0-regularly varying convergence moduli of Fourier and Fourier-Stieltjes series (Q1085369)

Let \(f\in L^ 1(T)\), \(\mu \in M(T)1<p\leq 2\), and let m be a positive integer. Through the representation of \(\sum _{| k| \leq n}| k| ^{p-1} | \Delta \hat f(k)| ^ p,\sum _{| k| \leq n}| \Delta ^ m\hat f(k)|\) and \(\Delta\) \({\hat \mu}\), via 0- regular variation, new Tauberian theorems for \(L^ 1\)-convergence and pointwise convergence are obtained, and new succinct proofs are given.