraria est, ${$`= -\tfrac{1}{2}t`} ponimus, summam autem ${$`a + b + c + d + \dots = s`} (exclusa ${$`t`}), prodit, multiplicando per ${$` \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm`} ${$$`\begin{aligned} \rule{}{2em} T = \quad&\frac{\small{ \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm }} A \cos\!\tfrac{1}{2}(t - a) \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm }} B \cos\!\tfrac{1}{2}(t - b) \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm }} C \cos\!\tfrac{1}{2}(t - c) \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm }} D \cos\!\tfrac{1}{2}(t - d) \\[1.0em] {}+{}&\:\mathrm{etc.}\\[.2em] {}+{}& 2^{2m-1}i^{2m} \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d) \dotsm \\[.3em] & \qquad \qquad \quad \times \left\{ \alpha^m\cos{\tfrac{1}{2}s} + \beta^m\sin{\tfrac{1}{2}s} \right\} \end{aligned}`} Circa hanc formulam quasdam adhuc annotationes adiicimus I. Est ${$`i^{2m} = +1`}, vel ${$`-1`}, prout ${$`m`} par est vel impar. II. Quum formula indefinite pro quovis valore ipsius ${$`t`} valeat, necessario, evolutis sinuum productis in sinus et cosinus arcuum multiplicium, cum formula pro ${$`X`}, si pro ${$`x`} scribitur ${$`t`}, identica erit, i. e. cum ${$$`\begin{alignedat}{4} \alpha &{}+ \alpha'& \cos{t} +{}& \alpha''& \cos{2t} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{t} +{}& \beta'' & \sin{2t} +{} \mathrm{etc.} \end{alignedat}`} III. Determinantur itaque omnes coefficientes ${$`\alpha`}, ${$`\alpha'`}, ${$`\beta'`}, ${$`\alpha''`}, ${$`\beta''`} etc. per quantitates cognitas, atque unicam incognitam ${$`\alpha^m \cos{\!\tfrac{1}{2}s} + \beta^m \sin{\!\tfrac{1}{2}s}`}, quamobrem, si qua inter illos relatio insuper datur, omnes ex asse facile poterunt assignari. IV. Quodsi autem tantummodo expressio eiusdem formae ut ${$$`\begin{alignedat}{4} \alpha &{}+ \alpha'& \cos{t} +{}& \alpha''& \cos{2t} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{t} +{}& \beta'' & \sin{2t} +{} \mathrm{etc.} \end{alignedat}`} quaeritur (sive cum hac identica sit , sive non), per quam valores propositi ${$`A`}, ${$`B`}, ${$`C`}, ${$`D`} etc. repraesententur, si pro ${$`t`} resp. ${$`a`}, ${$`b`}, ${$`c`}, ${$`d`} etc. substituitur: quantitas ${$`\alpha^m \cos{\!\tfrac{1}{2}s} + \beta^m \sin{\!\tfrac{1}{2}s}`} ex incognita *arbitraria* evadit, quam si placet etiam ${$`= 0`} statuere licebit; reveraque formula nostra pro ${$`T`} ita comparata est, ut absque calculo sponte pateat, hacce eam proprietate praeditam esse. `}

` 30:T16f3,md` My attempt at a translation which is inelegant and likely inaccurate – I don’t speak Latin:

${md` 10\. Let ${$`X`} be a function of angle measure ${$`x`} of the form ${$$`\begin{alignedat}{4} \alpha &{}+ \alpha'& \cos{x} +{}& \alpha''& \cos{2x} +{}& \alpha'''& \cos{3x} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{x} +{}& \beta'' & \sin{2x} +{}& \beta''' & \sin{3x} +{} \mathrm{etc.} \end{alignedat}`} which does not extend infinitely, but terminates with ${$`\cos{mx}`} and ${$nb`\sin{mx}`}, so that the number of (unknown) coefficients is ${$`2m + 1`}. For several values of ${$`x`}, say ${$nb`a`}, ${$nb`b`}, ${$nb`c`}, ${$`d`} etc. there are given corresponding values of the function ${$`X`}, say ${$nb`A`}, ${$nb`B`}, ${$nb`C`}, ${$nb`D`} . . . (Furthermore these values of ${$`x`} are distinct, not differing by an integer multiple of the circumference). From these given values we seek a formula for the value ${$nb`T`}, the value of function ${$`X`} at an arbitrary ${$`x`} value, say ${$`t`}. We have ${$nb`2m + 2`} equations ${$$`\begin{alignedat}{4} A = \alpha &{}+ \alpha'& \cos{a} +{}& \alpha''& \cos{2a} +{}& \alpha'''& \cos{3a} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{a} +{}& \beta'' & \sin{2a} +{}& \beta''' & \sin{3a} +{} \mathrm{etc.} \\[.3em] B = \alpha &{}+ \alpha'& \cos{b} +{}& \alpha''& \cos{2b} +{}& \alpha'''& \cos{3b} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{b} +{}& \beta'' & \sin{2b} +{}& \beta''' & \sin{3b} +{} \mathrm{etc.} \\[.3em] C = \alpha &{}+ \alpha'& \cos{c} +{}& \alpha''& \cos{2c} +{}& \alpha'''& \cos{3c} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{c} +{}& \beta'' & \sin{2c} +{}& \beta''' & \sin{3c} +{} \mathrm{etc.} \\[.3em] D = \alpha &{}+ \alpha'& \cos{d} +{}& \alpha''& \cos{2d} +{}& \alpha'''& \cos{3d} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{d} +{}& \beta'' & \sin{2d} +{}& \beta''' & \sin{3d} +{} \mathrm{etc.} \\ \quad \mathrm{etc.} \\ T = \alpha &{}+ \alpha'&{} \cos{t} +{}& \alpha''&{} \cos{2t} +{}& \alpha'''&{} \cos{3t} +{} \mathrm{etc.} \\ &{}+ \beta' &{} \sin{t} +{}& \beta'' &{} \sin{2t} +{}& \beta''' &{} \sin{3t} +{} \mathrm{etc.} \end{alignedat}`} Multiplying these equations respectively by ${$$`\begin{aligned} &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm \sin\!\tfrac{1}{2}(a - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm \sin\!\tfrac{1}{2}(b - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm \sin\!\tfrac{1}{2}(c - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm \sin\!\tfrac{1}{2}(d - t)}} \\[1.2em] &\:\mathrm{etc.}\\ &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} \end{aligned}`} and then summing the products makes ${$$`\begin{aligned} &\frac{\small{A}}{\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm \sin\!\tfrac{1}{2}(a - t)}} \\[.8em] {}+{}&\frac{\small{B}}{\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm \sin\!\tfrac{1}{2}(b - t)}} \\[.8em] {}+{}&\frac{\small{C}}{\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm \sin\!\tfrac{1}{2}(c - t)}} \\[.8em] {}+{}&\frac{\small{D}}{\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm \sin\!\tfrac{1}{2}(d - t)}} \\[1.2em] {}+{}&\:\mathrm{etc.}\\ {}+{}&\frac{\small{T}}{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} = W \end{aligned}`} Then a result from section 2, whose ${$`m`} here is ${$`2m + 2`} so the second case applies, shows that ${$`W = 0`}; therefore multiplying by ${$$` \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm `} makes ${$$`\begin{aligned} T = \quad&\frac{\small{ \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm }} A \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm }} B \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm }} C \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm }} D \\[1.2em] {}+{}&\:\mathrm{etc.}\\ \end{aligned}`} Since this is an indefinite formula for any value ${$`t`} whatsoever, it is clear that the product of sines in the numerator becomes a sum of cosines and sines of multiples of angle measure, so it can be written as ${$$`\begin{alignedat}{4} \alpha &{}+ \alpha'& \cos{t} +{}& \alpha''& \cos{2t} +{}& \alpha'''& \cos{3t} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{t} +{}& \beta'' & \sin{2t} +{}& \beta''' & \sin{3t} +{} \mathrm{etc.} \end{alignedat}`} which must be *identical*, for some coefficients ${$`\alpha, \alpha', \beta', \alpha'', \beta''`}. Furthermore, the formula for ${$`T`} shown here, when we substitute respectively ${$`a`}, ${$`b`}, ${$`c`}, ${$`d`} etc. for ${$`t`}, attains the given values ${$`A`}, ${$`B`}, ${$`C`}, ${$`D`} etc. naturally and without calculation, satisfying our condition. `}

