Riesz type kernels over the ring of integers of a local field

[摘要] Let O denote the ring of integers of a local field. In this note we prove an approximation theorem for the Riesz type kernels {gamma(mu,lambda,n)}(n=1)(x) over O. The proof requires a sharp estimate of the Dirichlet kernel D-n(x) on P-1\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. Sec. 65, 1949, 372-414) concerning rates of convergence of the Fejer sum on the Walsh system. Our approach has potential applications to other operators such as Cesaro means and Abel-Poisson means of ''type I.'' (C) 1997 Academic Press.

[发布日期] 1997-04-15