Abstract. In this paper we examine approximation on $B_{pq}^{\alpha }$, Besov spaces over the $2$-series field, for ${1\over2} < p< 1$. Namely, we shall obtain strong and weak estimates for certain operators associated with strong approximation by the Cesàro means, the strong Cesàro means, and a transplantation theorem for maximal multipliers on $B_{pq}^{\alpha }$. We also estimate the operator norms of partial sums and Cesàro means, and use this estimate to characterize Besov Walsh--Fourier series.

AMS Subject Classification (1991): 42C10, 26A16, 42A24, 42A45

Keyword(s): Walsh functions, Besov spaces, strong approximation, strong Cesàro means, transplantation

Received August 22, 1994. (Registered under 5659/2009.)