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Izv. RAN. Ser. Mat., 2000, Volume 64, Issue 1, Pages 95–122 (Mi izv275)  

This article is cited in 32 scientific papers (total in 32 papers)

On the coefficients of multiple Fourier series in $L_p$-spaces

E. D. Nursultanov

Institute of Applied Mathematics National Academy of Sciences of Kazakhstan

Abstract: In this paper we use new function spaces and interpolation methods to study the dependence of the properties of summable multiple Fourier series on their coefficients. We obtain theorems for multiple orthogonal series that reinforce the Hardy–Littlewood theorem for trigonometric series. We prove inequalities of Hardy–Littlewood–Paley type for multiple orthogonal series that refine certain well-known inequalities of this kind.


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English version:
Izvestiya: Mathematics, 2000, 64:1, 93–120

Bibliographic databases:

MSC: 41A17, 41A58, 41A65, 42A32, 42A20, 42C15, 42C30, 46E15, 46E30, 41A65, 41A17
Received: 05.02.1998

Citation: E. D. Nursultanov, “On the coefficients of multiple Fourier series in $L_p$-spaces”, Izv. RAN. Ser. Mat., 64:1 (2000), 95–122; Izv. Math., 64:1 (2000), 93–120

Citation in format AMSBIB
\by E.~D.~Nursultanov
\paper On the coefficients of multiple Fourier series in $L_p$-spaces
\jour Izv. RAN. Ser. Mat.
\yr 2000
\vol 64
\issue 1
\pages 95--122
\jour Izv. Math.
\yr 2000
\vol 64
\issue 1
\pages 93--120

