Multivariate approximation in φ-variation for nonlinear integral operators via
summability methods

ASLAN, ISMAIL

doi:10.3906/mat-2109-77

Multivariate approximation in φ-variation for nonlinear integral operators via summability methods

İsmail ASLAN (Hacettepe Üniversitesi, Fen Fakültesi, Matematik Bölümü, Ankara, Türkiye)

Turkish Journal of Mathematics

0 0

Yıl: 2022 Cilt: 46 Sayı: 1 Sayfa Aralığı: 277 - 298 Metin Dili: İngilizce DOI: 10.3906/mat-2109-77 İndeks Tarihi: 19-07-2022

Multivariate approximation in φ-variation for nonlinear integral operators via summability methods

Öz:

We consider convolution-type nonlinear integral operators endowed with Musielak-Orlicz φ-variation. Our aim is to get more powerful approximation results with the help of summability methods. In this study, we use φabsolutely continuous functions for our convergence results. Moreover, we study the order of approximation using suitable Lipschitz class of continuous functions. A general characterization theorem for φ-absolutely continuous functions is also obtained. We also give some examples of kernels in order to verify our approximations. At the end, we indicate our approximations in figures together with some numerical computations.

Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık

[1] Alemdar ME, Duman O. General summability methods in the approximation by Bernstein-Chlodovsky operators. Numerical Functional Analysis and Optimization 2021; 42 (5): 497-509.
[2] Angeloni L. A characterization of a modulus of smoothness in multidimensional setting. Bollettino dell’Unione Matematica Italiana 2011; 4 (1): 79-108.
[3] Angeloni L. Approximation results with respect to multidimensional φ-variation for nonlinear integral operators. Zeitschrift für Analysis und ihre Anwendungen 2013; 32 (1): 103-128.
[4] Angeloni L, Vinti G. Approximation by means of nonlinear integral operators in the space of functions with bounded φ-variation. Differential and Integral Equations 2007; 20: 339-360.
[5] Angeloni L, Vinti G. Convergence and rate of approximation for linear integral operators in BV φ -spaces in multidimensional setting. Journal of Mathematical Analysis and Applications 2009; 349: 317-334.
[6] Angeloni L, Vinti G. Errata corrige to: Approximation by means of nonlinear integral operators in the space of functions with bounded φ-variation. Differential and Integral Equations 2010; 23: 795-799.
[7] Angeloni L, Vinti G. Convergence and rate of approximation in BVφ$(R^+_N)$for class of Mellin integral operators. Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematische e Naturali, Rendiconti Lincei Matematica e Applicazioni 2014; 25 (3): 217-232.
[8] Angeloni L, Vinti G. Discrete operators of sampling type and approximation in φ-variation. Mathematische Nachrichten 2018; 291 (4): 546-555.
[9] Aslan I. Convergence in Phi-variation and rate of approximation for nonlinear integral operators using summability process. Mediterranean Journal of Mathematics 2021; 18 (1): paper no. 5.
[10] Aslan I. Approximation by sampling-type nonlinear discrete operators in φ-variation. Filomat 2021; 35 (8): 2731- 2746.
[11] Aslan I, Duman O. Nonlinear approximation in N-dimension with the help of summability methods. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 2021; 115 (3): paper no. 105.
[12] Aslan I, Duman O. Approximation by nonlinear integral operators via summability process. Mathematische Nachrichten 2020; 293 (3): 430-448.
[13] Aslan I, Duman O. Characterization of absolute and uniform continuity. Hacettepe Journal of Mathematics and Statistics 2020; 49 (5): 1550-1565.
[14] Bardaro C. Alcuni teoremi di convergenza per l’integrale multiplo del Calcolo delle Variazioni (in Italian). Atti del Seminario Matematico e Fisico dell’Universita di Modena e Reggio Emilia 1982; 31: 302-324.
[15] Bardaro C, Sciamannini S, Vinti G. Convergence in BV φ by nonlinear Mellin-type convolution operators. Functiones et Approximatio Commentarii Mathematici 2001; 29: 17-28.
[16] Bell HT. A-summability. Thesis (PhD.)–Lehigh University 1971; 82 pp.
[17] Bell HT. Order summability and almost convergence. Proceedings of the American Mathematical Society 1973; 38: 548-552.
[18] Boni M. Sull’approssimazione dell’integrale multiplo del Calcolo delle Variazioni (in Italian). Atti del Seminario Matematico e Fisico dell’Universita di Modena e Reggio Emilia 1971; 20 (1): 187-211.
[19] Braha NL, Mansour T, Mursaleen M, Acar T. Convergence of λ Î»-Bernstein operators via power series summability method. Journal of Applied Mathematics and Computing 2021; 65 (1): 125-146.
[20] Demirci K, Karakuş S. Statistical A-summability of positive linear operators. Mathematical and Computer Modelling 2011; 53 (1-2): 189-195.
[21] Goffman C, Serrin J. Sublinear functions of measures and variational integrals. Duke Mathematical Journal 1964; 31 (1): 159-178.
[22] Gökçer TY, Duman O. Summation process by max-product operators. In: Anastassiou G., Duman O. (eds) Computational Analysis. Springer Proceedings in Mathematics & Statistics 2016; 155: 59-67.
[23] Gökçer TY, Duman O. Regular summability methods in the approximation by max-min operators. Fuzzy Sets Systems 2022; 426: 106-120.
[24] Hardy GH. Divergent series. Oxford University Press, London, 1949.
[25] Jordan C. Sur la serie de Fourier. Comptes Rendus de l’Académie des Sciences 1881; 92: 228-230.
[26] Lorentz GG. A contribution to the theory of divergent sequences. Acta Mathematica 1948; 80: 167-190.
[27] Love ER, Young LC. Sur une classe de fonctionnelles linéaires. Fundamenta Mathematicae 1937; 28 (1): 243-257 (in French).
[28] Serrin J. On the differentiability of functions of several variables. Archive for Rational Mechanics and Analysis 1961; 7: 359-372.
[29] Musielak J, Orlicz W. On generalized variations (I). Studia Mathematica 1959; 18: 11-41.
[30] Söylemez D, Ünver M. Korovkin type theorems for Cheney-Sharma operators via summability methods. Results in Mathematics 2017; 72 (3): 1601-1612.
[31] Tonelli L. Su alcuni concetti dell’analisi moderna (in Italian). Annali della Scuola Normale Superiore di Pisa-Classe di Scienze 1942; 11 (1-2): 107-118.
[32] Turkun C, Duman O. Modified neural network operators and their convergence properties with summability methods. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 2020; 114: paper no. 132.
[33] Wiener N. The quadratic variation of a function and its Fourier coefficients. Journal of Mathematics and Physics 1924; 3 (2): 72-94.
[34] Young LC. An inequality of the Hölder type, connected with Stieltjes integration. Acta Mathematica 1936; 67: 251-282.
[35] Young LC. Sur une généralisation de la notion de variation de puissance $p^{ieme}$ bornée au sens de M. Wiener, et sur la convergence des séries de Fourier. Comptes rendus de l’Académie des Sciences 1937; 204: 470-472 (in French).
[36] Yurdakadim T, Ünver M. Some results concerning the summability of spliced sequences. Turkish Journal of Mathematics 2016; 40 (5): 1134-1143.

