Barycentric formulas for interpolating trigonometric polynomials and their conjugates

SummaryThe trigonometric polynomial of minimum degree assuming at the pointsφk := 2πk/n (k=0, 1, ...,n−1) given valuesfk is forn even andφ ≠φk represented by(*) $$t(\phi ) = \frac{{\sum\limits_{k = 0}^{n - 1} {( - 1)^k f_k ctg\frac{{\phi - \phi _k }}{2}} }}{{\sum\limits_{k = 0}^{n - 1} {( - 1)^k ctg\frac{{\phi - \phi _k }}{2}} }}.$$ Similar formulas hold forn odd, and for the conjugate polynomialt*(ϕ). A simple recursive algorithm exists forn=2l. This method of evaluatingt ort* is numerically stable even for every largen, and for values of ϕ arbitrarily close to someφk. Inasmuch as the evaluation of (*) requires a mereO(n) operations, our formulas are more advantageous than the Fast Fourier Transform ift ort* is to be evaluated only for a small number of values of ϕ.