Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces
Bollettino della Unione matematica italiana, Série 9, Tome 4 (2011) no. 3, pp. 445-468.

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In this paper we study a linear version of the sampling Kantorovich type operators in a multivariate setting and we show applications to Image Processing. By means of the above operators, we are able to reconstruct continuous and uniformly continuous signals/images (functions). Moreover, we study the modular convergence of these operators in the setting of Orlicz spaces $L^\varphi(\mathbb{R}^n)$ that allows us to deal the case of not necessarily continuous signals/images. The convergence theorems in $L^p(\mathbb{R}^n)$- spaces, $L^\alpha\log^\beta L(\mathbb{R}^n)$-spaces and exponential spaces follow as particular cases. Several graphical representations, for the various examples and Image Processing applications are included.
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Costarelli, Danilo; Vinti, Gianluca. Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces. Bollettino della Unione matematica italiana, Série 9, Tome 4 (2011) no. 3, pp. 445-468. http://geodesic.mathdoc.fr/item/BUMI_2011_9_4_3_a7/

[1] C. Bardaro - P. L. Butzer - R. L. Stens - G. Vinti, Kantorovich-Type Generalized Sampling Series in the Setting of Orlicz Spaces, Sampling Theory in Signal and Image Processing, 6, No. 1 (2007), 29-52. | MR | Zbl

[2] C. Bardaro - I. Mantellini, Modular Approximation by Sequences of Nonlinear Integral Operators in Musielak-Orlicz Spaces, Atti Sem. Mat. Fis. Univ. Modena, special issue dedicated to Professor Calogero Vinti, suppl., vol. 46, (1998), 403-425. | MR

[3] C. Bardaro - J. Musielak - G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, New York, Berlin, 9, 2003. | DOI | MR

[4] C. Bardaro - G. Vinti, Modular convergence in generalized Orlicz spaces for moment type operators, Applicable Analysis, 32 (1989), 265-276. | DOI | MR | Zbl

[5] C. Bardaro - G. Vinti, A general approach to the convergence theorems of generalized sampling series, Applicable Analysis, 64 (1997), 203-217. | DOI | MR | Zbl

[6] C. Bardaro - G. Vinti, An Abstract Approach to Sampling Type Operators Inspired by the Work of P. L. Butzer - Part I - Linear Operators, Sampling Theory in Signal and Image Processing, 2 (3) (2003), 271-296. | MR | Zbl

[7] L. Bezuglaya - V. Katsnelson, The sampling theorem for functions with limited multi-band spectrum I, Zeitschrift für Analysis und ihre Anwendungen, 12 (1993), 511-534. | DOI | MR | Zbl

[8] P. L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition, 3 (1983), 185-212. | MR | Zbl

[9] P. L. Butzer - W. Engels - S. Ries - R. L. Stens, The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines, SIAM J. Appl. Math., 46 (1986), 299-323. | DOI | MR | Zbl

[10] P. L. Butzer - A. Fisher - R. L. Stens, Generalized sampling approximation of multivariate signals: theory and applications, Note di Matematica, 10, Suppl. n. 1 (1990), 173-191. | MR | Zbl

[11] P. L. Butzer - G. Hinsen, Reconstruction of bounded signal from pseudo-periodic, irregularly spaced samples, Signal Processing, 17 (1989), 1-17. | DOI | MR

[12] P. L. Butzer - R. J. Nessel, Fourier Analysis and Approximation, I, Academic Press, New York-London, 1971. | MR | Zbl

[13] P. L. Butzer - S. Ries - R. L. Stens, Shannon's sampling theorem, Cauchy's integral formula, and related results, In: Anniversary Volume on Approximation Theory and Functional Analysis, (Proc. Conf., Math. Res. Inst. Oberwolfach, Black Forest, July 30-August 6, 1983), P. L. Butzer, R. L. Stens and B. Sz.-Nagy (Eds.), Internat. Schriftenreihe Numer. Math., 65, Birkhäuser, Basel, 1984, 363-377. | MR

