Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T19:23:51.900Z Has data issue: false hasContentIssue false
  • English
  • Français

Strong summability for the Marcinkiewicz means in the integral metric and related questions

Part of: Harmonic analysis in one variable

Published online by Cambridge University Press:  09 April 2009

E. S. Belinsky
E. S. Belinsky
Affiliation:
Department of Mathematics, University of Zimbabwe, Harare, Zimbabwe e-mail: belinsky@maths.uz.ac.zw
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The inequality of strong summability for the Marcinkiewicz means of multiply Fourier series is proved. The inequalities of strong summability with gaps for the different classes of integrable functions are established. The Bernstein inequality for the fractional derivative of analytic polynomials is proved.

Keywords

Marcinkiewicz meansstrong summabilityBernstein inequality

MSC classification

Secondary: 42A24: Summability and absolute summability of Fourier and trigonometric series
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 3 , December 1998 , pp. 303 - 312
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Aleksandrov, A. B., Essays on non locally convex Hardy classes, Lecture Notes in Math. 864 (Springer, New York, 1981) pp. 1–89.Google Scholar
[2]Arestov, V., ‘On integral inequalities for trigonometric polynomials and their derivatives’, Izv. Acad. Nauk SSSR Ser. Mat. 18 (1982), 1–17 (in Russian).Google Scholar
[3]Belinskii, E. S., ‘Strong summability by the method of lacunary arithmetical means’, Anal. Math. 10 (1984), 275–282.CrossRefGoogle Scholar
[4]Belinskii, E. S., ‘Strong summability of the periodic functions from H p’, Constr. Approx. 12 (1996). 187–195.Google Scholar
[5]Belinsky, E. and Liflyand, E., ‘Approximation properties in Lp. 0 < p < 1’, Funct. Approx. Comment. Math. 22 (1993), 189–200.Google Scholar
[6]Busko, E., ‘Fonctions continues et fonctions bornees non adherentes dans L ∞(T) a la suite de leurs sommes partielles de Fourier’, Studia Math. 34 (1970), 319–337.CrossRefGoogle Scholar
[7]Chen, G. and Jiang, S. Lu Y., ‘Strong approximation of Riesz means at critical index on Hp (0 < p ≦ 1)’, Appr. Theory Appl. 5 (1989), 39–49.Google Scholar
[8]Duren, P. L. and Shields, A. L., ‘Coefficient multipliers of Hp and Bp spaces’, Pacific J. Math. 32 (1970), 69–78.CrossRefGoogle Scholar
[9]Fefferman, C. and Stein, E., ‘Hp spaces of several variables’, Acta Math. 129 (1972), 137–193.CrossRefGoogle Scholar
[10]Ganzburg, M., ‘Inequalities for the entire functions of finite order in symmetric spaces’, in: Studies on theory of functions of several real variables (Yaroslavl State Univ., Yaroslavl, 1976) pp. 22–33. (in Russian).Google Scholar
[11]Garcia-Cuerva, J. and Rubio de Francia, J. L., Weighted norm inequalities and related topics (North Holland, Amsterdam, 1985).Google Scholar
[12]Gogoladze, D., ‘On strong means of Marcinkiewicz type’, Sooobshch. Akad. Nauk Gruzin. SSR 102 (1981), 293–295 (in Russian).Google Scholar
[13]Kislyakov, S., ‘Fourier coefficients of continuous functions and a class of multipliers’, Ann. Inst. Fourier (Grenoble) 38 (1989).Google Scholar
[14]Samko, S., Kilbas, A. and Marichev, O., Fractional Integrals and Derivatives. Theory and Applications (Gordon&Breach; Sci., New York, 1992).Google Scholar
[15]Smith, B., A strong convergence theorem for H 1(T), Lecture Notes in Math. 995 (Springer, New York, 1984) pp. 169–173.Google Scholar
[16]Weisz, F., ‘Strong convergence theorems for two-parameter Walsh-Fourier series and trigonometric Fourier series’, Studia Math. 117 (1996), 173–194.CrossRefGoogle Scholar
[17]Zygmund, A., Trigonometric Series (Cambridge Univ. Press, Cambridge, 1959).Google Scholar