Some Tauberian theorems for iterations of Hölder integrability method

Let f be a real or complex-valued function on [1 , ?) which is continuous over every finite interval [1, x) for 1 < x< ?. We set s(x):=?1xf(t)dt and define ? k (s(x)) by ?k(s(x))={1x?1x?k-1(s(t))dt,k?1s(x),k=0for each nonnegative integer k. An improper integral ?1?f(x)dxis said to be integrable to a finite number µ by the k-th iteration of Hölder or Cesàro mean method of order one, or for short, the (H, k) integrable to µ if limx›??k(s(x))=µ.In this case, we write s(x)›µ(H,k). It is clear that the (H, k) integrability method reduces to the ordinary convergence for k= 0 and the (H, 1) integrability method is (C, 1) integrability method. It is known that lim x › ? s(x) = µ implies lim x › ? ? k (s(x) ) = µ. But the converse of this implication is not true in general. In this paper, we obtain some Tauberian conditions for the iterations of Hölder integrability method under which the converse implication holds. © 2019, Springer Nature Switzerland AG.