`31:T2005,md`

${md` 11. If the number of given values ${$`A`}, ${$`B`}, ${$`C`}, ${$`D`} were one less than we supposed above, say ${$`= 2m`}, they would not be sufficient to uniquely determine the ${$`2m+1`} unknown coefficients, unless the given values also satisfy some other relation. We therefore cannot generally specify ${$`T`} in this case without employing an unknown constant. The reasoning from the previous section does not apply here, because the computation ${$`W = 0`} relied on the assumption of an even number of values ${$`A`}, ${$`B`} . . . ${$`T`}. A formula like the one found for ${$`T`} in the previous section which satisfies the values ${$`A`}, ${$`B`}, etc. – whether the number of values is even or odd – can be found with some effort, by expanding the product of the sines to make, in our case, a different expression of the form ${$$`\begin{alignedat}{4} &{}\mathfrak{A}& \cos{\tfrac{1}{2}t} +{}& \mathfrak{A}'& \cos{\tfrac{3}{2}t} +{}& \mathfrak{A}''& \cos{\tfrac{5}{2}t} +{} \mathrm{etc.} \\[.4em] {}+{}& \mathfrak{B}& \sin{\tfrac{1}{2}t} +{}& \mathfrak{B}'& \sin{\tfrac{3}{2}t} +{}& \mathfrak{B}''& \sin{\tfrac{5}{2}t} +{} \mathrm{etc.} \end{alignedat}`} Wherefore in this case we ought to follow another way. Multiply ${$`X`} by ${$`\cos\bigl(\tfrac{1}{2}x+k\bigr)`}, and the result, if the final terms in ${$`X`} are ${$`\alpha^m \cos{mx} + \beta^m \sin{mx}`}, is ${$$`\begin{alignedat}{4} & \alpha& \cos\bigl(\tfrac{1}{2}x + k\bigr) &{}+ \tfrac{1}{2}\alpha' & \cos\bigl(\tfrac{3}{2}x + k\bigr) &{}+ \tfrac{1}{2}\alpha'' \cos\bigl(\tfrac{5}{2}x + k\bigr) + \dots \\[.3em] &&&&&{}+ \tfrac{1}{2}\alpha^m \cos\bigl(\bigl(m - \tfrac{1}{2}\bigr)x + k\bigr) \\[.3em] {}+\tfrac{1}{2} & \alpha' & \cos\bigl(\tfrac{1}{2}x - k\bigr) &{}+ \tfrac{1}{2}\alpha'' & \cos\bigl(\tfrac{3}{2}x - k\bigr) &{}+ \tfrac{1}{2}\alpha''' \cos\bigl(\tfrac{5}{2}x - k\bigr) + \dots \\[.3em] &&&& &{}+ \tfrac{1}{2}\alpha^m \cos\bigl(\bigl(m - \tfrac{1}{2}\bigr)x - k\bigr) \\[.3em] && &{}+ \tfrac{1}{2}\beta' & \sin\bigl(\tfrac{3}{2}x + k\bigr) &{}+ \tfrac{1}{2}\beta'' \sin\bigl(\tfrac{5}{2}x + k\bigr) + \dots \\[.3em] &&&&&{}+ \tfrac{1}{2}\beta^m \sin\bigl(\bigl(m - \tfrac{1}{2}\bigr)x + k\bigr) \\[.3em] {}+\tfrac{1}{2} & \beta' & \sin\bigl(\tfrac{1}{2}x - k\bigr) &{}+ \tfrac{1}{2}\beta'' & \sin\bigl(\tfrac{3}{2}x - k\bigr) &{}+ \tfrac{1}{2}\beta''' \sin\bigl(\tfrac{5}{2}x - k\bigr) + \dots \\[.3em] &&&& &{}+ \tfrac{1}{2}\beta^m \sin\bigl(\bigl(m - \tfrac{1}{2}\bigr)x - k\bigr) \end{alignedat}`} Now if in this expression ${$`x`} is successively replaced by ${$`a`}, ${$`b`}, ${$`c`}, ${$`d`} . . . ${$`t`}, and then the resulting expressions are respectively multiplied by ${$$`\begin{aligned} &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm \sin\!\tfrac{1}{2}(a - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm \sin\!\tfrac{1}{2}(b - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm \sin\!\tfrac{1}{2}(c - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm \sin\!\tfrac{1}{2}(d - t)}} \\[1.2em] &\:\mathrm{etc.}\\ &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} \end{aligned}`} let the resulting sum of products ${$`= W`}; from section 2 (the first case, whose ${$`m`} here is ${$`2m + 1`}), it follows easily that this is ${$$`\begin{alignedat}{2} W = \quad&\tfrac{1}{2}(2i)^{2m}\alpha^m \cos &\bigl( \tfrac{1}{2}(a + b + c + d + \dots + t) + k \bigr) \\[.3em] {}+{}&\tfrac{1}{2}(2i)^{2m}\beta^m \sin &\bigl( \tfrac{1}{2}(a + b + c + d + \dots + t) + k \bigr) \end{alignedat}`} Which can be also be expressed as ${$$`\begin{aligned} W = \quad&\frac{\small{A\cos\bigl(\tfrac{1}{2}a+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm \sin\!\tfrac{1}{2}(a - t)}} \\[.8em] {}+{}&\frac{\small{B\cos\bigl(\tfrac{1}{2}b+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm \sin\!\tfrac{1}{2}(b - t)}} \\[.8em] {}+{}&\frac{\small{C\cos\bigl(\tfrac{1}{2}c+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm \sin\!\tfrac{1}{2}(c - t)}} \\[.8em] {}+{}&\frac{\small{D\cos\bigl(\tfrac{1}{2}d+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm \sin\!\tfrac{1}{2}(d - t)}} \\[1.2em] {}+{}&\:\mathrm{etc.}\\ {}+{}&\frac{\small{T\cos\bigl(\tfrac{1}{2}t+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} \end{aligned}`} But now if the quantity ${$`k`}, which can be chosen arbitrarily, is set to ${$`= -\tfrac{1}{2}t`}, the sum ${$`a + b + c + d + \dots = s`} (excluding ${$`t`}), makes, after multiplying by ${$` \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm`} ${$$`\begin{aligned} \rule{}{2em} T = \quad&\frac{\small{ \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm }} A \cos\!\tfrac{1}{2}(t - a) \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm }} B \cos\!\tfrac{1}{2}(t - b) \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm }} C \cos\!\tfrac{1}{2}(t - c) \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm }} D \cos\!\tfrac{1}{2}(t - d) \\[1.0em] {}+{}&\:\mathrm{etc.}\\[.2em] {}+{}& 2^{2m-1}i^{2m} \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d) \dotsm \\[.3em] & \qquad \qquad \quad \times \left\{ \alpha^m\cos{\tfrac{1}{2}s} + \beta^m\sin{\tfrac{1}{2}s} \right\} \end{aligned}`} Let us add some additional notes about this formula I. ${$`i^{2m} = +1`} or ${$`-1`}, if ${$`m`} is respectively even or odd. II. As this indefinite formula is valid for any value of ${$`t`}, expanding the product of sines necessarily makes a sum of sines and cosines of multiples of angle measure, as a formula for ${$`X`}, if for ${$`x`} we write ${$`t`}, the result is identical, i.e. as ${$$`\begin{alignedat}{4} \alpha &{}+ \alpha'& \cos{t} +{}& \alpha''& \cos{2t} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{t} +{}& \beta'' & \sin{2t} +{} \mathrm{etc.} \end{alignedat}`} III. Because all of the coefficients ${$`\alpha`}, ${$`\alpha'`}, ${$`\beta'`}, ${$`\alpha''`}, ${$`\beta''`} etc. are determined from known quantities, plus a single unknown ${$`\alpha^m \cos{\!\tfrac{1}{2}s} + \beta^m \sin{\!\tfrac{1}{2}s}`}, if there is a known relation between the data, all of the coefficients can be easily specified. IV. But if we merely copy this form as ${$$`\begin{alignedat}{4} \alpha &{}+ \alpha'& \cos{t} +{}& \alpha''& \cos{2t} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{t} +{}& \beta'' & \sin{2t} +{} \mathrm{etc.} \end{alignedat}`} and ask (whether it is identical or not), for each proposed value ${$`A`}, ${$`B`}, ${$`C`}, ${$`D`} etc. found by substituting respectively ${$`a`}, ${$`b`}, ${$`c`}, ${$`d`} etc. for ${$`t`}: the quantity ${$`\alpha^m \cos{\!\tfrac{1}{2}s} + \beta^m \sin{\!\tfrac{1}{2}s}`} comes out from these *arbitrary* unknowns, because we can set it ${$`= 0`} if we like; in fact our formula for ${$`T`}, naturally and without calculation, satisfies our previously mentioned condition. `}

`32:T7e6,md` Rewritten in more modern notation:

${md` 10\. Let ${$`y`} be a function of angle measure ${$`x \in (-\pi, \pi]`} of the form ${$$`\begin{alignedat}{4} y(x) = \alpha_0 + \sum_{k=1}^m \alpha_k \cos kx + \beta_k \sin kx \end{alignedat}`} so that the number of (unknown) coefficients is ${$`2m + 1`}. For several distinct values ${$`x_j`} (the “nodes”), with ${$`x_j \neq x_k`}, there are given corresponding values ${$nb`y_k = y(x_k)`}. From these given values we seek a formula for the value ${$nb`T`}, the value of function ${$`X`} at an arbitrary ${$`x`} value, say ${$`t`}. We have ${$nb`2m + 2`} equations ${$$`\begin{alignedat}{4} y(x) &= \alpha_0 + \sum_{k=1}^m \alpha_k \cos kx + \beta_k \sin kx, \\ y_k &= \alpha_0 + \sum_{k=1}^m \alpha_k \cos kx_k + \beta_k \sin kx_k \quad 1 \leq k \leq 2m + 1 \end{alignedat}`} Multiplying these equations respectively by ${$$`\begin{aligned} &\frac1{\prod_j \sin\tfrac12(x - x_j)}, \\[.8em] &\frac1{\sin\tfrac12(x_k - x) \prod_{j\neq k} \sin\tfrac12(x_k - x_j)} \end{aligned}`} and then summing the products makes ${$$`\begin{aligned} \frac{y(x)}{\prod_{j} \sin\tfrac12(x - x_j)} + \sum_{k=1}^{2m+1} \frac{y_k}{\sin\tfrac12(x_k - x) \prod_{j\neq k} \sin\tfrac12(x_k - x_j)} = 0 \end{aligned}`} by a result from section 2; therefore multiplying by ${$$` \prod_{j} \sin\tfrac12(x - x_j) `} makes ${$$`\begin{aligned} y(x) = \sum_{k=1}^{2m+1} \frac{\prod_{j} \sin\tfrac12(x - x_j)}{\prod_{j\neq k} \sin\tfrac12(x_k - x_j)}y_k. \end{aligned}`} Since this is an indefinite formula for any value ${$`t`} whatsoever, it is clear that the product of sines in the numerator becomes a sum of cosines and sines of multiples of angle measure, so it can be written as ${$$`\begin{alignedat}{4} y(x) = \alpha_0 + \sum_{k=1}^m \alpha_k \cos kx + \beta_k \sin kx \end{alignedat}`} which must be *identical*, for some coefficients ${$`\alpha_k, \beta_k`}. Furthermore, the formula attains the given values ${$`y_k = y(x_k)`}. `}

