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Approximate properties of discrete Fourier sums. (Russian) Zbl 0718.42002

Let \(\omega_ N\) be a set \(\{t_ 0,...,t_{N-1}\}\), \(t_ j\in {\mathbb{R}}\), of N distinct points. Every discrete function \(f: \omega_ N\mapsto {\mathbb{R}}\) admits a representation as \(f(t)=\sum^{N- 1}_{k=0}c_ k\phi_ k(t),\quad t\in \omega_ N,\) where \(c_ k=c_ k(t)\) are the Fourier coefficients of f with respect to the orthonormal system \(\phi^ N=\{\phi_ 0(t),...,\phi_{N-1}(t)\},\) \(t\in \omega_ N.\)
The author estimates \(\max \{| f(t)-S_{n,N}(f;t)|:\) \(t\in \omega_ N\}\), where \(S_{n_ N}(f;t:=\sum^{N}_{k=0}c_ k(f)\phi_ k(t),\quad t\in \omega_ N,\) in the particular cases \(\omega_ N=\{u+j(2\pi)/N,\quad j=0,...,N-1\},\quad u\in {\mathbb{R}},\phi^ N=\{1;\quad \cos x,\sin x;...,\cos nx,\sin nx\}\quad (2n\leq N),\) respectively \(\omega_ N=\{0,...,N-1\}\) and \(\phi^ N\) is the system of Hahn polynomials of a discrete variable.
Reviewer: I.Badea (Craiova)

MSC:

42A10 Trigonometric approximation
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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