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    This publication is cited in the following articles:
    1. N. T. Tleukhanova, “On Hardy and Bellman Transformations for Orthogonal Fourier Series”, Math. Notes, 70:4 (2001), 577–579  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. E. D. Nursultanov, “Application of Interpolational Methods to the Study of Properties of Functions of Several Variables”, Math. Notes, 75:3 (2004), 341–351  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. Nursultanov E.D., “Interpolation theorems for anisotropic function spaces and their applications”, Dokl. Math., 69:1 (2004), 16–19  mathnet  mathscinet  zmath  isi
    4. G. A. Akishev, “Approximation of function classes in spaces with mixed norm”, Sb. Math., 197:8 (2006), 1121–1144  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    5. E. D. Nursultanov, “Nikol'skii's Inequality for Different Metrics and Properties of the Sequence of Norms of the Fourier Sums of a Function in the Lorentz Space”, Proc. Steklov Inst. Math., 255 (2006), 185–202  mathnet  crossref  mathscinet
    6. G. A. Akishev, “O poryadkakh priblizheniya klassov polinomami po obobschennoi sisteme Khaara”, Sib. elektron. matem. izv., 3 (2006), 92–105  mathnet  mathscinet  zmath
    7. G. A. Akishev, “On Orders of Approximation of Function Classes in Lorentz spaces with Anisotropic Norm”, Math. Notes, 81:1 (2007), 3–14  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    8. G. A. Akishev, “Convergence of Double Fourier Series of Functions from Symmetric Spaces”, Math. Notes, 81:3 (2007), 287–290  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. A. P. Antonov, “Smoothness of sums of trigonometric series with monotone coefficients”, Russian Math. (Iz. VUZ), 51:4 (2007), 18–26  mathnet  crossref  mathscinet  zmath
    10. Simonov B., Tikhonov S., “Norm inequalities in multidimensional Lorentz spaces”, Math. Scand., 103:2 (2008), 278–294  crossref  mathscinet  zmath  isi  elib  scopus
    11. Dyachenko M., Tikhonov S., “A Hardy-Littlewood theorem for multiple series”, J. Math. Anal. Appl., 339:1 (2008), 503–510  crossref  mathscinet  zmath  isi  elib  scopus
    12. S. S. Volosivets, “Hardy and Bellman transformations of series with respect to multiplicative systems”, Sb. Math., 199:8 (2008), 1111–1137  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    13. G. A. Akishev, “The ortho-diameters of Nikol'skii and Besov classes in the Lorentz spaces”, Russian Math. (Iz. VUZ), 53:2 (2009), 21–29  mathnet  crossref  mathscinet  zmath  elib
    14. Bekmaganbetov K.A., Nursultanov E.D., “Neravenstvo raznykh metrik v anizotropnykh prostranstvakh Lorentsa”, Vestn. Rossiiskogo un-ta druzhby narodov. Ser.: Matem., inform., fiz., 2009, no. 3, 5–11  mathscinet  elib
    15. Nursultanov E., Tikhonov S., “Net Spaces and Boundedness of Integral Operators”, J Geom Anal, 21:4 (2011), 950–981  crossref  mathscinet  zmath  isi  elib  scopus
    16. Gorbachev D., Tikhonov S., “Moduli of Smoothness and Growth Properties of Fourier Transforms: Two-Sided Estimates”, J. Approx. Theory, 164:9 (2012), 1283–1312  crossref  mathscinet  zmath  isi  elib  scopus
    17. Jumabayeva A., Smailov E., Tleukhanova N., “On Spectral Properties of the Modified Convolution Operator”, J. Inequal. Appl., 2012, 146  crossref  mathscinet  zmath  isi  scopus
    18. A. M. Zhantakbayeva, E. D. Nursultanov, “Paley inequality for Bellman transform of multiple Fourier series”, Moscow University Mathematics Bulletin, 69:3 (2014), 94–99  mathnet  crossref  mathscinet
    19. Dyachenko M., Nursultanov E., Kankenova A., “On Summability of Fourier Coefficients of Functions From Lebesgue Space”, J. Math. Anal. Appl., 419:2 (2014), 959–971  crossref  mathscinet  zmath  isi  scopus
    20. Antonov A.P., “On Nikol'Skii Classes For Double Trigonometric Series With Monotone Coefficients”, Russ. J. Math. Phys., 21:2 (2014), 148–155  crossref  mathscinet  zmath  isi  scopus
    21. E. D. Nursultanov, N. T. Tleukhanova, “On reconstruction of multiplicative transformations of functions in anisotropic spaces”, Siberian Math. J., 55:3 (2014), 482–497  mathnet  crossref  mathscinet  isi  elib  elib
    22. A. U. Bimendina, E. S. Smailov, “Fourier–Price coefficients of class GM and best approximations of functions in the Lorentz space $L_{p\theta}[0,1)$, $1<p<+\infty$, $1<\theta<+\infty$”, Proc. Steklov Inst. Math., 293 (2016), 77–98  mathnet  crossref  crossref  mathscinet  isi  elib
    23. Ydyrys A., Sarybekova L., Tleukhanova N., “The multipliers of multiple trigonometric Fourier series”, Open Eng., 6:1 (2016), 367–371  crossref  isi  elib  scopus
    24. K. A. Bekmaganbetov, Ye. Toleugazy, “Order of the orthoprojection widths of the anisotropic Nikol'skii–Besov classes in the anisotropic Lorentz space”, Eurasian Math. J., 7:3 (2016), 8–16  mathnet  mathscinet
    25. Baladi V., “The Quest for the Ultimate Anisotropic Banach Space”, J. Stat. Phys., 166:3-4, SI (2017), 525–557  crossref  mathscinet  zmath  isi  scopus
    26. Kopezhanova A., Nursultanov E., Persson L.-E., “Some New Two-Sided Inequalities Concerning the Fourier Transform”, Math. Inequal. Appl., 20:3 (2017), 855–864  crossref  mathscinet  zmath  isi  scopus
    27. A. N. Kopezhanova, “Some new inequalities for the Fourier transform for functions in generalized Lorentz spaces”, Eurasian Math. J., 8:1 (2017), 58–66  mathnet
    28. Dyachenko M., Nursultanov E., Tikhonov S., “Hardy-Type Theorems on Fourier Transforms Revised”, J. Math. Anal. Appl., 467:1 (2018), 171–184  crossref  mathscinet  zmath  isi  scopus
    29. Kopezhanova A., Nursultanov E., Persson L.-E., “A New Generalization of Boas Theorem For Some Lorentz Spaces Lambda(Q)(Omega)”, J. Math. Inequal., 12:3 (2018), 619–633  crossref  mathscinet  isi
    30. Akishev G., Persson L.E., Seger A., “Some Fourier Inequalities For Orthogonal Systems in Lorentz-Zygmund Spaces”, J. Inequal. Appl., 2019, 171  crossref  isi
    31. Sadykova K.K., Tleukhanova N.T., “Estimates of the Norm of the Convolution Operator in Anisotropic Besov Spaces With the Dominated Mixed Derivative”, Bull. Karaganda Univ-Math., 95:3 (2019), 51–59  crossref  isi
    32. Akishev G., Lukkassen D., Persson L.E., “Some New Fourier Inequalities For Unbounded Orthogonal Systems in Lorentz-Zygmund Spaces”, J. Inequal. Appl., 2020:1 (2020), 77  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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