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Hausdorff-summability of power series II

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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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References

  1. Agmon. On the singularities of TAYLOR series with reciprocal coefficients. Pac. J. Math.2, (1952), 431–453

    Google Scholar 

  2. Aronszajn, Sur les décompositions des fonctions analytiques uniformes et sur leurs applications. Acta Math.65, (1935), 1–152

    Google Scholar 

  3. BIEBERBACH, Analytische Fortsetzung Berlin 1955

  4. Okada, Über die Annäherung analytischer Funktionen Math. Z.23, (1925), 62–71

    Google Scholar 

  5. PEYERIMHOFF, Lectures on Summability Lecture Notes in Mathematics 107, Berlin 1959

  6. Rogosinski, On Hausdorff 's method of summability I Proc. Cambridge Phil. Soc.38 (1942), 166–192

    Google Scholar 

  7. Soula, Sur les points singulaires des deux fonctions ∑anzn et ∑zn/an Bull. Soc. Math. France56 (1928), 36–49

    Google Scholar 

  8. TITCHMARSH, The theory of functions Oxford 1939

  9. Trautner, Hausdorff summability of power series I Man. math7 (1972), 1–12

    Google Scholar 

  10. TRAUTNER, Summierbarkeit von Potenzreihen Habilitationsschrift Ulm 1973

  11. TRAUTNER, Wachstumseigenschaften von Mellin-Transformationen in Winkelräumen (to appear)

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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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