Barycentric formulas for interpolating trigonometric polynomials and their conjugates

Summary

The trigonometric polynomial of minimum degree assuming at the pointsφ _k := 2πk/n (k=0, 1, ...,n−1) given valuesf _k is forn even andφ ≠φ _k 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=2^l. This method of evaluatingt ort ^* is numerically stable even for every largen, and for values of ϕ arbitrarily close to someφ _k . 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.