Semi-discrete sampling operators acting on function spaces
DOI:
https://doi.org/10.64700/altay.25Keywords:
Orlicz spaces, \(L^p\)-approximation, Modular convergence, Durrmeyer sampling operators, Lipschitz classes, modulus of continuityAbstract
In this paper, we present an overview of recent advances in the study of the approximation properties of a family of semi-discrete sampling operators in several function spaces. We investigate their convergence properties in the space of continuous functions, providing an approximation result in the uniform norm and both quantitative and qualitative estimates. Here, we also establish a regularization result for functions in $L^p$-spaces. We then extend the study to the broader framework of Orlicz spaces, which allows the treatment of functions that are not necessarily continuous, as is often the case for real-world signals. In this setting, besides convergence, we also study the rate of approximation in terms of the $\varphi$-modulus of continuity defined by the modular functional. This unified approach yields approximation results in several particular cases, including Zygmund spaces, exponential-type spaces, and $L^p$-spaces. In the last setting, we are also able to achieve a sharper rate of convergence.
References
[1] U. Abel, O. Agratini and R. P˘alt˘anea: Szász-Mirakjan-Durrmeyer operators defined by multiple Appell polynomials, Positivity, 29 (2025), Article ID: 17.
[2] T. Acar, A. Aral, S. Kursun: Approximation Properties of Modified Durrmeyer Forms of Exponential Sampling Series, Results Math., 80 (2025), Article ID: 187.
[3] T. Acar, D. Costarelli, G. Vinti: Linear prediction and simultaneous approximation by m-th order Kantorovich type sampling series, Banach J. Math. Anal., 14 (4) (2020), 1481–1508.
[4] G. Aiello, L. Angeloni, and G. Vinti: Durrmeyer sampling-type operators: approximation in variation, submitted, 2025.
[5] O. Alagoz, M. Turgay, T. Acar and M. Parlak: Approximation by Sampling Durrmeyer Operators in Weighted Space of Functions, Numer. Funct. Anal. Optim., 43 (10) (2022), 1223–1239.
[6] F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Ra¸sa: A generalization of Kantorovich operators for convex compact subsets, Banach J. Math. Anal., 11 (3) (2017), 591–614.
[7] L. Angeloni, D. Costarelli and G. Vinti: A characterization of the convergence in variation for the generalized sampling series, Ann. Acad. Sci. Fenn., 43 (2018), 755–767.
[8] S. Bajpeyi, A. S. Kumar and I. Mantellini: Approximation by Durrmeyer Type Exponential Sampling Operators, Numer. Funct. Anal. Optim., 43 (1) (2022), 16–34.
[9] C. Bardaro, G. Vinti, P. L. Butzer and R. L. Stens: Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampling Theory in Signal and Image Processing, 6 (1) (2007), 29–52.
[10] C. Bardaro, L. Faina and I. Mantellini: Quantitative Voronovskaja formulae for generalized Durrmeyer sampling type series, Math. Nachr., 289 (2016), 1702–1720.
[11] C. Bardaro, I. Mantellini: Asymptotic expansion of generalized Durrmeyer sampling type series, J. J. Approx., 6 (2) (2014), 143–165.
[12] C. Bardaro, I. Mantellini: On pointwise approximation properties of multivariate semi-discrete sampling type operators, Results Math., 72 (2017), 1449–1472.
[13] C. Bardaro, J. Musielak and G. Vinti: Nonlinear integral operators and applications, de Gruyter Series in Nonlinear Analysis and Applications, 9,Walter de Gruyter & Co., Berlin (2003).
[14] P. L. Butzer, A. Fisher and R. L. Stens: Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx. Theory, 50 (1987), 25–39.
[15] P. L. Butzer, A. Fisher and R. L. Stens: Generalized sampling approximation of multivariate signals, Atti Sem. Mat. Fis. Univ. Modena, 41 (1993), 17–37.
[16] P. L. Butzer, J. R. Higgins and R. L. Stens: Sampling theory of signal analysis, in Development of Mathematics 1950–2000, Birkhauser, Basel, (2000), 193–234.
