On a new kind of Birkhoff type trigonometric interpolation
type: Item
instance of: scholarly article
title: On a new kind of Birkhoff type trigonometric interpolation
zbMATH Open document ID: 0849.42003
DOI: 10.1007/BF01739825
author: Tianzi Jiang
published in: BIT
publication date: 1996-11-11T00:00:00Z
review text: This paper deals with the interpolation at the nodes \(x_k= {2k\pi\over n}\), \(k= 0,\dots, n- 1\), \(n\in \mathbb{N}\), of continuous \(2\pi\)-periodic functions. For a function \(f\in C_{2\pi}\) its natural difference of degree \(m\) \((m\in \mathbb{N})\) with step \(h\in (0, {\pi\over n})\) is defined by \[ \delta^m f(x)= \sum^m_{k= 0} (- 1)^k (\begin{smallmatrix} m\\ k\end{smallmatrix}) f((x+ (m- 2k)h). \] In the classical Hermite and Birkhoff interpolations the conditions are all put on the derivatives of the functions. Here the purpose of the author is to replace the derivatives by the natural differences in these interpolation problems. Denote by \({\mathcal T}_n\) the collection of all trigonometric polynomials of degree \(n\) and define \[ {\mathcal T}_{n\varepsilon}= \{T+ a_n \cos(nx+ \varepsilon \textstyle{{\pi\over 2}}): T\in {\mathcal T}_{n- 1}\},\quad \varepsilon= 0, 1. \] Let \(p_e\) and \(p_0\) be real even and odd algebraic polynomials, respectively, and let \(p= p_e+ p_0\). The interpolation problem proposed by the author can be stated as follows: (P) Given complex numbers \(\{\alpha_k\}\) and \(\{\beta_k\}\), \(k= 0,\dots, n- 1\) \((n\in \mathbb{N})\) find such polynomial \(t_n\in {\mathcal T}_{n\varepsilon}\) that \[ t_n(x_k)= \alpha_k,\quad {(p(\delta) t_n)(x_k)\over p(2h)}= \beta_k,\quad k= 0,\dots, n- 1. \] If for any given \(\alpha_k\), \(\beta_k\) problem (P) has always a unique solution then (P) is said to be regular. The author finds the necessary and sufficient conditions on the polynomial \(p\) for the regularity of (P). And he finds also expressions of the fundamental polynomials of the interpolation when (P) is regular.
Mathematics Subject Classification ID: 42A15
zbMATH DE Number: 844667
zbMATH Keywords: trigonometric interpolation
zbMATH Keywords: regularity of interpolation problem
zbMATH Keywords: natural differences
zbMATH Keywords: trigonometric polynomials
reviewed by: Wang Kunyang
MaRDI profile type: Publication
cites work: Q3972192
cites work: On the rate of convergence of a lacunary trigonometric interpolation process
cites work: Q5758642
full work available at URL: https://doi.org/10.1007/bf01739825
OpenAlex ID: W2073123672
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description: scientific article; zbMATH DE number 844667