[14] P. L. Butzer - S. Ries - R. L. Stens, Approximation of continuous and discountinuous functions by generalized sampling series, J. Approx. Theory, 50 (1987), 25-39. | DOI | MR | Zbl

[15] P. L. Butzer - W. Splettstoßer - R. L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein, 90 (1988), 1-70. | MR | Zbl

[16] P. L. Butzer - R. L. Stens, Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Review, 34 (1) (1992), 40-53. | DOI | MR | Zbl

[17] P. L. Butzer - R. L. Stens, Linear prediction by samples from the past, Advanced Topics in Shannon Sampling and Interpolation Theory, (editor R. J. Marks II), Springer-Verlag, New York, 1993. | MR

[18] M. M. Dodson - A. M. Silva, Fourier Analysis and the Sampling Theorem, Proc. Ir. Acad., 86, A (1985), 81-108. | MR | Zbl

[19] G. B. Folland, Real Analysis: Modern techniques and their applications, Wiley and Sons, 1984. | MR | Zbl

[20] J. R. Higgins, Five short stories about the cardinal series, Bull. Amer. Math. Soc., 12 (1985), 45-89. | DOI | MR | Zbl

[21] J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford Univ. Press, Oxford, 1996. | Zbl

[22] J. R. HIGGINS - R. L. STENS (Eds.), Sampling Theory in Fourier and Signal Analysis: advanced topics, Oxford Science Publications, Oxford Univ. Press, Oxford, 1999.

[23] A. J. Jerry, The Shannon sampling-its various extensions and applications: a tutorial review, Proc. IEEE, 65 (1977), 1565-1596.

[24] W. M. Kozlowski, Modular Function Spaces, (Pure Appl. Math.) Marcel Dekker, New York and Basel, 1988. | MR | Zbl

[25] M. A. Krasnosel'Skǐi - Ya. B. Rutickǐi, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd. - Groningen - The Netherlands, 1961. | MR

[26] L. Maligranda, Orlicz Spaces and Interpolation, Seminarios de Matematica, IMECC, Campinas, 1989. | MR | Zbl

[27] I. Mantellini - G. Vinti, Approximation results for nonlinear integral operators in modular spaces and applications, Ann. Polon. Math., 81 (1) (2003), 55-71. | fulltext EuDML | DOI | MR | Zbl

[28] J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Lecture Notes in Math., 1034, 1983. | DOI | MR | Zbl

[29] J. Musielak - W. Orlicz, On modular spaces, Studia Math., 28 (1959), 49-65. | fulltext EuDML | DOI | MR | Zbl

[30] M. M. Rao - Z. D. Ren, Theory of Orlicz Spaces, Pure and Appl. Math., Marcel Dekker Inc. New York-Basel-Hong Kong, 1991. | MR

[31] M. M. Rao - Z. D. Ren, Applications of Orlicz Spaces, Monographs and Textbooks in Pure and applied Mathematics, vol. 250, Marcel Dekker Inc., New York, 2002. | DOI | MR | Zbl

[32] S. Ries - R. L. Stens, Approximation by generalized sampling series, Constructive Theory of Functions '84, Sofia (1984), 746-756.

[33] C. E. Shannon, Communication in the presence of noise, Proc. I.R.E., 37 (1949), 10-21. | MR

[34] C. Vinti, A Survey on Recent Results of the Mathematical Seminar in Perugia, inspired by the Work of Professor P. L. Butzer, Result. Math., 34 (1998), 32-55. | DOI | MR | Zbl

[35] G. Vinti, Approximation in Orlicz spaces for linear integral operators and applications, Rendiconti del Circolo Matematico di Palermo, Serie II, N. 76 (2005), 103-127. | MR | Zbl

[36] G. Vinti - L. Zampogni, A Unifying Approach to Convergence of Linear Sampling Type Operators in Orlicz Spaces, Advances in Differential Equations, Vol. 16, Numbers 5-6 (2011), 573-600. | MR | Zbl