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Azerbaijan; Turkey Baku; Muş
Year
2024
Volume
34
Issue
1
Pages
19-32
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Section Mathematics
Title Approximation by Nörlund type means in the grand Lebesgue spaces with variable exponent
Author(-s) Jafarov S.Z.ab
Affiliations Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciencesa, Muş Alparslan Universityb
Abstract In the present paper the approximation of functions by Nörlund type means in the generalized grand Lebesgue spaces with variable exponent is studied.
Keywords grand variable exponent Lebesgue spaces, modulus of smoothness, Lipschitz classes, trigonometric approximation, Nörlund means
UDC 517.518.8
MSC 40A05, 40A30, 41A10, 41A17, 42A10, 42A24
DOI 10.35634/vm240102
Received 17 March 2023
Language English
Citation Jafarov S.Z. Approximation by Nörlund type means in the grand Lebesgue spaces with variable exponent, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2024, vol. 34, issue 1, pp. 19-32.
References
  1. Akgün R., Kokilashvili V. On converse theorems of trigonometric approximation in weighted variable exponent Lebesgue spaces, Banach Journal of Mathematical Analysis, 2011, vol. 5, no. 1, pp. 70–82. https://doi.org/10.15352/bjma/1313362981
  2. Akgün R. Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent, Ukrainian Mathematical Journal, 2011, vol. 63, no. 1, pp. 1–26. https://doi.org/10.1007/s11253-011-0485-0
  3. Akgün R., Kokilashvili V. The refined direct and converse inequalities of trigonometric approximation in weighted variable exponent Lebesgue spaces, Georgian Mathematical Journal, 2011, vol. 18, issue 3, pp. 399–423. https://doi.org/10.1515/GMJ.2011.0037
  4. Akgün R., Kokilashvili V. Some approximation problems for $(\alpha,\psi)$-differentiable functions in weighted variable exponent Lebesgue spaces, Journal of Mathematical Sciences, 2012, vol. 186, no. 2, pp. 139–152. https://doi.org/10.1007/s10958-012-0980-3
  5. Armitage D.H., Maddox I.J. A new type of Cesàro mean, Analysis, 1989, vol. 9, issues 1–2, pp. 195–204. https://doi.org/10.1524/anly.1989.9.12.195
  6. Chandra P. Approximation bu Nörlund operators, Matematički Vesnik, 1986, vol. 38, issue 3, pp. 263–269.
  7. Chandra P. Functions of classes $L_{p}$ and $\mathrm{Lip}(\alpha,p)$ their Riesz means, Rivista di Matematica della Università di Parma. Serie 4, 1986, vol. 12, pp. 275–282. http://rivista.math.unipr.it/fulltext/1986-12/1986-12-275.pdf
  8. Chandra P. A note on degree of approximation by Nörlund and Riesz operators, Matematički Vesnik, 1990, vol. 42, issue 1, pp. 9–10.
