On regularity of linear summation methods of Taylor series
DOI:
Keywords:
Summation method, infinite matrix, regularity, Taylor seriesAbstract
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.References
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
S. Lozinski, On convergence and summability of Fourier series and interpolation processes, Rec. Math. [Mat. Sbornik] N.S. 14(56) (1944), 175-268. MathSciNet
S. M. Nikol′skii, On linear methods of summation of Fourier series, Izvestiya Akad. Nauk SSSR. Ser. Mat. 12 (1948), 259-278. MathSciNet
L. V. Taikov, On summation methods for Taylor series, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 625-630. MathSciNet
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)
Friedrich Riesz, Uber Potenzreihen mit vorgeschriebenen Anfangsgliedern, Acta Math. 42 (1920), no. 1, 145-171. MathSciNet CrossRef
L. V. Kantorovich, G. P. Akilov, Funktsionalnyi analiz, ``Nauka'', Moscow, 1984. MathSciNet
Frigyes Riesz, Bela Sz.-Nagy, Lektsii po funktsionalnomu analizu, ``Mir'', Moscow, 1979. MathSciNet
A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, Izdat. ``Nauka'', Moscow, 1976. MathSciNet
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)
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
A. Zigmund, Trigonometricheskie ryady. Tomy I, II, Izdat. ``Mir'', Moscow, 1965. MathSciNet
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
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)
