Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by de la Vallée-Poussin means

Abstract

We consider the space formed by -periodic real measurable functions for which the integral exists and is finite, where , , is a -periodic measurable function (a variable exponent). If , then the space can be endowed with the structure of Banach space with the norm 0: \\int_{-\\pi}^{\\pi}\\biggl|\\frac{f(x)}{\\alpha}\\biggr|^{p(x)}\\,dx\\le1\\biggr\\}. \\end{equation*} ?> In the space we distinguish a subspace of Sobolev type. We investigate the approximation properties of the de la Vallée-Poussin means for trigonometric Fourier sums for functions in the space . In particular, we prove that if the variable exponent satisfies the Dini-Lipschitz condition and if , then the de la Vallée-Poussin means with satisfy where is a modulus of continuity of the function defined in terms of the Steklov functions. It is proved that if , , and the Dini-Lipschitz condition holds, then where stands for the best approximation to by trigonometric polynomials of order . Bibliography: 19 titles.