  9. Chandra P. Trigonometric approximation of functions in $L_{p}$-norm, Journal of Mathematical Analysis and Applications, 2002, vol. 275, issue 1, pp. 13–26. https://doi.org/10.1016/S0022-247X(02)00211-1
  10. Cruz-Uribe D.V., Fiorenza A. Variable Lebesgue spaces. Foundation and harmonic analysis, Basel: Birkhäuser, 2013. https://doi.org/10.1007/978-3-0348-0548-3
  11. Diening L., Harjulehto P., Hästö P., Růžička M. Lebesgue and Sobolev spaces with variable exponents, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-18363-8
  12. Danelia N., Kokilashvili V. Approximation by trigonometric polynomials in subspace of weighted grand Lebesgue spaces, Bulletin of the Georgian National Academy of Sciences, 2013, vol. 7, no. 1, pp. 11–15. http://science.org.ge/old/moambe/7-1/Danelia%2011-15.pdf
  13. Danelia N., Kokilashvili V. Approximation of periodic functions in grand variable exponent Lebesgue spaces, Proceedings of A. Razmadze Mathematical Institute, 2014, vol. 164, pp. 100–103. https://zbmath.org/1296.41004
  14. Danelia V., Kokilashvili V., Tsanava Ts. Some approximation results in subspace of weighted grand Lebesgue spaces, Proceedings of A. Razmadze Mathematical Institute, 2014, vol. 164, pp. 104–108. https://zbmath.org/1297.42003
  15. Danelia N., Kokilashvili V. Approximation by trigonometric polynomials in the framework of variable exponent grand Lebesgue spaces, Georgian Mathematical Journal, 2016, vol. 23, issue 1, pp. 43–53. https://doi.org/10.1515/gmj-2015-0059
  16. Sbordone C., Greco L., Iwaniec T. Inverting the $p$-harmonic operators, Manuscripta Mathematica, 1997, vol. 92, issue 2, pp. 249–258. https://eudml.org/doc/156263
  17. Guven A., Israfilov D.M. Trigonometric approximation in generalized Lebesgue spaces $L$${p(x)}$ , Journal of Mathematical Inequalities, 2010, vol. 4, no. 2, pp. 285–299. https://doi.org/10.7153/jmi-04-25
  18. Güven A., Israfilov D.M. Approximation by means of Fourier trigonometric series in weighted Orlicz spaces, Advanced Studies in Contemporary Mathematics (Kyungshang), 2009, vol. 19, issue 2, pp. 283–295.
  19. Il'yasov N.A. Approximation of periodic functions by Zygmund means, Mathematical Notes of the Academy of Sciences of the USSR, 1986, vol. 39, issue 3, pp. 200–209. https://doi.org/10.1007/BF01170248
  20. Iwaniec T., Sbordone C. On the integrability of the Jacobian under minimal hypotheses, Archive for Rational Mechanics and Analysis, 1992, vol. 119, no. 2, pp. 129–143. https://doi.org/10.1007/BF00375119
  21. Iwaniec T., Sbordone C. Riesz transform and elliptic PDEs with VMO coefficients, Journal d'Analyse Mathématique, 1998, vol. 74, no. 1, pp. 183–212. https://doi.org/10.1007/BF02819450
  22. Jafarov S.Z. Linear methods for summing Fourier series and approximation in weighted Lebesgue spaces with variable exponents, Ukrainian Mathematical Journal, 2015, vol. 66, no. 10, pp. 1509–1518. https://doi.org/10.1007/s11253-015-1027-y
  23. Jafarov S.Z. Linear methods of summing Fourier series and approximation in weighted Orlicz spaces, Turkish Journal of Mathematics, 2018, vol. 42, no. 6, article 6, pp. 2916–2925. https://doi.org/10.3906/mat-1804-31
  24. Jafarov S.Z. Approximation by linear means of Fourier series in weighted Orlicz spaces, Proceedings of the Institute of Mathematics and Mechanics, 2017, vol. 43, no. 2, pp. 175–187. https://proc.imm.az/volumes/43-2/43-02-01.pdf
  25. Jafarov S.Z. On approximation of a weighted Lipschitz class functions by means $t_{n}(f;x)$, $N_{n}$${\beta}$ $(f;x)$ and $R_{n}$${\beta}$ $(f,x)$ of Fourier series, Transactions Issue Mathematics. National Academy of Sciences of Azerbaijan. Series of Physical-Technical and Mathematical Sciences, 2020, vol. 40, no. 4, pp. 118–129. https://doi.org/10.29228/proc.50
  26. Jafarov S.Z. On approximation of functions by means of Fourier trigonometric series in weighted generalized grand Lebesgue spaces, Advanced Studies: Euro-Tbilisi Mathematical Journal, 2022, special issue 10, pp. 277–291. https://tcms.org.ge/Journals/ASETMJ/Special%20issue/10/PDF/asetmj_SpIssue_10_20.pdf
  27. Kováčik O., Rákosník J. On spaces $L$${p(x)}$ and $W$${k,p(x)}$, Czechoslovak Mathematical Journal, 1991, vol. 41, issue 4, pp. 592–618. https://doi.org/10.21136/CMJ.1991.102493
  28. Kokilashvili V. On a progress in the theory of integral operators in weighted Banach function spaces, Function Spaces, Differential Operators and Nonlinear Analysis: Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands, May 28–June 2, 2004, Praha: Mathematical Institute of the Academy of Sciences of the Czech Republic, 2005, pp. 152–174.
