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

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).