The approximate identity kernels of product type for the Walsh system

Abstract

At present there are only a few approximate identity kernels for the Walsh system, for example, the p N -truncated Dirichlet kernel D p N − 1 ( t) = ∑ j = 0 p N − 1 w j ( t) [6]; the Abel-Poisson kernel λ γ ( t) = ∑ k = 0 ∞ γ k w k ( t) [3], and so on. In [6], Zheng has introduced a new kind of approximate identity kernels for the Walsh system—the kernels of product type. In the present paper we discuss the approximation properties of such product type kernels. Estimates of their moments as well as a direct approximation theorem are obtained. Then, to establish an inverse approximation theorem, we need the p-adic derivative of product type kernels and we estimate this derivative in L 1-norm.