  29. Kokilashvili V.M., Samko S.G. Operators of harmonic analysis in weighted spaces with non-standard growth, Journal of Mathematical Analysis and Applications, 2009, vol. 352, issue 1, pp. 15–34. https://doi.org/10.1016/j.jmaa.2008.06.056
  30. Kokilashvili V., Meskhi A. Maximal and Calderón–Zygmund operators in grand variable exponent Lebesgue spaces, Georgian Mathematical Journal, 2014, vol. 21, issue 4, pp. 447–461. https://doi.org/10.1515/gmj-2014-0047
  31. Kokilashvili V., Meskhi A., Rafeiro H., Samko S. Integral operators in non-standard function spaces. Volume 1: Variable exponent Lebesgue and amalgam spaces, Cham: Birkhäuser, 2016. https://doi.org/10.1007/978-3-319-21015-5
  32. Kokilashvili V., Meskhi A., Rafeiro H., Samko S. Integral operators in non-standard function spaces. Volume 2: Variable exponent Hölder, Morrey–Campanato and grand spaces, Cham: Birkhäuser, 2016. https://doi.org/10.1007/978-3-319-21018-6
  33. Krasniqi X.Z. Trigonometric approximation of (signals) functions by Nörlund type means in the variable space $L$${p(x)}$, Palestine Journal of Mathematics, 2017, vol. 6, no. 1, pp. 84–93. https://zbmath.org/1352.42004
  34. Krasniqi X.Z. Approximation of periodic functions by sub-matrix means of their Fourier series, TWMS Journal of Applied and Engineering Mathematics, 2020, vol. 10, no. 1, pp. 279–287. http://jaem.isikun.edu.tr/web/index.php/archive/104-vol10no1/513-approximation-of-periodic-functions-by-sub-matrix-means-of-their-fourier-series
  35. Krasniqi X.Z. On degree of approximation of continuous functions by a linear transformation of their Fourier series, Communications in Mathematics, 2022, vol. 30, issue 1, pp. 37–46. https://doi.org/10.46298/cm.9273
  36. Krasniqi X.Z., Łenski W., Szal B. Seminormed approximation by deferred matrix means of integrable functions in $H_{P}$${(\omega)}$ space, Results in Mathematics, 2022, vol. 77, issue 4, article number: 145. https://doi.org/10.1007/s00025-022-01696-3
  37. Leindler L. Trigonometric approximation in $L_{p}$-norm, Journal of Mathematical Analysis and Applications, 2005, vol. 302, issue 1, pp. 129–136. https://doi.org/10.1016/J.JMAA.2004.07.049
  38. Mittal M.L., Mradul Veer Singh. Approximation of signals (functions) by trigonometric polynomials in $L_{p}$-norm, International Journal of Mathematics and Mathematical Sciences, 2014, vol. 2014, article ID: 267383. https://doi.org/10.1155/2014/267383
  39. Mohaparta R.N., Russell D.C. Some direct and inverse theorems in approximation of functions, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 1983, vol. 34, issue 2, pp. 143–154. https://doi.org/10.1017/S144678870002317X
  40. Osikiewicz J.A. Equivalence results for Cesàro submethods, Analysis, 2000, vol. 20, issue 1, pp. 35–43}. https://doi.org/10.1524/anly.2000.20.1.35
  41. Quade E.S. Trigonometric approximation in the mean, Duke Mathematical Journal, 1937, vol. 3, no. 3, pp. 529–543. https://doi.org/10.1215/S0012-7094-37-00342-9
  42. Sbordone C. Grand Sobolev spaces and their applications to variational problems, Le Matematiche, 1996, vol. 51, no. 2, pp. 335–347. https://lematematiche.dmi.unict.it/index.php/lematematiche/article/view/443
  43. Sbordone C. Nonlinear elliptic equations with right hand side in nonstandard spaces, Atti del Seminario Matematico e Fisico dell'Università di Modena, 1998, vol. 46 suppl., pp. 361–368. https://zbmath.org/0913.35050
