Riesz Type Kernels over the Ring of Integers of a Local Field

Let O denote the ring of integers of a local field. In this note we prove an approximation theorem for the Riesz type kernels g 4` over O. The proof m, l, n ns1 requires a sharp estimate of the Dirichlet kernel Dn x. on Py1_O, which may also have independent interest. As a consequence we solve the local field analog of N. J. Fine’s problem Trans. Amer. Math. Soc. 65, 1949, 372]414.concerning rates of convergence of the Fej´er sum on the Walsh system. Our approach has potential applications to other operators such as Ces´aro means and Abel]Poisson means of ‘‘type I.’’