On the estimate of deviations of partial sums of a multiple Fourier-Walsh series of the form S 2 j , ⋯ , 2 j f ( x ) of a function in the metric L 1(Q k )

In this paper, deviations of the partial sums of a multiple Fourier-Walsh series of a function in the metric L 1(Q k ) on a dyadic group are investigated. This estimate plays an important role in the study of equivalent normalizations in this space by means of a difference, oscillation, and best approximation by polynomials in the Walsh system. The classical classical Besov space and its equivalent normalizations are set forth in the well-known monographs of Nikolsky S.M., Besov O.V., Ilyin V.P., Triebel H.; in the works of Kazakh scientists such as Amanov T.I., Mynbaev K.T., Otelbaev M.O., Smailov E.S.. The Besov spaces on the dyadic group and the Vilenkin groups in the one-dimensional case are considered in works by Ombe H., Bloom Walter R, Fournier J., Onneweer C.W., Weyi S., Jun Tateoka.