`33:T1fba,md`

${md` 11. If the number of given values ${$`y_k`} were one less than we supposed above, say ${$`= 2m`}, they would not be sufficient to uniquely determine the ${$`2m+1`} unknown coefficients, unless the given values also satisfy some other relation. We therefore cannot generally specify ${$`y(x)`} in this case without employing an unknown constant. The reasoning from the previous section does not apply here, because the result that the sum of products ${$`= 0`} relied on the assumption of an odd number of values ${$`y_k`}. A formula like the one found for ${$`y(x)`} in the previous section which satisfies the values ${$`y_k`} – whether the number of values is even or odd – can be found with some effort, by expanding the product of the sines to make, in the even case, a different expression of the form ${$$`\begin{alignedat}{4} y(x) = \sum_{k=1}^{m}\mathfrak{A}_k \cos\big({\big(k-\tfrac12\big)x\big)} + \mathfrak{B}_k \sin\big({\big(k-\tfrac12\big)x\big)} \end{alignedat}`} Wherefore in this case we ought to follow another way. Multiply ${$`X`} by ${$`\cos\bigl(\tfrac{1}{2}x+\phi\bigr)`}, and the result, if the final terms in ${$`y(x)`} are ${$`\alpha_m \cos{mx} + \beta_m \sin{mx}`}, is ${$$`\begin{alignedat}{4} \sum_{k=1}^m \alpha_m \cos\bigl(\tfrac{1}{2}x + \phi\bigr) \end{alignedat}`} ${$$`\begin{alignedat}{4} & \alpha& \cos\bigl(\tfrac{1}{2}x + k\bigr) &{}+ \tfrac{1}{2}\alpha' & \cos\bigl(\tfrac{3}{2}x + k\bigr) &{}+ \tfrac{1}{2}\alpha'' \cos\bigl(\tfrac{5}{2}x + k\bigr) + \dots \\[.3em] &&&&&{}+ \tfrac{1}{2}\alpha^m \cos\bigl(\bigl(m - \tfrac{1}{2}\bigr)x + k\bigr) \\[.3em] {}+\tfrac{1}{2} & \alpha' & \cos\bigl(\tfrac{1}{2}x - k\bigr) &{}+ \tfrac{1}{2}\alpha'' & \cos\bigl(\tfrac{3}{2}x - k\bigr) &{}+ \tfrac{1}{2}\alpha''' \cos\bigl(\tfrac{5}{2}x - k\bigr) + \dots \\[.3em] &&&& &{}+ \tfrac{1}{2}\alpha^m \cos\bigl(\bigl(m - \tfrac{1}{2}\bigr)x - k\bigr) \\[.3em] && &{}+ \tfrac{1}{2}\beta' & \sin\bigl(\tfrac{3}{2}x + k\bigr) &{}+ \tfrac{1}{2}\beta'' \sin\bigl(\tfrac{5}{2}x + k\bigr) + \dots \\[.3em] &&&&&{}+ \tfrac{1}{2}\beta^m \sin\bigl(\bigl(m - \tfrac{1}{2}\bigr)x + k\bigr) \\[.3em] {}+\tfrac{1}{2} & \beta' & \sin\bigl(\tfrac{1}{2}x - k\bigr) &{}+ \tfrac{1}{2}\beta'' & \sin\bigl(\tfrac{3}{2}x - k\bigr) &{}+ \tfrac{1}{2}\beta''' \sin\bigl(\tfrac{5}{2}x - k\bigr) + \dots \\[.3em] &&&& &{}+ \tfrac{1}{2}\beta^m \sin\bigl(\bigl(m - \tfrac{1}{2}\bigr)x - k\bigr) \end{alignedat}`} Now if in this expression ${$`x`} is successively replaced by ${$`a`}, ${$`b`}, ${$`c`}, ${$`d`} . . . ${$`t`}, and then the resulting expressions are respectively multiplied by ${$$`\begin{aligned} &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm \sin\!\tfrac{1}{2}(a - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm \sin\!\tfrac{1}{2}(b - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm \sin\!\tfrac{1}{2}(c - t)}} \\[.8em] &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm \sin\!\tfrac{1}{2}(d - t)}} \\[1.2em] &\:\mathrm{etc.}\\ &\frac{\small{1}}{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} \end{aligned}`} let the resulting sum of products ${$`= W`}; from section 2 (the first case, whose ${$`m`} here is ${$`2m + 1`}), it follows easily that this is ${$$`\begin{alignedat}{2} W = \quad&\tfrac{1}{2}(2i)^{2m}\alpha^m \cos &\bigl( \tfrac{1}{2}(a + b + c + d + \dots + t) + k \bigr) \\[.3em] {}+{}&\tfrac{1}{2}(2i)^{2m}\beta^m \sin &\bigl( \tfrac{1}{2}(a + b + c + d + \dots + t) + k \bigr) \end{alignedat}`} Which can be also be expressed as ${$$`\begin{aligned} W = \quad&\frac{\small{A\cos\bigl(\tfrac{1}{2}a+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm \sin\!\tfrac{1}{2}(a - t)}} \\[.8em] {}+{}&\frac{\small{B\cos\bigl(\tfrac{1}{2}b+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm \sin\!\tfrac{1}{2}(b - t)}} \\[.8em] {}+{}&\frac{\small{C\cos\bigl(\tfrac{1}{2}c+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm \sin\!\tfrac{1}{2}(c - t)}} \\[.8em] {}+{}&\frac{\small{D\cos\bigl(\tfrac{1}{2}d+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm \sin\!\tfrac{1}{2}(d - t)}} \\[1.2em] {}+{}&\:\mathrm{etc.}\\ {}+{}&\frac{\small{T\cos\bigl(\tfrac{1}{2}t+k\bigr)}}{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} \end{aligned}`} But now if the quantity ${$`k`}, which can be chosen arbitrarily, is set to ${$`= -\tfrac{1}{2}t`}, the sum ${$`a + b + c + d + \dots = s`} (excluding ${$`t`}), makes, after multiplying by ${$` \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm`} ${$$`\begin{aligned} \rule{}{2em} T = \quad&\frac{\small{ \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(a - b) \sin\!\tfrac{1}{2}(a - c) \sin\!\tfrac{1}{2}(a - d) \dotsm }} A \cos\!\tfrac{1}{2}(t - a) \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(b - a) \sin\!\tfrac{1}{2}(b - c) \sin\!\tfrac{1}{2}(b - d) \dotsm }} B \cos\!\tfrac{1}{2}(t - b) \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - d)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(c - a) \sin\!\tfrac{1}{2}(c - b) \sin\!\tfrac{1}{2}(c - d) \dotsm }} C \cos\!\tfrac{1}{2}(t - c) \\[1.2em] {}+{}&\frac{\small{ \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c)\dotsm}} {\small{ \sin\!\tfrac{1}{2}(d - a) \sin\!\tfrac{1}{2}(d - b) \sin\!\tfrac{1}{2}(d - c) \dotsm }} D \cos\!\tfrac{1}{2}(t - d) \\[1.0em] {}+{}&\:\mathrm{etc.}\\[.2em] {}+{}& 2^{2m-1}i^{2m} \sin\!\tfrac{1}{2}(t - a) \sin\!\tfrac{1}{2}(t - b) \sin\!\tfrac{1}{2}(t - c) \sin\!\tfrac{1}{2}(t - d) \dotsm \\[.3em] & \qquad \qquad \quad \times \left\{ \alpha^m\cos{\tfrac{1}{2}s} + \beta^m\sin{\tfrac{1}{2}s} \right\} \end{aligned}`} Let us add some additional notes about this formula I. ${$`i^{2m} = +1`} or ${$`-1`}, if ${$`m`} is respectively even or odd. II. As this indefinite formula is valid for any value of ${$`t`}, expanding the product of sines necessarily makes a sum of sines and cosines of multiples of angle measure, as a formula for ${$`X`}, if for ${$`x`} we write ${$`t`}, the result is identical, i.e. as ${$$`\begin{alignedat}{4} \alpha &{}+ \alpha'& \cos{t} +{}& \alpha''& \cos{2t} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{t} +{}& \beta'' & \sin{2t} +{} \mathrm{etc.} \end{alignedat}`} III. Because all of the coefficients ${$`\alpha`}, ${$`\alpha'`}, ${$`\beta'`}, ${$`\alpha''`}, ${$`\beta''`} etc. are determined from known quantities, plus a single unknown ${$`\alpha^m \cos{\!\tfrac{1}{2}s} + \beta^m \sin{\!\tfrac{1}{2}s}`}, if there is a known relation between the data, all of the coefficients can be easily specified. IV. But if we merely copy this form as ${$$`\begin{alignedat}{4} \alpha &{}+ \alpha'& \cos{t} +{}& \alpha''& \cos{2t} +{} \mathrm{etc.} \\ &{}+ \beta' & \sin{t} +{}& \beta'' & \sin{2t} +{} \mathrm{etc.} \end{alignedat}`} and ask (whether it is identical or not), for each proposed value ${$`A`}, ${$`B`}, ${$`C`}, ${$`D`} etc. found by substituting respectively ${$`a`}, ${$`b`}, ${$`c`}, ${$`d`} etc. for ${$`t`}: the quantity ${$`\alpha^m \cos{\!\tfrac{1}{2}s} + \beta^m \sin{\!\tfrac{1}{2}s}`} comes out from these *arbitrary* unknowns, because we can set it ${$`= 0`} if we like; in fact our formula for ${$`T`}, naturally and without calculation, satisfies our previously mentioned condition. `}

