Geodesic
Approximation of Functions by Fourier--Haar Sums in Weighted Variable Lebesgue and Sobolev Spaces
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 3, pp. 295-304.

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It is considered weighted variable Lebesgue $L^{p(x)}_w$ and Sobolev $W_{p(\cdot),w}$ spaces with conditions on exponent $p(x) \ge 1$ 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). 
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M. G. Magomed-Kasumov. Approximation of Functions by Fourier--Haar Sums in Weighted Variable Lebesgue and Sobolev Spaces. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 3, pp. 295-304. http://geodesic.mathdoc.fr/item/ISU_2014_14_3_a7/

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