Hardy Type Asymptotics for Cosine Series in Several Variables with Decreasing Power-Like Coefficients

Kozyakin, Victor

doi:10.18052/www.scipress.com/IJARM.5.35

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Hardy Type Asymptotics for Cosine Series in Several Variables with Decreasing Power-Like Coefficients

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Abstract:

The investigation of the asymptotic behavior of trigonometric series near the origin is a prominent topic in mathematical analysis. For trigonometric series in one variable, this problem was exhaustively studied by various authors in a series of publications dating back to the work of G. H. Hardy, 1928. Trigonometric series in several variables have got less attention. The aim of the work is to partially fill this gap by finding the asymptotics of trigonometric series in several variables with the terms, having a form of `one minus the cosine' up to a decreasing power-like factor.The approach developed in the paper is quite elementary and essentially algebraic. It does not rely on the classic machinery of the asymptotic analysis such as slowly varying functions, Tauberian theorems or the Abel transform. However, in our case, it allows obtaining explicit expressions for the asymptotics and extending to the general case of trigonometric series in several variables classical results of G. H. Hardy and other authors known for the case of one variable.

Info:

Periodical:

International Journal of Advanced Research in Mathematics (Volume 5)

Pages:

35-51

DOI:

https://doi.org/10.18052/www.scipress.com/IJARM.5.35

Citation:

V. Kozyakin, "Hardy Type Asymptotics for Cosine Series in Several Variables with Decreasing Power-Like Coefficients", International Journal of Advanced Research in Mathematics, Vol. 5, pp. 35-51, 2016

Online since:

June 2016

Authors:

Victor Kozyakin

Keywords:

Asymptotic Behavior at Zero, Decreasing Coefficients, Hardy Type Asymptotics, Power-Like Coefficients, Trigonometric Series in Several Variables

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Open Access

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Creative Commons Attribution 4.0 International License

References:

[1] G. H. Hardy, A theorem concerning trigonometrical series, J. London Math. Soc. S1-3 (1) (1928) 12-13. doi: 10. 1112/jlms/s1-3. 1. 12.

[2] G. H. Hardy, Some theorems concerning trigonometrical series of a special type, Proc. London Math. Soc. S2-32 (1) (1931) 441-448. doi: 10. 1112/plms/s2-32. 1. 441.

[3] S. Aljančić, R. Bojanić, M. Tomić, Sur le comportement asymtotique au voisinage de zéro des séries trigonométriques de sinus à coefficients monotones, Acad. Serbe Sci., Publ. Inst. Math. 10 (1956) 101-120.

[4] C. -H. Yong, On the asymptotic behavior of trigonometric series. I, J. Math. Anal. Appl. 33 (1971) 23-34. doi: 10. 1016/0022-247X(71)90178-8.

[5] C. -H. Yong, On the asymptotic behavior of trigonometric series. II, J. Math. Anal. Appl. 38 (1972) 1-14. doi: 10. 1016/0022-247X(72)90111-4.

[6] G. H. Hardy, W. W. Rogosinski, Notes on Fourier series. III. Asymptotic formulae for the sums of certain trigonometrical series, Quart. J. Math., Oxford Ser. 16 (1945) 49-58.

[7] J. R. Nurcombe, On trigonometric series with quasimonotone coefficients, J. Math. Anal. Appl. 178 (1) (1993) 63-69. doi: 10. 1006/jmaa. 1993. 1291.

[8] C. -P. Chen, L. Chen, Asymptotic behavior of trigonometric series with O-regularly varying quasimonotone coefficients, J. Math. Anal. Appl. 250 (1) (2000) 13-26. doi: 10. 1006/jmaa. 2000. 6952.

[9] C. -P. Chen, L. Chen, Asymptotic behavior of trigonometric series with O-regularly varying quasimonotone coefficients. II, J. Math. Anal. Appl. 245 (1) (2000) 297-301. doi: 10. 1006/jmaa. 2000. 6745.

[10] S. Tikhonov, Trigonometric series with general monotone coefficients, J. Math. Anal. Appl. 326 (1) (2007) 721-735. doi: 10. 1016/j. jmaa. 2006. 02. 053.

[11] A. Zygmund, Trigonometric series. Vol. I, II, 3rd Edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002, with a foreword by Robert A. Fefferman.

[12] E. Yarovaya, Branching random walks with heavy tails, Comm. Statist. Theory Methods 42 (16) (2013) 3001-3010. doi: 10. 1080/03610926. 2012. 703282.

[13] E. B. Yarovaya, Criteria for transient behavior of symmetric branching random walks on Z and Z2, in: J. R. Bozeman, V. Girardin, C. H. Skiadas (Eds. ), New Perspectives on Stochastic Modeling and Data Analysis, ISAST Athens Greece, 2014, pp.283-294.

[14] V. S. Kozyakin, On the asymptotics of cosine series in several variables with power coefficients, Journal of Communications Technology and Electronics 60 (12) (2015) 1441-1444. doi: 10. 1134/S1064226915120165.

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[1] A. Rytova, E. Yarovaya, "Survival analysis of particle populations in branching random walks", Communications in Statistics - Simulation and Computation, p. 1, 2019

DOI: https://doi.org/10.1080/03610918.2019.1618870