Riesz type kernels over the ring of integers of a local field
The Riesz kernels of order \((\mu, \lambda)\) over the ring of integers of a local field \(O\) are studied in this paper: \[ \gamma_{\mu, \lambda,n} (x)= \sum^{n-1}_{k=0} \left(1- {k \choose n}^\mu \right)^\lambda \chi_k(x), \] for \(\mu>0\), \(\lambda>0\), and \[ \widetilde \gamma_{\mu,n} (x)= \sum^{n-1}_{k=0} \biggl(1- \bigl|u(k)u(n)^{-1}\bigr |^\mu \biggr) \chi_k(x), \] for \(\mu>0\). Also, the convolution Riesz means operators and the Riesz means of type I \[ \begin{gathered} R_{\mu, \lambda,n} f(x)= \gamma_{\mu, \lambda, n}*f(x), \\ \widetilde R_{\mu,n} (x)f(x)= \widetilde\gamma_{\mu,n}*f(x)\end{gathered} \] are considered. The main results in this note are approximation theorems for these two kinds of convolution Riesz means and the approximation orders for the Lip-functions are given.
scientific article; zbMATH DE number 1031588
scientific article