On the proof of the main Tauberian theorem for the πΆ_{π}-and π»_{π}-methods
Konrad Knopp Β· Proceedings of the American Mathematical Society Β· 1954
The proofs are, for general k, not quite easy. The following inductive proof, taking the theorem for granted for k = 1-this is the fundamental Hardy-Landau theorem3 from 1910-seems to me of simpler structure than all of them, its main difference from these lying in the fact that it uses the series-to-series instead of the sequence-to-sequence transform. We prove it first for the Hk-method, both theorems being identical for k =1: