×

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

Hermite interpolation problems with equally spaced nodes \(\{z_j=e^{i2\pi j/n}\}_{j=0}^{n-1}\) 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 \(H_{I,2n-1}(z)\in\mathbb P_{2n-1}\) such that \(H_{I,2n-1}(z_j)=u_j\) and \(H^{(1)}_{I,2n-1}(z_j)=0\) for \(j=0,\dots,n-1\). In Theorem 2 it is proved that a solution of the Hermite interpolation problem of type I is given by \(H_{I,2n-1}(z)=\sum_{k=0}^{2n-1}c_kZ_k(z)\), where
\[ \{Z_k(z)\}_{k=0}^{2n-1}=\{z^k\}_{k=0}^{n-1} \cup\{z^{n+k}-z^{k}(1+k(n+k))/(1+k^2)\}_{k=0}^{n-1} \]
and
\[ c_k=\frac1{1+k^2}\frac1n\sum_{j=0}^{n-1}u_j\bar z_j^k,\qquad c_{k+n}=\frac{-k}{n}\frac1n\sum_{j=0}^{n-1} u_j \bar z_j^k,\quad k=0,\dots,n-1. \]
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 \([-1,1]\) based on the Tchebycheff nodes \(\{\cos(\pi(2j+1)/2n)\}_{j=0}^{n-1}\) and the Hermite trigonometric interpolation problems on \([0,2\pi]\) are considered.

MSC:

33C47 Other special orthogonal polynomials and functions
65D05 Numerical interpolation
33C52 Orthogonal polynomials and functions associated with root systems
65T50 Numerical methods for discrete and fast Fourier transforms
65D15 Algorithms for approximation of functions
65T40 Numerical methods for trigonometric approximation and interpolation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Faber, G., Über die Interpolatorische Darstellung stetiger Funktionen, Jber. Deutsch. Math. Verein, 23, 192-210 (1914) · JFM 45.0381.04
[2] Fejér, L., Über interpolation, Gött. Nachr., 66-91 (1916) · JFM 46.0419.01
[3] Hermite, C., Sur la formula d’interpolation de Lagrange, J. Math., 84, 70-79 (1878) · JFM 09.0312.02
[4] Daruis, L.; González-Vera, P., A note on Hermite-Fejér interpolation for the unit circle, Appl. Math. Letters, 14, 997-1003 (2001) · Zbl 0982.41003
[5] Kincaid, D.; Cheney, W., Numerical Analysis: Mathematics of Scientific Computation (1991), Brooks/Cole Publishing Company
[6] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (1996), Springer: Springer New York · Zbl 1004.65001
[8] Schoenberg, I. J., On Hermite-Birkhoff interpolation, J. Math. Anal. Appl., 16, 538-543 (1966) · Zbl 0156.28702
[9] Rivlin, T., (The Chebyshev Polynomials. The Chebyshev Polynomials, Pure and Applied Mathematics (1974), John Wiley & Sons: John Wiley & Sons New York) · Zbl 0299.41015
[10] Davis, P. J., Interpolation and Approximation (1975), Dover Publications: Dover Publications New York · Zbl 0111.06003
[11] Szabados, J.; Vértesi, P., A survey on mean convergence of interpolatory processes, J. Comput. Appl. Math., 43, 3-18 (1992) · Zbl 0761.41004
[12] Berrut, J.-P.; Welscher, A., Fourier and barycentric formulae for equidistant Hermite trigonometric interpolation, Appl. Comput. Harmon. Anal., 23, 307-320 (2007) · Zbl 1215.41006
[13] Quak, E., Trigonometric wavelets for Hermite Interpolation, Math. Comput., 65, 214, 683-722 (1996) · Zbl 0873.42024
[14] Sahakyan, K. P., Hermite trigonometric interpolation, East J. Approx., 12, 4, 441-449 (2006) · Zbl 1487.42002
[15] Kress, R., On general Hermite trigonometric interpolation, Numer. Math., 20, 125-138 (1972) · Zbl 0231.65009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.