The approximate identity kernels of product type for the Walsh system (Q1089562)

At present there are only a few approximate identity kernels for the Walsh system such as the \(p^ N\)-truncated Dirichlet kernel, the Abel- Poisson kernel. In Chin. Ann. Math. Ser. A 4, 177-184 (1983; Zbl 0489.42025) \textit{W. Zheng} introduced a new kind of approximate identity kernels for the Walsh system, known as product type kernels. In this paper the author discusses the approximation properties of these product kernels. Estimates of their moments as well as a direct approximation theorem are obtained. In order to establish an inverse approximation theorem, the p-adic derivative of product type kernels are needed. The author estimates this derivative in \(L^ 1\)-norm.