[17] P.L. Butzer, R.J. Nessel: Fourier Analysis and Approximation I, Academic Press, New York (1971).
[18] Q. B. Cai, E. Kangal, Ü. Dinlemez Kantar: On the Convergence Properties of Durrmeyer Type Exponential Sampling Series in (Mellin) Orlicz Spaces, J. Math. Inequal., 18 (3) (2024), 1135–1152.
[19] Q. B. Cai, G. Zhou: Approximation Properties of (λ, μ)-Bernstein-Durrmeyer Operators, Math. Methods Appl. Sci., 48 (5) (2025), 5946–5953.
[20] M. Campiti, C. Tacelli: Perturbations of Bernstein-Durrmeyer operators on the simplex and best approximation properties, Commun. Appl. Anal., 13 (2009), 597–607.
[21] M. Cappelletti Montano, V. Leonessa: A generalization of Bernstein-Durrmeyer operators on hypercubes by means of an arbitrary measure, Stud. Univ. Babes-Bolyai Math., 64 (2) (2019), 239-252.
[22] D. Cardenas-Morales, P. Garrancho and I. Ra¸sa: Approximation properties of Bernstein-Durrmeyer type operators, Appl. Math. Comput., 232 (2014), 1–8.
[23] D. Costarelli, M. Piconi and G. Vinti: The multivariate Durrmeyer-sampling type operators in functional spaces, Dolomites Res. Notes Approx., 15 (5) (2022), 128–144.
[24] D. Costarelli, M. Piconi and G. Vinti: On the convergence properties of sampling-Durrmeyer-type operators in Orlicz spaces, Mathematische Nachrichten, 296 (2022), 588–609.
[25] D. Costarelli, M. Piconi and G. Vinti: Quantitative estimates for Durrmeyer-sampling series in Orlicz spaces, Sampl. Theory Signal Process. Data Anal., 21 (2022), Article ID: 3.
[26] D. Costarelli, M. Piconi, G. Vinti: On the Regularization by Durrmeyer-Sampling Type Operators in Lp-Spaces via a Distributional Approach, J. Fourier Anal. Appl., 31 (2025), Article ID: 11.
[27] D. Costarelli, M. Piconi and G. Vinti: A characterization of generalized Lipschitz classes by the rate of convergence of semi-discrete operators, submitted, 2025.
[28] D. Costarelli, G. Vinti: Order of approximation for sampling Kantorovich operators, J. Integral Equ. Appl., 26 (2014), 345–368.
[29] D. Costarelli, G. Vinti: An inverse result of approximation by sampling Kantorovich series, Proc. Edinb. Math. Soc., 62 (1) (2019), 265–280.
[30] D. Costarelli, G. Vinti: Inverse results of approximation and saturation order for the sampling Kantorovich series, J. Approx. Theory, 242 (2019), 64–82.
[31] D. Costarelli, G. Vinti: Approximation properties of the sampling Kantorovich operators: regularization, saturation, inverse results and Favard classes in Lp-spaces, J. Fourier Anal. Appl., 28 (2022), Article ID: 49.
[32] M. M. Derriennic: Sur l’approximation de fonctions intégrables sur [0, 1] par des polynômes de Bernstein modifiés, J. Approx. Theory, 31 (4) (1981), 325–343.
[33] J. L. Durrmeyer: Une formule d’inversion de la transformée de Laplace: applications á la théorie des moments, Thèse de 3ème cycle, Université de Paris (1967).
[34] T. Garg, A. M. Acu, P. N. Agrawal: Further results concerning some general Durrmeyer type operators, Rev. Real Acad. Cienc. Exactas Fis. Nat., Serie A. Matematicas, 113 (2019), 2373–2390.
[35] H. Gonska, X. Zhou: A global inverse theorem on simultaneous approximation by Bernstein-Durrmeyer operators, J. Approx. Theory, 67 (1991), 284–302.
