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Convergence of interpolation by translation. (English) Zbl 0638.42003
A. Haar Mem. Conf., Budapest/Hung. 1985, Colloq. Math. Soc. János Bolyai 49, 273-287 (1987).
[For the entire collection see Zbl 0607.00008.]
Let $$C_{2\pi}$$ denote the linear space of complex valued continuous functions on R with period $$2\pi$$. $$A_{2\pi}$$ denotes the subspace consisting of functions which have absolutely convergent Fourier series. Consider a function $$g\in A_{2\pi}$$ such that $$g(t)=\sum^{\infty}_{k=-\infty}d_ ke^{ikt},$$ $$d_{-k}=d_ k$$, $$d_ k\geq d_{k+1}>0$$ $$(k>1)$$. The author [Delay Equations, Approximation and Applications, Int. Symp. Mannheim/Ger. 1984, 74, 105- 121 (1985; Zbl 0578.41012)] has recently studied the n-dimensional space $$V_ n(g)=\lim \{1,g(\cdot -t_ 1)-g,...,g(\cdot -t_{n-1})-g\}$$ where $$t_ k=2\pi k/n$$, $$k\in Z$$ which is translation invariant with respect to $$t_ 1$$, For a given $$f\in C_{2\pi}$$ it is shown that there exists a unique $$Q_ n(f)\in V_ n(g)$$ such that $$Q_ n(f)(t_ k)=f(t_ k),$$ $$k=0,...,n-1$$. Periodic spline interpolation of odd degree [see M. Golomb: J. Approximation Theory 1, 26-65 (1968; Zbl 0185.309)] follows as a special case of the foregoing result. The author studies here converence of $$Q_ n(f)$$ to f for functions of certain subclasses of $$A_{2\pi}$$ in the uniform norm as n tends to infinity.
Reviewer: H.P.Dikshit

MSC:
 42A15 Trigonometric interpolation 42A05 Trigonometric polynomials, inequalities, extremal problems 41A05 Interpolation in approximation theory 41A15 Spline approximation 65D07 Numerical computation using splines