`34:T99e,Salzer_1960 = md` * * * ### Herbert Salzer (1960)
[“New Formulas for Trigonometric Interpolation”](https://doi.org/10.1002/sapm196039183)

${md` A well-known trigonometric interpolation formula for ${$`f(x)`} given at an odd number ${$`n`} of arbitrary points ${$`x_j`}, ${$`j = 1, 2, \dots, n`} is due to Gauss. His formula for the trigonometric sum ${$`S_{\tfrac{1}{2}(n-1)}(x) = a_0 + \sum_{k=1}^{(n-1)/2} a_k \cos{kx} + b_k \sin{kx}`} which is determined uniquely by ${$`S_{\tfrac{1}{2}(n-1)}(x_j) = f(x_j)`}, ${$`j = 1, 2, \dots, n`}, is given by ${$$` S_{\tfrac{1}{2}(n-1)}(x) = \sum_{j=1}^{n} \dfrac {{\prod\!'\,}_{k=1}^n \sin{\tfrac{1}{2}(x - x_k)}} {{\prod\!'\,}_{k=1}^n \sin{\tfrac{\vphantom{\bar{1}}1}{2}(x_j - x_k)}} f(x_j) \tag{1} `} where ${$`k = j`} is absent from ${$`\prod\!'`}. Obviously some modification is required in (1) in order to give an interpolating trigonometric sum for n an even number of points. Gauss ([1], [p. 281](https://archive.org/details/werkecarlf03gausrich/page/n294/mode/2up)) gave a formula for n an even number of points by inserting a factor of ${$`\cos{\tfrac{1}{2}(x - x_k)}`} into each term of (1). The resulting trigonometric sum ${$`S_{\tfrac{1}{2}n}(x)`}, having ${$`n + 1`} coefficients, is then not uniquely determined. Here we shall modify Gauss’s even-point formula to obtain a uniquely determined interpolating trigonometric sum. [...] In numerical approximation, as borne out by experience with polynomials and elementary functions, a unique representation by a naturally appearing sequence of simple functions that does not go further than necessary in degree, tends to be more accurate than one of a family of ${$`\infty^M`} possibilities of higher degree, which enter when there are ${$`M`} coefficients that may be fixed arbitrarily. Thus, we are concerned here only with formulas for the lowest degree trigonometric sums which are uniquely determined, so that the number of coefficients ${$`a_k`} and ${$`b_k`} is exactly equal to the number of conditions. An even number of interpolations points, say ${$`n = 2m`}, suggests a lowest degree trigonometric sum, not going beyond ${$`\cos{mx}`} or ${$`\sin{mx}`} terms, where either one or the other is missing, so that exactly ${$`2m`} coefficients are involved. The uniqueness may be established by an adaptation of an argument employed in proving the uniqueness of ${$`S_{\tfrac{1}{2}(n-1)}(x)`} in (1). `}

`35:T770,md` * * * ### Herbert Salzer (1976)
[“A New Form of Trigonometric Orthogonality and Gaussian-Type Quadrature”][salzer1976] > The uniqueness of the interpolating ${$`S_{2n+1}(x)`} is needed in the proof of these factor theorem lemmas. Gauss gave also an even-point formula for trigonometric interpolation by inserting a factor of ${$`\cos{\tfrac{1}{2}(x - x_j)}`} into each coefficient of ${$`f(x_i)`} in (2), with ${$`2n`} in place of ${$`2n+1`}; but since the resulting ${$`S_{2n+1}(x)`} still has ${$`2n+1`} coefficients for only ${$`2n`} values of ${$`f(x_i)`}, and thus is not uniquely determined, we cannot complete the argument leading to (14) and (15). In [2] it is shown how to further modify Gauss's formula by adding to his modified (2) a term of the form ${$`\lambda\prod_{j=-n}^n \sin{\tfrac{1}{2}(x-x_j)}`} which will produce an ${$`S_{2n+1}(x)`} that lacks either the ${$`\cos{nx}`} or ${$`\sin{nx}`} term and is uniquely determined by ${$`f(x_i)`}, ${$`i = 1(1)2n`}. The artificiality of a trigonometric sum which is prescribed to be without ${$`\cos{nx}`} or ${$`\sin{nx}`} is likely to make it no more accurate than (2) for ${$`2n-1`} points. But the main difficulty in obtaining a factor theorem like (14) when ${$`\cos{nx}`} or ${$`\sin{nx}`} is missing would occur in that part of the above argument where an ${$`S_{2n+1}(x)`} lacking a ${$`\cos{(n-1)x}`} or ${$`\sin{(n-1)x}`} term would be multiplied by ${$`1 + \alpha_1 \cos{x} + \beta_1 \sin{x}`} to give a product ${$`\bar{S}_{2n+1}(x)`} having both ${$`\cos{nx}`} and ${$`\sin{nx}`} terms, so that it could not be a factorization of an ${$`S_{2n+1}(x)`} lacking a ${$`\cos{nx}`} or ${$`\sin{nx}`} term. These properties of even-point interpolation justify the restriction of the present investigation to just the well-known odd-point case given by (2). [salzer1976]: https://doi.org/10.1016/0771-050X(76)90045-0 `36:T1358,Henrici_1979 = md` * * * ### Peter Henrici (1979)
[“Barycentric formulas for interpolating trigonometric polynomials and their conjugates.”](https://doi.org/10.1007/BF01399556)

${md` Here we present a formula for the evaluation of ${$`t(\phi)`} which is patterned after the well-known barycentric formula for polynomial interpolation [13; 1; 12; 10, p. 106]. It evaluates ${$`t(\phi)`} for any integer ${$`n`} and at any ${$`\phi`} in only ${$`O(n)`} operations and thus is preferable to the evaluation via FFT in situations where the evaluation is to be performed for a few values of ${$`\phi`} only. Contrary to a more classical representation of ${$`t(\phi)`}, our barycentric representation is numerically stable even for very large values of ${$`n`}. At the same time, our formula is much simpler than a barycentric representation for ${$`t(\phi)`} that was already given [11]. [...] #### 2. Barycentric Formula for ${$`\bm{t(\phi)}`} To begin with, let ${$`n`} be even, ${$`n=2m`}. For the special case where ${$`f_0=1`}, all other ${$`f_k=0`}, the interpolating trigonometric polynomial evidently is ${$`\ell_0 e^{i\phi}`}, where ${$$` \ell_0(z) = \frac{1}{n}\Bigl\{ \tfrac{1}{2}z^{-m} + z^{-m+1} + \dots + 1 + z + \dots + z^{m-1} + \tfrac{1}{2}z^{m} \Bigr\}. \tag{2.1a} `} By summing a terminating geometric series, this is brought into the form ${$$` \ell_0(z) = \frac{1}{2n} \frac{z^n-1}{z^m} \frac{z + 1}{z - 1}. `} For arbitrary data ${$`f_0`}, ${$`f_1`}, ..., ${$`f_{n-1}`}, the interpolating trigonometric polynomial thus is ${$`t(\phi) = r(e^{i\phi})`}, ${$$` r(z) \mathrel{\vcentcolon=} \sum_{k=0}^{n-1} \ell_k(z) f_k, `} where, letting ${$`z_k \mathrel{\vcentcolon=} e^{i\phi_k}`}, ${$$` \ell_k(z) \mathrel{\vcentcolon=} \ell_0(z_k^{-1}z) = \frac{1}{2n} \frac{z^n-1}{{z_k}^{\!m}z^m} \frac{z + z_k}{z - z_k}, `} ${$`k = 0, 1, \dots, n-1`}. Using ${$`{z_k}^{\!m} = (-1)^k`} we thus have ${$$` r(z) = \frac{1}{2n} \frac{z^n-1}{z^m}\sum_{k=0}^{n-1} (-1)^k \frac{z + z_k}{z - z_k} f_k. \tag{2.2} `} Letting ${$`z = e^{i\phi}`} and using ${$$` \frac{z + z_k}{z - z_k} = \frac{1}{i} \operatorname{ctg}{\frac{\phi - \phi_k}{2}}, `} there results the classical formula ${$$` t(\phi) = \frac{\sin(n\phi/2)}{n}\sum_{k=0}^{n-1} (-1)^k f_k \operatorname{ctg}{\frac{\phi - \phi_k}{2}}, \tag{2.3} `} [...] Dividing (2.3) by the theoretically correct relation ${$$` 1 = \frac{\sin(n\phi/2)}{n}\sum_{k=0}^{n-1} (-1)^k \operatorname{ctg}{\frac{\phi - \phi_k}{2}}, `} we obtain __Theorem 1.__ *If ${$`n`} is even, the trigonometric polynomial *(1.4)* interpolating given data ${$`f_k`} at the equidistant points ${$`\phi_k = 2k/n (k = 0, 1, \dots, n-1)`} is given by* ${$$` t(\phi) = \frac {\displaystyle{\sum_{k=0}^{n-1} (-1)^k f_k \operatorname{ctg}{\frac{\phi - \phi_k}{2}}}} {\displaystyle{\sum_{k=0}^{n-1} (-1)^k \operatorname{ctg}{\frac{\phi - \phi_k}{2}}} \rule{}{34mu}} \tag{2.4a} `} If ${$`n`} is odd, ${$`n = 2m + 1`}, a similar calculation carried through with the function ${$$`\begin{aligned} \ell_0(z) &\mathrel{\vcentcolon=} \frac{1}{n}\bigl\{ z^{-m} + z^{-m+1} + \dots + 1 + z + \dots + z^{m-1} + z^{m} \bigr\} \\[.8em] &\hphantom{\vcentcolon}= \frac{1}{n} \frac{z^n-1}{z^m} \frac{1}{z - 1} \tag{2.1b} \end{aligned}`} yields __Theorem 2.__ *If ${$`n`} is odd, the interpolating trigonometric polynomial *(1.1)* is represented by* ${$$` t(\phi) = \frac {\displaystyle{\sum_{k=0}^{n-1} (-1)^k f_k \csc{\frac{\phi - \phi_k}{2}}}} {\displaystyle{\sum_{k=0}^{n-1} (-1)^k \csc{\frac{\phi - \phi_k}{2}}} \rule{}{34mu}} \tag{2.4a} `} [${$`\csc{\phi} =(\sin{\phi})^{-1}`}]. Rutishauser would have called the formulas (2.4) barycentric because they are formally identical with the formulas for the center of gravity (barycenter) of a system of masses ${$`(-1)^k \operatorname{ctg}{\tfrac{1}{2}(\phi - \phi_k)}`} ${$`\bigl(`}or ${$`(-1)^k \operatorname{ctg}{\tfrac{1}{2}(\phi - \phi_k)}\bigr)`} attached to the points ${$`f_k`}. The analogy is incomplete inasmuch as the ‘masses’ may have either sign. However, in view of the fact that the masses are large if ${$`\phi_k`} is near ${$`\phi`}, and have the same sign for the two ${$`\phi_k`} nearest to ${$`\phi`}, the formulas naturally emphasize the corresponding values of ${$`f_k`}, which already demosntrates their basic stability. It should be noted that aready Salzer [11] has obtained a barycentric formula for ${$`t(\phi)`}. He started from the classical formula of Gauss [6] for the trigonometric interpolant for general (not necessarily equidistant) interpolating points, and he made the crucial observation that for ${$`f(\phi)=1`} an identity results which can be used to eliminate an obnoxious factor, as above. However, Salzer did not notice a considerable simplification which results when the interpolating points are equally spaced; as a consequence, his formulas are much more complicated than (2.4). `}

