THE DYADIC INTEGRAL MODULE OF CONTINUITY AND BEST APPROXIMATIONS OF A FUNCTION OF MULTIPLE VARIABLES BY WALSH POLYNOMIALS - Igenberlina, А.; Keulimzhayeva, Ж. (Bulletin of Abai KazNPU. Series of Physical and Mathematical sciences, 2022)
In the theory of orthogonal series, along with trigonometric systems, Haar and Walsh systems and their generalizations are widely used. The classical theory of Fourier series deals with the expansion of functions in sinusoidal harmonics. Unlike these continuous harmonics, Walsh functions are "square" waves. It turned out that in some cases they are preferable to sinusoidal waves. The study of classical function spaces is based on the approximation of functions by trigonometric polynomials, and in this paper, functional spaces are considered from the point of view of approximation of functions by partial Fourier-Walsh sums on a dyadic group: a connection is established between the dyadic integral modulus of continuity and the best approximations of a function of many variables by Walsh polynomials. In addition, an integral estimate of the partial sums of the multiple Fourier-Walsh series is given and the relationship between the deviations of such sums of the multiple series from the function and the group modulus of continuity is studied. In the one-dimensional case, dyadic modules of continuity are considered in the monograph by Golubov B.I. «Walsh series and transformations. Theory and applications».