Approximation of Functions by Fourier--Haar Sums in Weighted Variable Lebesgue and Sobolev Spaces

M. G. Magomed-Kasumov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 3, pp. 295-304

Résumé

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).