[36] V. Gupta, G. S. Srivastava: Approximation by Durrmeyer-type operators, Ann. Polon. Math., 64 (2) (1996), 153–159.
[37] P. Harjulehto, P. Hästö: Orlicz Spaces and Generalized Orlicz Spaces, vol. 2236, Springer (2019).
[38] M. Heilmann, I. Raşa: A nice representation for a link between Baskakov- and Szász-Mirakjan-Durrmeyer operators and their Kantorovich variants, Results Math., 74 (2019), Article ID: 9.
[39] S. Hencl: A sharp form of an embedding into exponential and double exponential spaces, J. Funct. Anal., 204 (1) (2003), 196–227.
[40] H. Hudzik, J. Musielak, E. Tirbanski: Linear operators in modular spaces, Annales Societatis Mathematicae Polonae, Series I: Commentationes Mathematicae, Vol. XXIII (1983).
[41] E. Kangal, Ü. Dinlemez Kantar: Estimates for Durrmeyer-type Exponential Sampling Series in Mellin–Orlicz Spaces, Demonstr. Math., 58 (1) (2025), Article ID: 20250155.
[42] A. Kivinukk, G. Tamberg: On window methods in generalized Shannon sampling operators, In: Zayed, A., Schmeisser, G. (eds) New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham., (2014), 63–85.
[43] V. A. Kotel’nikov: On the carrying capacity of "ether" and wire in elettrocommunications, in Material for The First All-Union Conference on Questions of Communications, Izd. Red. Upr. Svyazi RKKA, Moscow, 1933 (in Russian).
[44] L. Maligranda: Orlicz Spaces and Interpolation, Seminarios de Matemática, vol. 5, Universidad Nacional del Litoral, Santa Fe (1989).
[45] J. Musielak: Orlicz spaces and Modular spaces, Lecture Notes in Math., 1034, Springer-Verlag Berlin (1983).
[46] W. Orlicz: Über eine gewisse Klasse von Räumen vom Typus B, Bull. Acad. Polon. Sci. Lett. Ser. A, (1932), 207–220.
[47] W. Orlicz: Über Räume LM, Bull. Acad. Polon. Sci. Lett. Ser. A, (1936), 93–107.
[48] O. Orlova, G. Tamberg: On approximation properties of generalized Kantorovich-type sampling operators, J. Approx. Theory, 201 (2016), 73–86.
[49] M. Piconi, G. Vinti: Semi-discrete Sampling in Sobolev-Orlicz Spaces, submitted, 2025.
[50] M. M. Rao, Z. D. Ren: Theory of Orlicz Spaces, Pure and Applied Mathematics, Marcel Dekker Inc., New York–Basel–Hong Kong (1991).
[51] M. M. Rao, Z. D. Ren: Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 250, Marcel Dekker Inc., New York (2002).
[52] L. Schwartz: Théorie des distributions, Hermann, Paris (1966).
[53] C. E. Shannon: Communication in the presence of noise, Proc. I.R.E., 37 (1949), 10–21.
[54] V. Sharma, V. Gupta: Convergence properties of Durrmeyer-type sampling operators, Comput. Appl. Math., 43 (2024), Article ID: 403.
[55] E. M. Stein: Note on the class Llog L, Studia Mathematica, 32 (1969), 305–310.
[56] A. Travaglini, G. Vinti: Nonlinear sampling Durrmeyer operators: approximation results in function spaces, submitted, 2025.
[57] G. Vinti: A general approximation result for nonlinear integral operators and applications to signal processing, Appl. Anal., 79 (1-2) (2001), 217–238.
[58] G. Vinti, L. Zampogni: A unifying approach to convergence of linear sampling type operators in Orlicz spaces, Adv. Differ. Equ., 16 (5-6) (2011), 573–600.
[59] E. T. Whittaker: On the functions which are represented by the expansion of the interpolation theory, Proc. Roy. Soc. Edinburgh, 35 (1915), 181–194.
[60] A. Zygmund: Trigonometric Series, Cambridge University Press, Cambridge (1959).
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