  44. Samko S.G. Convolution type operators in $L$${p(x)}$ , Integral Transforms and Special Functions, 1998, vol. 7, issues 1–2, pp. 123–144. https://doi.org/10.1080/10652469808819191
  45. Sharapudinov I.I. The topology of the space $\mathscr{L}$${p(t)}$ $([0,1])$ , Mathematical Notes of the Academy of Sciences of the USSR, 1979, vol. 26, issue 4, pp. 796–806. https://doi.org/10.1007/BF01159546
  46. Sharapudinov I.I. Nekotorye voprosy teorii priblizhenii v prostranstvakh Lebega s peremennym pokazatelem (Some questions of approximation theory in the Lebesgue spaces with variable exponent), Vladikavkaz: Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences and the Government of the Republic of North Ossetia–Alania, 2012. https://www.elibrary.ru/item.asp?id=22887342
  47. Sharapudinov I.I. Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by finite Fourier–Haar series, Sbornik: Mathematics, 2014, vol. 205, issue 2, pp. 291–306. https://doi.org/10.1070/SM2014v205n02ABEH004376
  48. Szal B. Trigonometric approximation by Nörlund type means in $L$${p}$ -norm, Commentationes Mathematicae Universitatis Carolinae, 2009, vol. 50, issue 4, pp. 575–589. http://dml.cz/dmlcz/137448
  49. Sonker S., Singh U. Approximation of signals (functions) belonging to $\mathrm{Lip}(\alpha,p,\omega)$-class using trigonometric polynomials, Procedia Engineering, 2012, vol. 38, pp. 1575–1585. https://doi.org/10.1016/j.proeng.2012.06.193
  50. Testici A., Israfilov D.M. Approximation by matrix transforms in generalized grand Lebesgue spaces with variable exponent, Applicable Analysis, 2021, vol. 100, issue 4, pp. 819–834. https://doi.org/10.1080/00036811.2019.1622680
  51. Testici A. Approximation by Nörlund and Riesz means in weighted Lebesgue spaces with variable exponent, Communications Faculty Of Science University of Ankara Series A1 Mathematics and Statistics, 2019, vol. 68, no. 2, pp. 2014–2025. https://doi.org/10.31801/cfsuasmas.460449
  52. Testici A., Israfilzade D.M. Linear methods of approximation in weighted Lebesgue spaces with variable exponent, Hacettepe Journal of Mathematics and Statistics, 2021, vol. 50, issue 3, pp. 744–753. https://doi.org/10.15672/hujms.798028
  53. Testici A., Israfilov D.M. Approximation by matrix transforms in Morrey spaces, Problemy Analiza — Issues of Analysis, 2021, vol. 10 (28), issue 2, pp. 79–98. https://doi.org/10.15393/j3.art.2021.9635
  54. Testici A., Israfilov D.M. Approximation by matrix transforms in generalized grand Lebesgue spaces with variable exponent, Journal of Numerical Analysis and Approximation Theory, 2021, vol. 50, no. 1, pp. 60–72. https://doi.org/10.33993/jnaat501-1234
  55. Volosivets S.S. Approximation of functions and their conjugate in variable Lebesgue spaces, Sbornik: Mathematics, 2017, vol. 208, issue 1, pp. 44–59. https://doi.org/10.1070/SM8636
  56. Volosivets S.S. Modified modulus of smoothness and approximation in weighted Lorentz spaces by Borel and Euler means, Problemy Analiza — Issues of Analysis, 2021, vol. 10 (28), issue 1, pp. 87–100. https://doi.org/10.15393/j3.art.2021.8950
  57. Volosivets S.S. Approximation by Vilenkin polynomials in weighted Orlizc spaces, Analysis Mathematica, 2021, vol. 47, issue 2, pp. 437–449. https://doi.org/10.1007/s10476-021-0086-6
  58. Zygmund A. Trigonometric series, Cambridge: Cambridge University Press, 2003. https://doi.org/10.1017/CBO9781316036587
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