`37:T1062,Berrut_1984 = md` * * * ### Jean-Paul Berrut 1984
[“Baryzentrische Formeln zur Trigonometrischen Interpolation (I)”](https://doi.org/10.1007/BF00945179)

${md` > **Definition.** ${$`\quad \displaystyle{ t(\varphi) = \tfrac{1}{2} a_0 + \sum_{k=1}^p{a_k \cos{k\varphi} + b_k \sin{k\varphi}}} `}
${$$`\tag{1.1}`}
nennt man *trigonometrisches Polynom vom Grade* ${$`p`}. ${$`t(\varphi)`} heißt *ausgewogen*, wenn ${$`b_p = 0`}. [...] Sei ${$`f`} eine ${$`2\pi`}-periodische, komplexwertige Funktion auf ${$`\mathbb{R}`}. In dieser Arbeit suchen wit baryzentrische Formeln für trigonometrische Polynome minimalen Grades, die ${$`f`} an ${$`n`} vorgegebenen Stützstellen ${$`\varphi_k, k = 0(1)n - 1`}, interpolieren: ${$$`t(\varphi_k) = f(\varphi_k) \tag{1.2}`} [...] Wie groß muß ${$`p`} sein, damit jede periodische Funktion ${$`f`} an ${$`n`} beliebigen Stützstellen durch ein Polynom vom Grade ${$`p`} interpoliert werden kann? Falls ${$`n`} ungerade ist, ${$`n = 2m + 1`}, gibt es genau ein trigonometrisches Polynom vom Grade ${$`m`}, das ${$`f`} an den ${$`n`} vorgegebenen Stützstellen ${$`S_n`} interpoliert ([14], p. 1). Wenn aber ${$`n`} gerade ist, ${$`n = 2m`}, reichen die ${$`2m`} Bedingungen (1.2) nicht aus, um ein interpolierendes Polynom vom Grade ${$`m`} eindeutig festzulegen; man braucht eine zusätzliche Bedingung. Falls z. B. ${$`S_n = S_n^g`} mit ${$nb`S_n^g = \Bigl\{ \varphi_k \,|\, \varphi_k = \dfrac{2\pi}{n}, k = 0(1)n - 1\Bigr\}`}, gilt ${$`\sin(m \varphi_k) = 0`}; ${$`b_m`} ist frei wählbar, und man setzt meistens ${$`b_m \mathrel{\vcentcolon=} 0`}, was zu Eindeutigkeit führt ([6]). Damit wir später die üblichen Arbeitsmittel der diskreten Fourieranalyse (insbesondere die FFT) benützen können, wollen wir hier auch bei beliebigen Stützstellen das ausgewogene trigonometrische Interpolationspolynom bestimmen. [...] #### 2. Eine baryzentrische Formel für ${$`\bm{t(\varphi)}`} bei ${$`\bm{n = 2m}`} beliebigen Stützstellen Die Formel (1.5) von Gauß versagt bei gerader Anzahl ${$`\varphi_k`}: ihre rechte Seite ist eine gewichtete Summe von Produkten einer ungeraden Anzahl von ${$`\sin{\tfrac{1}{2}(\varphi - \varphi_k)}`}, ergibt also nach den Formeln (1.6) einen Ausdruck der Form ${$$` \sum_{k=1}^{n-1}\left( c_k \cos{\frac{k}{2}\varphi} + d_k \sin{\frac{k}{2}\varphi}\right) `} und somit kein trigonometrisches Polynom in ${$`\varphi`}. Gauss ([4], [p. 283](https://archive.org/details/werkecarlf03gausrich/page/n296/mode/2up)) hat aber weiter gezeigt, daß jedes an ${$`n = 2m`} Stützstellen ${$`\varphi_k`} interpolierende trigonometrische Polynom (1.1) sich schreiben 1äßt als ${$$` t(\varphi) = \sum_{k=1}^{n-1} \left\{\rule{}{2.5em}\right. \prod_{\substack{i = 0\\[.1em] i \neq k}}^{n - 1} \dfrac{\sin{\tfrac{1}{2}(\varphi - \varphi_i)}} {\rule{}{1em}\sin{\tfrac{1}{2}(\varphi_k - \varphi_i)}} \!\left.\rule{}{2.5em}\right\} \cos{\tfrac{1}{2}(\varphi - \varphi_k)} f_k + c \cdot \prod_{i=0}^{n-1} \sin{\tfrac{1}{2}(\varphi - \varphi_i)} `}
${$$`\tag{2.1}`}
wobei ${$`\displaystyle{\; c \mathrel{\vcentcolon=} (-1)^{m}2^{n-1} \Bigl( a_m \cos{\frac{\sigma}{2}} + b_m \sin{\frac{\sigma}{2}} \Bigr)\;}`} mit ${$`\displaystyle{\; \sigma \mathrel{\vcentcolon=} \sum_{i=0}^{n-1} \varphi_i}`} Die Formeln (1.6) zeigen, daß die rechte Seite von (2.1) die Summe von zwei trigonometrischen Polynomen vom Grade ${$`m`} ist, wobei das erste ${$`f`} an den Stützstellen ${$`p_k`} interpoliert („interpolierender Teil“), das zweite dort identisch ${$`0`} ist („verschwindender Teil“). Das interpolierende Polynom ist also bis auf die Konstante ${$`c`} eindeutig dargestellt. Falls man irgendein interpolierendes Polynom berechnen will, wird man ${$`c = 0`} wählen. Hier wollen wir ${$`c`} so bestimmen, daß ${$`t(\varphi)`} ausgewogen wird. Dazu werden wir natürlich den Koeffizienten von ${$`\sin{m\varphi}`} im verschwindenden Teil entgegengesetzt gleich dem entsprechenden Koeffizienten im interpolierenden Teil machen. `}

`38:Tf24,md` Very rough English translation (I don’t speak German):

