Approximations by Walsh and Haar polynomials in variable exponent Lebesgue space

In recent years the interest to the investigation of variable exponent Lebesgue spaces has grown. The growing of the interest was motivated by their applications in various practical tasks in fluid dynamics, partial differential equations withnon-standard growth conditions, calculus of variations, image processing, etc. Approximation theory plays an important role in harmonic analysis. Theory of functions in classical functional spaces is given in many well-known monographs and scientific activities in this direction are still intensive. For the research of the approximation of functions in variable exponent Lebesgue spaces should be noted Sharapudinov’s pioneering works. In one of his work Sharapudinov proved that the best approximation by trigonometric polynomials in variable exponent Lebesgue space can be estimated by Butzer-Nesseltype generalized smoothness module of the function. The best approximation by trigonometric polynomials of the continuous function can be estimated by modulus of continuity of the function (Jackson's theorem). The analogue of the Jackson's theorem for the Walsh and Haar polynomials obtained by Golubov.From the given function the estimation of the deviation of partial sums of Fourier-Haar series for the functions of L^p space with integral modulus of continuity was given by Ulyanov.

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