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Algorithms for solving Hermite interpolation problems using the fast Fourier transform. (English) Zbl 1215.33009

Hermite interpolation problems with equally spaced nodes ${\left\{{z}_{j}={e}^{i2\pi j/n}\right\}}_{j=0}^{n-1}$ on the unit circle are considered. This problem is decomposed into two problems I and II. The Hermite interpolation problem of type I with equally spaced nodes on the unit circle (well-known as Hermite-Fejér interpolation problem) is said to be a problem to determine a polynomial ${H}_{I,2n-1}\left(z\right)\in {ℙ}_{2n-1}$ such that ${H}_{I,2n-1}\left({z}_{j}\right)={u}_{j}$ and ${H}_{I,2n-1}^{\left(1\right)}\left({z}_{j}\right)=0$ for $j=0,\cdots ,n-1$. In Theorem 2 it is proved that a solution of the Hermite interpolation problem of type I is given by ${H}_{I,2n-1}\left(z\right)={\sum }_{k=0}^{2n-1}{c}_{k}{Z}_{k}\left(z\right)$, where

${\left\{{Z}_{k}\left(z\right)\right\}}_{k=0}^{2n-1}={\left\{{z}^{k}\right\}}_{k=0}^{n-1}\cup {\left\{{z}^{n+k}-{z}^{k}\left(1+k\left(n+k\right)\right)/\left(1+{k}^{2}\right)\right\}}_{k=0}^{n-1}$

and

${c}_{k}=\frac{1}{1+{k}^{2}}\frac{1}{n}\sum _{j=0}^{n-1}{u}_{j}{\overline{z}}_{j}^{k},\phantom{\rule{2.em}{0ex}}{c}_{k+n}=\frac{-k}{n}\frac{1}{n}\sum _{j=0}^{n-1}{u}_{j}{\overline{z}}_{j}^{k},\phantom{\rule{1.em}{0ex}}k=0,\cdots ,n-1·$

The Hermite interpolation problem of type II is solved in Theorem 3. To compute the coefficients ${c}_{k}$, the fast Fourier transform (FFT) is used. The general Hermite interpolation problems with an arbitrary number of derivatives in the case of algebraic and Laurent polynomials are also considered. In Section 4 algorithms for computing the Hermite interpolation polynomials in the interval $\left[-1,1\right]$ based on the Tchebycheff nodes ${\left\{cos\left(\pi \left(2j+1\right)/2n\right)\right\}}_{j=0}^{n-1}$ and the Hermite trigonometric interpolation problems on $\left[0,2\pi \right]$ are considered.

##### MSC:
 33C47 Other special orthogonal polynomials and functions 65D05 Interpolation (numerical methods) 33C52 Orthogonal polynomials and functions associated with root systems 65T50 Discrete and fast Fourier transforms (numerical methods) 65D15 Algorithms for functional approximation 65T40 Trigonometric approximation and interpolation (numerical methods)
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