${md` > **Definition.** ${$`\quad \displaystyle{ t(\varphi) = \tfrac{1}{2} a_0 + \sum_{k=1}^p{a_k \cos{k\varphi} + b_k \sin{k\varphi}}} `}
${$$`\tag{1.1}`}
is called a *trigonometric polynomial of degree* ${$`p`}. ${$`t(\varphi)`} is called *balanced* when ${$`b_p = 0`}. [...] Let ${$`f`} be a ${$`2\pi`}-periodic, complex-valued function on ${$`\mathbb{R}`}. In this paper we are searching for barycentric formulas for trigonometric polynomials of minimal degree, which interpolate ${$`f`} at ${$`n`} given support points ${$`\varphi_k, k = 0(1)n - 1`}: ${$$`t(\varphi_k) = f(\varphi_k) \tag{1.2}`} [...] How big must ${$`p`} be, so that every periodic function ${$`f`} at ${$`n`} arbitrary support points can be interpolated by a polynomial of degree ${$`p`}? If ${$`n`} is odd, ${$`n = 2m + 1`}, there is exactly one trigonometric polynomial of degree ${$ `m`} which will interpolate the values of ${$`f`} at the ${$`n`} interpolation points ${$`S_n`} ([14], p. 1). But if n is even, ${$`n = 2m`}, the ${$`2m`} conditions (1.2) are not sufficient to uniquely specify an interpolating polynomial of degree ${$`m`}; we need an additional condition. If ${$`S_n = S_n^g`} with ${$nb`S_n^g = \bigl\{ \varphi_k \,|\, \varphi_k = 2\pi k/n, k = 0(1)n - 1\bigr\}`}, then ${$`\sin(m \varphi_k) = 0`}; ${$`b_m`} can be chosen freely and is typically set to ${$`b_m \mathrel{\vcentcolon=} 0`}, which leads to uniqueness ([6]). So that we can later use the usual tools of discrete Fourier analysis (particularly the FFT), we want to find the balanced trigonometric interpolation polynomial for arbitrary support points. [...] #### 2. A barycentric formula for ${$`\bm{t(\varphi)}`} with ${$`\bm{n = 2m}`} arbitrary support points Gauss’s formula (1.5) fails with an even number of nodes ${$`\varphi_k`}: its right-hand side is a weighted sum of products of an odd number of terms ${$`\sin{\tfrac{1}{2}(\varphi - \varphi_k)}`} so following identity (1.6) it has the form ${$$` \sum_{k=1}^{n-1}\left( c_k \cos{\tfrac{1}{2}k\varphi} + d_k \sin{\tfrac{1}{2}k\varphi} \right) `} which is not a trigonometric polynomial in ${$`\varphi`}. Gauss ([4], [p. 283](https://archive.org/details/werkecarlf03gausrich/page/n296/mode/2up)) has further shown that for degree ${$`n = 2m`} every trigonometric polynomial (1.1) interpolating the support points ${$`\varphi_k`} can be written as: ${$$` t(\varphi) = \sum_{k=1}^{n-1} \left(\rule{}{2.5em}\right. \prod_{\substack{i = 0\\[.1em] i \neq k}}^{n - 1} \dfrac{\sin{\tfrac{1}{2}(\varphi - \varphi_i)}} {\rule{}{1em}\sin{\tfrac{1}{2}(\varphi_k - \varphi_i)}} \!\left.\rule{}{2.5em}\right) \cos{\tfrac{1}{2}(\varphi - \varphi_k)} f_k + c \cdot \prod_{i=0}^{n-1} \sin{\tfrac{1}{2}(\varphi - \varphi_i)} `}
${$$`\tag{2.1}`}
where ${$`\displaystyle{\; c \mathrel{\vcentcolon=} (-1)^{m}2^{n-1} \bigl( a_m \cos{\tfrac{1}{2}\sigma} + b_m \sin{\tfrac{1}{2}\sigma} \bigr)\;}`} with ${$`\displaystyle{\; \sigma \mathrel{\vcentcolon=} \sum_{i=0}^{n-1} \varphi_i}`} The identities (1.6) show that the right-hand side of (2.1) is the sum of two trigonometric polynomials of degree ${$`m`}, with the first interpolating ${$`f`} at the support points ${$`p_k`} (“interpolating part”), and the second identically ${$`0`} there (“vanishing part”). The interpolating polynomial is thereby uniquely specified up to a choice of the constant ${$`c`}. If any interpolating polynomial will do, choose ${$`c = 0`}. Here we intend to choose ${$`c`} so that ${$`t(\varphi)`} is balanced. For that we naturally want the coefficient of ${$`\sin{m\varphi}`} in the vanishing part to cancel the corresponding coefficient in the interpolating part. `}

`39:Tdaa,Baltensperger_2000 = md` * * * ### Richard Baltensperger (2000)
[“Some Results on Linear Rational Trigonometric Interpolation”](https://core.ac.uk/download/pdf/82015379.pdf)

${md` Let ${$`f`} be a complex valued function on the real line that is periodic with period ${$`2\pi`}, and let ${$`\varphi_k, k = 0(1)n - 1`}, be ${$`n`} distinct points of the interval ${$`[0,2\pi]`}, such that ${$`\sigma \mathrel{\vcentcolon=} \sum_{i=0}^{n-1} \varphi_i \neq 2k\pi`}, ${$`\forall k \in \mathbb{Z}`}. The (balanced [1]) trigonometric polynomial of degree ${$`\lt n/2`} which interpolates the ${$nb`f_k \mathrel{\vcentcolon=} f(\varphi_k)`}, ${$nb`k = 0(1)n-1`}, between the points ${$`\varphi_k`} is given by, see [1,2], ${$$` t[f](\varphi) \mathrel{\vcentcolon=} \sum_{k=0}^{n-1} f_k \ell_k(\varphi) \tag{1} `} where we have introduced the trigonometric Lagrange polynomials ${$nb`\ell_k(\varphi)`}, ${$$` \ell_k(\varphi) \mathrel{\vcentcolon=} \sum_{k=0}^{n-1} a_k\ell(\varphi) \biggl( \operatorname{cst} \biggl( \frac{\varphi - \varphi_k}{2} \biggr) + c \biggr), `} with ${$$` \ell(\varphi) \mathrel{\vcentcolon=} \prod_{i=0}^{n-1} \sin \biggl( \frac{\varphi - \varphi_i}{2} \biggr), \quad \operatorname{cst} \varphi \mathrel{\vcentcolon=} \begin{cases} \,\csc \varphi, & \text{if $n$ is {\bf odd},} \\ \,\cot \varphi, & \text{if $n$ is {\bf even},} \end{cases} `} and ${$$` c \mathrel{\vcentcolon=} \begin{cases} \,0, & \text{if $n$ is {\bf odd},} \\ \,\cot \frac{\sigma}{2}, & \text{if $n$ is {\bf even}.} \end{cases} `} the ${$`a_k`} are given by ${$$`(a_k)^{-1} \mathrel{\vcentcolon=} \prod_{i=0,\,i \neq k}^{n-1} \sin \biggl( \frac{\varphi_k - \varphi_i}{2} \biggr), \tag{2} `} and are called the *weights* of representation (1).
${$`t[f]`} can be written in *barycentric form* by making use of the relation (which follows by interpolation of ${$`f(\varphi) = 1)`} ${$$` 1 = \ell(\varphi)\sum_{k=0}^{n-1} a_k \biggl( \operatorname{cst} \biggl( \frac{\varphi - \varphi_k}{2} \biggr) + c \biggr). \tag{3} `} Dividing (1) by (3) and simplifying, we obtain ${$$` t[f](\varphi) = \frac {\displaystyle \sum_{k=0}^{n-1} a_k \csc((\varphi - \varphi_k) / 2 ) \, f_k } {\displaystyle \sum_{k=0}^{n-1} a_k \csc((\varphi - \varphi_k) / 2 ) \rule{}{34mu}} \tag{4} `} when the number of interpolating points is **odd** (${$`n = 2m + 1`}), see [1,2]. In the case of an **even** number of interpolating points (${$`n = 2m`}), the barycentric form of the (balanced) trigonometric polynomial is given by [1,3] ${$$` t[f](\varphi) = \frac {\displaystyle \sum_{k=0}^{n-1} a_k (\cot((\varphi - \varphi_k) / 2 ) + \cot \sigma/2) \, f_k } {\displaystyle \sum_{k=0}^{n-1} a_k (\cot((\varphi - \varphi_k) / 2 ) + \cot \sigma/2) \rule{}{34mu}} \tag{5} `} with ${$`\sigma`} defined as before. For equidistant points ${$`\varphi_k = 2k\pi/n`}, ${$`k = 0(1)n-1`}, the weights (2) satisfy (after simplification, see [1]) ${$$`a_k = (-1)^ka_0`} so that the barycentric formulae (4) and (5) become [4] ${$$` t[f](\varphi) = \frac {\displaystyle \sum_{k=0}^{n-1} (-1)^k \operatorname{cst}((\varphi - \varphi_k) / 2 ) \, f_k } {\displaystyle \sum_{k=0}^{n-1} (-1)^k \operatorname{cst}((\varphi - \varphi_k) / 2 ) \rule{}{34mu}} \tag{6} `} with ${$`\operatorname{cst}`} defined as before. This formula holds for both **odd** and **even** ${$`n`}, as ${$nb`\operatorname{cst} \sigma/2 = 0`}. Representation (6) is known to be well conditioned [4]. `}

`3a:Tebc,Wright_et_al_2015 = md` * * * ### Wright, Javed, Montanelli, & Trefethen (2015)
[“Extension of Chebfun to Periodic Functions”](https://arxiv.org/pdf/1511.00166.pdf) §3

