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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