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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

MSC:
 41A15 Spline approximation 41A05 Interpolation in approximation theory
Citations:
Zbl 0554.00008; Zbl 0517.42004