Open Access

# On regularity of linear summation methods of Taylor series

### Abstract

The paper specifies necessary and sufficient conditions for regularity of an infinite matrix of real numbers, which determines some summation method for a class of functions that are analytic on the unit disk and continuous on the closed circle.

Key words: Summation method, infinite matrix, regularity, Taylor series.

### Article Information

 Title On regularity of linear summation methods of Taylor series Source Methods Funct. Anal. Topology, Vol. 21 (2015), no. 1, 56-68 MathSciNet MR3407920 zbMATH 06533467 Milestones Received 15/06/2013; Revised 10/11/2014 Copyright The Author(s) 2015 (CC BY-SA)

### Authors Information

M. V. Gaevskij
Kirovohrad Volodymyr Vynnychenko State Pedagogical University, Kirovohrad, Ukraine

Kyiv National University of Technology and Design, Kyiv, Ukraine

### Citation Example

M. V. Gaevskij and P. V. Zaderey, On regularity of linear summation methods of Taylor series, Methods Funct. Anal. Topology 21 (2015), no. 1, 56-68.

### BibTex

@article {MFAT736,
AUTHOR = {Gaevskij, M. V. and Zaderey, P. V.},
TITLE = {On regularity of linear summation methods of Taylor series},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {21},
YEAR = {2015},
NUMBER = {1},
PAGES = {56-68},
ISSN = {1029-3531},
MRNUMBER = {MR3407920},
ZBLNUMBER = {06533467},
URL = {http://mfat.imath.kiev.ua/article/?id=736},
}

### References

1. J. Karamata, M. Tomi, Sur la sommation des series de Fourier des fonctions continues, Acad. Serbe Sci. Publ. Inst. Math. 8 (1955), 123-138.  MathSciNet
2. S. Lozinski, On convergence and summability of Fourier series and interpolation processes, Rec. Math. [Mat. Sbornik] N.S. 14(56) (1944), 175-268.  MathSciNet
3. S. M. Nikol′skii, On linear methods of summation of Fourier series, Izvestiya Akad. Nauk SSSR. Ser. Mat. 12 (1948), 259-278.  MathSciNet
4. L. V. Taikov, On summation methods for Taylor series, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 625-630.  MathSciNet
5. N. V. Gaevskij, T. V. Gorislavec, P. V. Zaderej, On the regularity of certain methods of summation of the Taylor series, International Conference "Theory of Approximation of Functions and its Applications" dedicated to the 70th Anniversary of Corresponding Member of NAS of Ukraine, Professor A. I. Stepanets (May 28-June 3, 2012, Kamianets-Podilsky, Ukraine). Abstracts. Kyiv, Institute of Mathematics of NAS of Ukraine, 2012, p. 35. (Russian)
6. Friedrich Riesz, Uber Potenzreihen mit vorgeschriebenen Anfangsgliedern, Acta Math. 42 (1920), no. 1, 145-171.  MathSciNet CrossRef
7. L. V. Kantorovich, G. P. Akilov, Funktsionalnyi analiz, Nauka'', Moscow, 1984.  MathSciNet
8. Frigyes Riesz, Bela Sz.-Nagy, Lektsii po funktsionalnomu analizu, Mir'', Moscow, 1979.  MathSciNet
9. A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, Izdat. Nauka'', Moscow, 1976.  MathSciNet
10. A. V. Efimov, Estimate of integral of modulus of polynomial on the unit circle, Uspekhi Mat. Nauk 15 (1960), no. 4(94), 215-218. (Russian)
11. P. V. Zaderei, B. A. Smal′, On the convergence of Fourier series in the space $L_ 1$, Ukrain. Mat. Zh. 54 (2002), no. 5, 639-646.  MathSciNet CrossRef
12. A. Zigmund, Trigonometricheskie ryady. Tomy I, II, Izdat. Mir'', Moscow, 1965.  MathSciNet
13. S. A. Teljakovskii, Integrability conditions for trigonometrical series and their application to the study of linear summation methods of Fourier series, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 1209-1236.  MathSciNet
14. S. A. Telyakovskii, An estimate, useful in problems of approximation theory, of the norm of a function by means of its Fourier coefficients, Proc. Steklov Inst. Math. 109 (1971), 73-109. (Russian)