Interpolation of odd periodic functions on uniform meshes. (English) Zbl 0578.41012

Delay equations, approximation and application, Int. Symp. Mannheim/Ger. 1984, ISNM 74, 105-121 (1985).
[For the entire collection see Zbl 0554.00008.]
From the introduction: ”Let \(g\in C_{2\pi}\) have an absolutely convergent Fourier series. For \(n\in {\mathbb{N}}\) we define the uniform mesh \(t_ k=2\pi k/n\), \(k\in {\mathbb{Z}}\), and the translates \(g_ k=g(\cdot - t_ k)\), \(0\leq k<n\), of g. F. Locher [Math. Comput. 37, 403-416 (1981; Zbl 0517.42004)] presented a method of interpolation of periodic functions f at the uniform mesh \(t_ k\), \(k\in {\mathbb{Z}}\), by functions h from the linear space \(V_ n(g)=lin\{g_ 0,g_ 1,...,g_{n-1}\}\) of translates of g. Locher’s method is only applicable if \(B_ k(0)\neq 0\) for \(k=0,1,...,n-1\) where the functions \(B_ k\), \(k=0,1,...,n-1\), are defined by \(B_ k(t)=\sum^{n-1}_{j=0}g(t-t_ j)\exp (ikt_ j).\)
It is the objective of this paper to develop a related method of interpolation of odd periodic functions: Let us define \(S_ j(t)=(B_ j(t)/B_ j(0)-B_{n-j}(t)/B_{n-j}(0))/(2i)\), \(n=2m\) and \(V^ s_ n(g)=lin\{S_ 1,...,S_{m-1}\}.\) Theorem. Assume \(B_ j(0)\neq 0\), \(0<j<m\). Then for any odd \(f\in C_{2\pi}\) there is a unique function \(R_ m(f)\in V^ s_ n(g)\), \(n=2m\), which interpolates f at the points \(t_ k\), \(k\in {\mathbb{Z}}.''\)
Reviewer: M.Kinukawa


41A15 Spline approximation
41A05 Interpolation in approximation theory