${md` Suppose one wishes to evaluate the interpolant ${$`p_n(t)`} at certain points ${$`t`}. One good algorithm is to compute the discrete Fourier coefficients and then apply them. Alternatively, another good approach is to perform interpolation directly by means of the barycentric formula for trigonometric interpolation, introduced by Salzer [26] and later simplified by Henrici [16]: ${$$` p_n(t) = \left. \sum_{k=0}^{N-1}{(-1)^kf_k\csc\biggl(\frac{t - t_k}{2}\biggr)} \middle/ \sum_{k=0}^{N-1}{(-1)^k\csc\biggl(\frac{t - t_k}{2}\biggr)} \right. \tag{3.8} `} (If ${$`t`} happens to be exactly equal to a grid point ${$`t_k`}, one takes ${$`p_n(t) = f_k`}.) The work involved in this formula is just ${$`O(N)`} operations per evaluation, and stability has been established (after a small modification) in [3]. In practice, we find the Fourier coefficients and barycentric formula methods equally effective. In the above discussion, we have assumed that the number of interpolation points, ${$`N`}, is odd. However, trigonometric interpolation, unlike trigonometric projection, makes sense for an even number of degrees of freedom too (see e.g. [14, 19, 30]); it would be surprising if FFT codes refused to accept input vectors of even lengths! Suppose ${$`n \geq 1`} is given and we wish to interpolate ${$`f`} in ${$`N = 2n`} trigonometric points (3.1) rather than ${$nb`N = 2n+1`}. This is one data value less than usual for a trigonometric polynomial of this degree, and we can lower the number of degrees of freedom in (3.2) by imposing the condition ${$$`\tilde{c}_{-n} = \tilde{c}_{n} \tag{3.9}`} or equivalently in (3.3) by imposing the condition ${$$`\tilde{b}_{n} = 0. \tag{3.10}`} This amounts to prescribing that the trigonometric interpolant through sawtoothed data of the form ${$`f_k = (−1)^k`} should be ${$`\cos(nt)`} rather than some other function such as ${$`\exp(int)`} – the only choice that ensures that real data will lead to a real interpolant. An equivalent prescription is that an arbitrary number ${$`N`} of data values, even or odd, will be interpolated by a linear combination of the first ${$`N`} terms of the sequence ${$$`1, \cos(t), \sin(t), \cos(2t), \sin(2t), \cos(3t), \dots \tag{3.11}`} In this case of trigonometric interpolation with ${$`N`} even, the formulas (3.1)–(3.7) still hold, except that (3.4) and (3.6) must be multiplied by ${$`1/2`} for ${$`k = \pm n`}. FFT codes, however, do not store the information that way. Instead, following (3.9), they compute ${$`\tilde{a}_{-n}`} by (3.6) with ${$`2/N`} instead of ${$`1/N`} out front – thus effectively storing ${$`\tilde{c}_{-n} + \tilde{c}_{n}`} in the place of ${$`\tilde{c}_{-n}`} – and then apply (3.2) with the ${$`k = n`} term omitted. This gives the right result for values of ${$`t`} on the grid, but not at points in-between. Note that the conditions (3.9)–(3.11) are very much tied to the use of the sample points (3.1). If the grid were translated uniformly, then different relationships between ${$`c_n`} and ${$`c_{−n}`} or ${$`a_n/b_n`} and ${$`a_{-n}/b_{-n}`} would be appropriate in (3.9)–(3.10) and different basis functions in (3.11), and if the grid were not uniform, then it would be hard to justify any particular choices at all for even ${$`N`}. For these reasons, even numbers of degrees of freedom make sense in equispaced interpolation but not in other trigonometric approximation contexts, in general. Henrici [16] provides a modification of the barycentric formula (3.8) for the equispaced case ${$`N = 2n`}. `}

`3b:T493,Javed_2016 = md` * * * ### Mohsin Javed (2016)
[*Algorithms for Trigonometric Polynomial and Rational Approximation*](https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.730457), §1.3.2 > If ${$`N`} is even, the situation is slightly complicated. Recall from (1.2.1) that a trigonometric polynomial ${$`t \in \mathcal{T}_m`} has ${$`2m + 1`} degrees of freedom. If we seek an interpolant ${$`t \in \mathcal{T}_m`} where ${$`m = N/2`} then ${$`t`} has ${$`2m + 1 = N + 1`} degrees of freedom and the problem is under-determined. Hence, if ${$`m = N/2`}, the interpolant is not unique. On the other hand, taking ${$`m = N/2 - 1`} allows ${$`t`} to have only ${$`2m + 1 = N - 1`} degrees of freedom and the problem is over-determined. > In practice, interpolation at an even number of arbitrary points is usually handled by taking ${$`m = N/2`} and then enforcing an extra condition on the interpolant. For example, given an even number of points, one can enforce the condition that ${$`t`} must be *balanced*. We will further discuss the case of an even number of interpolation points, but only in the context of interpolation at equispaced points. ${$``} ` 3c:T84f,Austin_2016 = md` * * * ### Anthony Austin (2016)
[*Some New Results on and Applications of Interpolation in Numerical Computation*](https://faculty.nps.edu/aaustin/pubs/DPhilThesis.pdf)

${md` Thus far, we have only spoken of trigonometric interpolants in an odd number of points. The reason for this that a general trigonometric polynomial (1.21) has an odd number of terms/coefficients and hence an odd number of degrees of freedom associated with it. With an even number of points, the interpolation problem is either overspecified, in which case it generally has no solution, or underspecified, in which case the solution is not unique. [...] When the interpolation points are the ${$`K = 2N`} equispaced points ${$nb`x_k = 2πk/K`}, ${$nb`z_0 \leq k \leq K - 1`}, the usual way to deal with this issue is to take the interpolant from ${$`\mathcal{T}_{\!N}`} but require that the coefficients ${$`c^{-N}`} and ${$`c^{N}`} of the highest-order terms ${$`e^{−iNx}`} and ${$`e^{iNx}`} be equal. This amounts to the requirement that the interpolant to the “sawtooth” data ${$`f_k = (−1)^k`} be a pure cosine. [...] One can write down barycentric formulas for even-length trigonometric interpolants similar to those for odd-length ones given above, though they are a little more complicated for general grids. For equispaced grids, however, they are simple to write down: just replace the sine with the tangent! [...] The second formula seems to have been first written down by Henrici. It is a special case of the more general second barycentric formula for even-length trigonometric interpolation introduced by Salzer. Just as in the odd-length case, Berrut has developed special versions for the cases in which the data possess symmetry. Even-length trigonometric interpolants in equispaced points are encountered frequently in practice through the fast Fourier transform (FFT); indeed, the natural length for the FFT is a power of ${$`2`}. For non-equispaced grids, they are much less natural than odd-length ones, and accordingly, we will not say much about this case here. `}

`3d:Tbe7,md` ### Cinzia Bandiziol (2018)
[*Trigonometric and Tensor-Product Floater-Hormann Rational Interpolant*](http://tesi.cab.unipd.it/60887/1/tesi_Bandiziol.pdf)

${md` ### 2.1 Trigonometric interpolation – Generality Suppose ${$`I = [0, b]`} with ${$`b < 2\pi`} and be ${$`f : I \to R`}. For arbitrary ${$`n + 1`} nodes ${$`0 = x_0 < \dots < x_n = b`} and some values ${$`f_i := f(x_i)`} the trigonometric interpolant is > if ${$`n = 2m`} with ${$`m \in \mathbb{Z}`} ${$$` p(x) = \frac{a_0}{2} + \sum_{k=1}^m\left( a_k \cos(kx) + b_k \sin(kx) \right) `} > and if ${$`n = 2m + 1`} with ${$`m \in \mathbb{Z}`} ${$$` p(x) = \frac{a_0}{2} + \sum_{k=1}^m\left( a_k \cos(kx) + b_k \sin(kx) \right) + a_{m+1} \cos(mx) `} In the case ${$`n = 2m`}, the previous formulas tell us that the trigonometric interpolant is a linear combination of ${$`2m+1`} functions, for instance ${$`1`}, ${$`\cos(x)`}, ${$`\sin(x)`}, . . . , ${$`\cos(mx)`}, ${$`\sin(mx)`}. ${$$` \operatorname{det}(V) = \left| \rule{}{2.6em} \begin{matrix} \,1 & \cos(x_0) & \sin(x_0) & \cdots & \cos(mx_0) & \sin(mx_0) \,\\[.3em] \,\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \,\\[.3em] \,1 & \cos(x_n) & \sin(x_n) & \cdots & \cos(mx_n) & \sin(mx_n) \, \end{matrix} \right| \neq 0 `} Instead in the case ${$`n = 2m + 1`}, where ${$`p`} is a linear combination of ${$`1`}, ${$`\cos(x)`}, ${$`\sin(x)`}, . . . , ${$`\cos(mx)`}, ${$`\sin(mx)`}, ${$`\cos((m+1)x)`}, the uniqueness of the interpolant isn’t directly guaranteed and the non-vanishing is to be checked case by case. Now the determinant has the following form [6] ${$$` \operatorname{det}(V) = (-1)^{\frac{3m^2 - m}{2}}2^{2m^2 - 2m+1} \sin\Biggl(\frac{\sum_{j=0}^{2m-1}x_j}{2}\Biggr) \prod_{0 \leq q \lt p \le 2m-1}\! \sin\biggl(\frac{x_p-x_q}{2}\biggr). `} Knowing that ${$`x_k`} are distinct and ${$`0 \lt \frac{x_p-x_q}{2} < \pi`}, it follows ${$$`\begin{aligned} \operatorname{det}(V) = 0 &\iff \sin\Biggl(\frac{\sum_{j=0}^{2m-1}x_j}{2}\Biggr) = 0 \\[1em] &\iff \sum_{j=0}^{2m-1}x_j = 2\nu\pi \; \exists \nu \in \mathbb{N} \end{aligned} \tag{2.1} `} For every ${$`n`}, adding the condition ${$`\sigma = \sum_{j=0}^{2m-1} x_j \neq 2\nu\pi \; \forall \nu \in \mathbb{Z}`} if ${$`n`} is odd, the trigonometric interpolant is given by [5] ${$$` p(x) = \sum_{k=0}^n \ell_k(x)f_k `} where ${$`\{\ell_k\}_{k=0,\dots,n}`} are the Lagrange trigonometric polynomials, ${$$` \ell_k = a_k \ell(x) \left( \operatorname{cst}\left( \frac{x - x_k}{2} \right) + c \right), \tag{2.2} `} ${$$` a_k = \frac{1}{\prod_{j=0, j \neq k}^n \sin\bigl(\tfrac{x_k-x_j}{2}\bigr)}, \;\; \ell(x) = \prod_{j=0}^n \sin\biggl(\frac{x-x_j}{2}\biggr) `} and ${$$` \operatorname{cst}(x) \mathrel{\vcentcolon=} \begin{cases} \,\csc(z) = \frac{1}{\sin(x)} & \text{if $n$ even} \\[.5em] \,\cot(z) = \frac{\cos(x)}{\sin(x)} & \text{if $n$ odd} \end{cases} \;,\; c \mathrel{\vcentcolon=} \begin{cases} \, 0 & \text{if $n$ even} \\[.5em] \,\cot(\tfrac{\sigma}{2}) & \text{if $n$ odd.} \end{cases} `} `}