APA	ASLAN I (2022). Multivariate approximation in φ-variation for nonlinear integral operators via summability methods. , 277 - 298. 10.3906/mat-2109-77
Chicago	ASLAN ISMAIL Multivariate approximation in φ-variation for nonlinear integral operators via summability methods. (2022): 277 - 298. 10.3906/mat-2109-77
MLA	ASLAN ISMAIL Multivariate approximation in φ-variation for nonlinear integral operators via summability methods. , 2022, ss.277 - 298. 10.3906/mat-2109-77
AMA	ASLAN I Multivariate approximation in φ-variation for nonlinear integral operators via summability methods. . 2022; 277 - 298. 10.3906/mat-2109-77
Vancouver	ASLAN I Multivariate approximation in φ-variation for nonlinear integral operators via summability methods. . 2022; 277 - 298. 10.3906/mat-2109-77
IEEE	ASLAN I "Multivariate approximation in φ-variation for nonlinear integral operators via summability methods." , ss.277 - 298, 2022. 10.3906/mat-2109-77
ISNAD	ASLAN, ISMAIL. "Multivariate approximation in φ-variation for nonlinear integral operators via summability methods". (2022), 277-298. https://doi.org/10.3906/mat-2109-77

APA	ASLAN I (2022). Multivariate approximation in φ-variation for nonlinear integral operators via summability methods. Turkish Journal of Mathematics, 46(1), 277 - 298. 10.3906/mat-2109-77
Chicago	ASLAN ISMAIL Multivariate approximation in φ-variation for nonlinear integral operators via summability methods. Turkish Journal of Mathematics 46, no.1 (2022): 277 - 298. 10.3906/mat-2109-77
MLA	ASLAN ISMAIL Multivariate approximation in φ-variation for nonlinear integral operators via summability methods. Turkish Journal of Mathematics, vol.46, no.1, 2022, ss.277 - 298. 10.3906/mat-2109-77
AMA	ASLAN I Multivariate approximation in φ-variation for nonlinear integral operators via summability methods. Turkish Journal of Mathematics. 2022; 46(1): 277 - 298. 10.3906/mat-2109-77
Vancouver	ASLAN I Multivariate approximation in φ-variation for nonlinear integral operators via summability methods. Turkish Journal of Mathematics. 2022; 46(1): 277 - 298. 10.3906/mat-2109-77
IEEE	ASLAN I "Multivariate approximation in φ-variation for nonlinear integral operators via summability methods." Turkish Journal of Mathematics, 46, ss.277 - 298, 2022. 10.3906/mat-2109-77
ISNAD	ASLAN, ISMAIL. "Multivariate approximation in φ-variation for nonlinear integral operators via summability methods". Turkish Journal of Mathematics 46/1 (2022), 277-298. https://doi.org/10.3906/mat-2109-77