Characterization of summability points of Nörlund methods

Grosse-Erdmann, Karl-Goswin; Stadtmüller, Karin

doi:10.1090/S0002-9947-1995-1260167-5

Characterization of summability points of Nörlund methods
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by Karl-Goswin Grosse-Erdmann and Karin Stadtmüller PDF

Trans. Amer. Math. Soc. 347 (1995), 2563-2574 Request permission

Abstract:

By a theorem of F. Leja any regular Nörlund method $(N,p)$ sums a given power series $f$ at most at countably many points outside its disc of convergence. This result was recently extended to a class of non-regular Nörlund methods by K. Stadtmüller. In this paper we obtain a more detailed picture showing how possible points of summability and the value of summation depend on $p$ and $f$.

References

R. P. Agnew, Summability of power series, Amer. Math. Monthly 53 (1946), 251–259. MR 15525, DOI 10.2307/2305093
David Borwein and Amnon Jakimovski, Matrix transformations of power series, Proc. Amer. Math. Soc. 122 (1994), no. 2, 511–523. MR 1198451, DOI 10.1090/S0002-9939-1994-1198451-0

Sur la comparaison de certains procédés de sommation des séries divergentes

26

G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620

Sur la sommation des séries entières par la méthode des moyennes

54

Karin Stadtmüller, Summability of power series by nonregular Nörlund-methods, J. Approx. Theory 68 (1992), no. 1, 33–44. MR 1143959, DOI 10.1016/0021-9045(92)90097-8
P. Vermes, The application of $\gamma$-matrices to Taylor series, Proc. Edinburgh Math. Soc. (2) 8 (1948), 43–49. MR 27349, DOI 10.1017/S0013091500024780