`3e:T7ac,Baddoo_2020 = md` * * * ### Peter Baddoo (2020)
[“The AAAtrig Algorithm for Rational Approximation of Periodic Functions”](https://arxiv.org/abs/2008.05446)

${md` A trigonometric polynomial of degree ${$`m`} with period ${$`2\pi`} is a function of the form [25] ${$$` r(z) = \sum_{k=-m}^m{c_k \mathrm{e}^{\mathrm{i}kz}}. \tag{2.1} `} Alternatively, ${$`r`} may be expressed in terms of sines and cosines as ${$$` r(z) = \frac{1}{2}a_0 + \sum_{k=1}^m{a_k \cos(kz) + b_k \sin(kz)}. \tag{2.2} `} [...] For ${$`m`} function values ${$`f_j`} at sample points ${$`z_j`}, the Lagrangian representation of trigonometric polynomial that interpolates ${$`f`} is ${$$` r(z) = \sum_{j=1}^m{f_ja_j\ell(z) \operatorname{cst}\biggl(\frac{z - z_j}{2}\biggr)}, \tag{2.3} `} where ${$`\ell`} is the trigonometric node polynomial ${$$` \ell(z) = \prod_{j=1}^m{\sin\biggl(\frac{z - z_j}{2} \biggr)}, \tag{2.4} `} and the coefficients are ${$$` a_j = \prod_{\substack{k \neq j \\[.2em] k = 0}}^m{ \csc \biggl( \frac{z_k - z_j}{2} \biggr)}, \tag{2.5} `} where ${$$` \operatorname{cst}(z) = \begin{cases} \,\csc(z) & \text{if $m$ is odd,} \\[.2em] \,\cot(z) + c & \text{if $m$ is even.} \end{cases} \tag{2.6} `} The distinction between odd and even numbers of sample points stems from the odd (${$`2m + 1`}) number of unknowns in (2.1). As such, the interpolation problem for an even number of interpolation points is under-determined. The problem is usually closed by specifying the form of the highest frequency of oscillation by requiring that the coefficient of ${$`\sin(mz)`} vanishes [24], and the trigonometric polynomial is ‘balanced’ [5]. In this case, that would mean ${$nb`c = \cot(\sum zj /2)`}, but we take ${$`c=0`} in our calculations. The reason for this choice is that we wish to consider arbitrary sample data; obviously there will be a problem if, for example, ${$`\sum z_j = 2πk`} for integer ${$`k`}. `}

`3f:T8fd, Problema. Invenire summam seriei ubi a, b, c, d, e sunt m quantitates diversae, atque n numerus integer quicunque positivus, negativus sive etiam 0. Solutio. Faciendo brevitatis caussa ita ut summa quaesita, quam per Sm denotabimus, fiat = αan + ßbn + γcn + δdn + etc.: manifestum est, si x exprimat quantitatem indeterminatam , ex evolutione aggregati in seriem secundum potestates ipsius x ascendentem, prodire Statuatur ..., eritque Q functio integra indeterminatae x, ad ordinem m^tum ascendens; PQ autem fiet functio integra ordinis m-1 puta = Qua propius considerata, patebit, per substitutionem x = 1/a omnes partes praeter primam evanescere, hanc vero abire in Simili modo per substitutionem x = 1/b evanescent omnes partes praeter secundam, quae fit = 1/b^{m-1} transit PQ in 1/c^{m-1}, 1/d^{m-1} etc. Hinc vero sequitur, PQ - x^{m-1} per omnes has substitutiones valorem 0 obtinere , quod fieri nequit , nisi fuerit identice = 0, sive PQ = x^{m-1}; alioquin enim aequatio PQ - x^{m-1} = 0, quae non maioris quam m—1 ordinis est, m radices diversas 1/a, 1/b, 1/c, 1/d, ... haberet. Iam sit ... nempe A summa quantitatum a, b, c, d . . .; B summa productorum e binis; C summa productorum e ternis etc., patetque, quum ex evolutione fractionis x^{m-1}/Q = P prodire debeat ... primo: esse debere S^0 = 0, S^1 = 0, S^2 = 0 etc. usque ad S^{m-2} = 0*), tunc vero fieri S^{m-1} = 1, S^m = A, tandemque terminos ulteriores tamquam membra seriei recurrentis per legem sequentem determinari: ... Facile quidem hinc colligitur S^{m+1} esse = aa + bb + cc + ... + B sive summam quadratorum cum summa omnium productorum e binis diversis quantitatum a, b, c, d . . .; sed quo clarius perspiciatur quonam modo termini sequentes ex elementis a, b, c, d ... formentur , observamus 1/Q esse productum e seriebus ... .... 2. Applicabimus disquisitionem praecedentem ad eum casum, ubi quantitatibus a, b, c, d valores imaginarii tribuuntur: hac ratione ad quasdam insignes relationes perfacile perveniemus, quae alia methodo tractatae maiores difficultates obiicerent. Sit E basis logarithmorum naturalium, i quantitas imaginaria √-1; consideremus loco quantitatum realium a, b, c... imaginarias E^ia, E^ib, E^ic, E^id, ..., statuamusque ... = S^n, atque ... = T^n ....... 40:T4b8,{ return 'gauss' md` Gauss: > Si multitudo valorum datorum ${$`A`}, ${$`B`}, ${$`C`}, ${$`D`} etc. unitate minor esset, quam supposuimus, puta ${$`= 2m`}, hi non sufficient ad determinationem ${$`2m + 1`} coëfficientium incognitorum, nisi inter eos relatio aliqua cognita fuerit. In hoc itaque casu expressio generalis pro ${$`T`} aliter exhiberi nequit, nisi ita, ut constantem incognitam implicet. Ratiocinia art praec. hic adhiberi non possunt, quoniam summatio ${$`W = 0`}, ex art. 2 petita suppositioni innixa est , multitudinem quantitatum ${$`A, B \dots T`} esse parem. [...] Rough translation (I don’t speak Latin so this might be horribly wrong): > If the number of data values ${$`A`}, ${$`B`}, ${$`C`}, ${$`D`} etc. were less, suppose ${$`2m`}, they would not be sufficient to determine the ${$`2m + 1`} unknown coefficients, unless one of those is specified by some relation. We can’t write down the general expression for ${$`T`} as we did before without use of an unknown implicit constant. The reasoning in the previous section can’t be used because the sum ${$`W = 0`} from section 2 is based on the assumption that the number of quantities ${$`A, B \dots T`} is equal. [...] ` }41:[] 2c:{"id":"f0785cf69f0f8166","slug":"eventrig","title":"Commentary on even-degree trigonometric interpolation","thumbnail":"8a7dc8abf9730aa828d5da4f3c80203513f04c395cf6d57eaed867d4e5848b73","access_level":"unlisted","create_time":"$D2020-10-10T21:35:23.940Z","update_time":"$D2023-03-22T05:36:48.411Z","view_time":null,"first_public_time":null,"has_unpublished_edits":null,"fork":false,"stars":3,"starred":false,"owner":{"type":"team","id":"9b8a8e177a49059a","login":"jrus","name":"Jacob Rus","avatar_url":"https://avatars-next.observableusercontent.com/c33dbeed76527c50e64208b0e005e0b6e54a3bef2a14ea6a823f1c7de4867662","tier":"starter_2024"},"version":1443,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"ddd7235babc79ce0","login":"jrus","name":"Jacob Rus","avatar_url":"https://avatars-next.observableusercontent.com/c33dbeed76527c50e64208b0e005e0b6e54a3bef2a14ea6a823f1c7de4867662","tier":null}],"members":[{"type":"individual","id":"ddd7235babc79ce0","login":"jrus","name":"Jacob Rus","avatar_url":"https://avatars-next.observableusercontent.com/c33dbeed76527c50e64208b0e005e0b6e54a3bef2a14ea6a823f1c7de4867662","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Commentary on even-degree trigonometric interpolation","cells":[{"id":0,"value":"md`# Commentary on even-degree trigonometric interpolation`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":523,"value":"$2e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":779,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":557,"value":"$30","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1023,"value":"$31","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1380,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1422,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1207,"value":"md`\n* * *\nIn these last two notes III and IV of section 11 Gauss is pointing out that the formula for the even interpolant does not define a unique solution, but rather a family of solutions with one extra degree of freedom. The trigonometric coefficients ${$`\\alpha`}, ${$`\\alpha'`}, ${$`\\beta'`}, etc. are not uniquely determined, but can sit anywhere along a line in the coefficient space ${$`\\mathbb{R}^{2m+1}`}.\n\nSince the trigonometric polynomial ${$`\\sin{\\tfrac{1}{2}(t - a)\\cdot \\sin{\\tfrac{1}{2} (t - b)} \\dotsm}`} is ${$`0`} at every support point ${$`a`}, ${$`b`}, etc., it establishes the direction of that line in coefficient space; any arbitrary multiple of it can be added to the solution without disrupting the interpolation property.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":797,"value":"$35","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1101,"value":"$36","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":149,"value":"$37","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":227,"value":"$38","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":1268,"value":"$39","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":30,"value":"$3a","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":2,"value":"$3b","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":24,"value":"$3c","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":808,"value":"$3d","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":333,"value":"$3e","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":692,"value":"md`

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