Abstract
Let H x a regular Hausdorff method and P(w)=∑ ak wk a power series with positive radius of convergence. A theorem of Okada states that P(w) is summable (H x ) for w in a certain starshaped region G(H x ,P). We call G=G(H x ,P) the exact region of summability for P if summability cannot hold for any w\( \in \bar G\) Okada's theorem is said to be sharp for Hx if G(Hx,P) is the exact region of summability for any P.
Three items are treated: 1. Criteria for Okada's theorem to be sharp are given in terms of the distribution function X (t) and the Mellin transform\(D(z) = \int\limits_0^1 {t^z d\chi (t)} \). 2. When is Okada's theorem sharp for product methods? 3. Special classes of functions P(w) are indicated such that G(Hx, P) is the exact region of summability for any Hx.
We use the notations of “Hausdorff-Summability of Power Series I” referred as “I”.
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Trautner, R. Hausdorff-summability of power series II. Manuscripta Math 15, 45–63 (1975). https://doi.org/10.1007/BF01168878
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DOI: https://doi.org/10.1007/BF01168878