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Jakimovski, A.; Meyer-Koenig, W.; Zeller, K.

Power series methods of summability: positivity and gap perfectness. (English) Zbl 0479.40005

Trans. Am. Math. Soc. 266, 309-317 (1981).
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Cited in 5 Documents

MSC:

40E15 Lacunary inversion theorems
40D09 Structure of summability fields
40G10 Abel, Borel and power series methods
40C15 Function-theoretic methods (including power series methods and semicontinuous methods) for summability
30B10 Power series (including lacunary series) in one complex variable

Keywords:

high indices theorems; gap Tauberian theorems; perfectness of FK-spaces; gap-perfectness of FK-spaces; power-series methods of summability

Citations:

Zbl 0075.258; Zbl 0335.40002; Zbl 0423.40006
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Full Text: DOI

References:

[1] Wolfgang Beekmann and Karl Zeller, Positive Operatoren in der Limitierung, Linear operators and approximation, II (Proc. Conf., Math. Res. Inst., Oberwolfach, 1974) Birkhäuser, Basel, 1974, pp. 559 – 564. Internat. Ser. Numer. Math., Vol. 25 (German). · Zbl 0298.40009
[2] G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. · Zbl 0032.05801
[3] Kazuo Ishiguro, A converse theorem on the summability methods, Proc. Japan Acad. 39 (1963), 38 – 41. · Zbl 0113.04801
[4] Kazuo Ishiguro, On the summability methods of divergent series, Acad. Roy. Belg. Cl. Sci. Mém. Coll. in-8 (2) 35 (1965), no. 1, 42 pp. (With errata).
[5] A. Jakimovski and H. Tietz, Regularly varying functions and power series methods, J. Math. Anal. Appl. 73 (1980), no. 1, 65 – 84. · Zbl 0435.40009 · doi:10.1016/0022-247X(80)90022-0
[6] Donald Kershaw, Regular and convergent Korovkin sequences, Linear operators and approximation, II (Proc. Conf., Math. Res. Inst., Oberwolfach, 1974) Birkhäuser, Basel, 1974, pp. 377 – 389. Internat. Ser. Numer. Math., Vol. 25. · Zbl 0297.41011
[7] W. Kołodziej, Über den Beweis eines Satzes aus der Limitierungstheorie, Colloq. Math. 23 (1971), 299 – 300 (German). · Zbl 0227.46021
[8] V. K. Krishnan, Gap Tauberian theorem for generalized Abel summability, Math. Proc. Cambridge Philos. Soc. 78 (1975), no. 3, 497 – 500. · Zbl 0335.40002 · doi:10.1017/S0305004100051975
[9] V. K. Krishnan, Gap Tauberian theorem for logarithmic summability (\?), Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 117, 77 – 87. · Zbl 0423.40006 · doi:10.1093/qmath/30.1.77
[10] S. Mazur und L. Sternbach, Über Konvergenzmengen von Folgen linearer Operationen, Studia Math. 4 (1933), 54-65. · Zbl 0008.31601
[11] W. Meyer-König und K. Zeller, Lückenumkehrsätze und Lückenperfektheit, Math. Z. 66 (1956), 203-224. · Zbl 0075.25801
[12] -, \( FK\)-Räume und Lückenperfektheit, Math. Z. 78 (1962), 143-148.
[13] Michael Stieglitz and Hubert Tietz, Matrixtransformationen von Folgenräumen. Eine Ergebnisübersicht, Math. Z. 154 (1977), no. 1, 1 – 16 (German). · Zbl 0331.40005 · doi:10.1007/BF01215107
[14] C. T. Rajagopal, On Tauberian theorems for some standard methods of summability, J. Indian Math. Soc. (N.S.) 39 (1975), 69 – 82 (1976). · Zbl 0443.40003
[15] Albert Wilansky and Karl Zeller, Summation of bounded divergent sequences, topological methods, Trans. Amer. Math. Soc. 78 (1955), 501 – 509. · Zbl 0065.04404
[16] L. Włodarski, Sur les méthodes continues de limitation. I. Application de l’espace \?\(_{0}\) de Mazur et Orlicz à l’étude des méthodes continues, Studia Math. 14 (1954), 161 – 187 (1955) (French). · Zbl 0064.05701
[17] K. Zeller and W. Beekmann, Theorie der Limitierungsverfahren, Zweite, erweiterte und verbesserte Auflage. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 15, Springer-Verlag, Berlin-New York, 1970 (German). · Zbl 0199.11301
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