On a new kind of Birkhoff type trigonometric interpolation
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.
scientific article; zbMATH DE number 844667