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Approximation of functions by Fourier-Haar sums in weighted variable Lebesgue and Sobolev spaces. (Russian. English summary) Zbl 1301.42008
Summary: It is considered weighted variable Lebesgue \(L^{p(x)}_w\) and Sobolev \(W_{p(\cdot), w}\) spaces with conditions on exponent \(p(x)\geq 1\) and weight \(w(x)\) that provide Haar system to be a basis in \(L^{p(x)}_w\). In such spaces there were obtained estimates of Fourier-Haar sums convergence speed. Estimates are given in terms of modulus of continuity \(\Omega(f, \delta)_{p(\cdot),w}\), based on mean shift (Steklov’s function).
MSC:
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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Full Text: MNR