A Note on Riesz Means Over the Ring of P−Adic Integers

Let K be a local field, V and P the ring of integers and its maximal ideal, respectively. The Riesz means is R μ,λ,n f(x)=∑ k=0 n-1 1-k n μ λ f ∧ (k)χ k (x), with kernel r μ,λ,n =∑ k=0 n-1 (1-(k n) μ ) λ χ k. The main result is: Suppose that f ∈ L p (V),1≤p<∞,C(V),p=∞· If λ≥1, μ>0, then R μ,λ,n f converges to f in L p (V) as n→∞.