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 {γμ, λ, }n∞ over O . The proof requires a sharp estimate of the Dirichlet kernel Dn ( x ) on P −1\ O , which may also have independent interest. As a consequence we solve the local field analog of 1 concerning rates of convergence of the Fejér sum on the Walsh system. Our approach has potential applications to other operators such as Cesáro means and Abel–Poisson means of “type I.”