Order of approximation for nonlinear sampling Kantorovich operators in Orlicz spaces

• Autorzy: Costarelli, D. , Vinti, G.
• Języki publikacji: EN

Abstrakty

EN

In this paper, we study the rate of approximation for the nonlinear sampling Kantorovich operators. We consider the case of uniformly continuous and bounded functions belonging to Lipschitz classes of the Zygmund-type, as well as the case of functions in Orlicz spaces. We estimate the aliasing errors with respect to the uniform norm and to the modular functional of the Orlicz spaces, respectively. The general setting of Orlicz spaces allows to deduce directly the results concerning the rate of convergence in L^p-spaces, 1 ≤ p < ∞, very useful in the applications to Signal Processing. Others examples of Orlicz spaces as interpolation spaces and exponential spaces are discussed and the particular cases of the nonlinear sampling Kantorovich series constructed using Fejér and B-spline kernels are also considered.

Słowa kluczowe

EN

Nonlinear sampling Kantorovich operators; Orlicz spaces; order of approximation; Lipschitz classes; irregular sampling

• Czasopismo: Commentationes Mathematicae
• Rocznik: 2013
• Tom: Vol. 53, No. 2
• Strony: 171--192
• Opis fizyczny: Bibliogr. 58 poz.

Twórcy

autor

Costarelli, D.

• Department of Mathematics and Physics, Section of Mathematics, University of Roma, Tre, Largo S. Leonardo Murialdo, 1, 00146 Rome, Italy,

autor

Vinti, G.

• Department of Mathematics and Computer Sciences, University of Perugia, Via Vanvitelli, 1, 06123 Perugia, Italy,

Bibliografia

• [1] L. Angeloni, G. Vinti, Rate of approximation for nonlinear integral operators with applications to signal processing, Differential and Integral Equations 18 (8) (2005), 855-890.
• [2] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Approximation of the Whittaker Sampling Series in terms of an Average Modulus of Smoothness covering Discontinuous Signals, J. Math. Anal. Appl. 316 (2006), 269-306.
• [3] 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 (1) (2007), 29-52.
• [4] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Prediction by samples from the past with error estimates covering discontinuous signals, IEEE, Transaction on Information Theory 56 (1) (2010), 614-633.
• [5] C. Bardaro, I. Mantellini, On convergence properties for a class of Kantorovich discrete operators, Numer. Funct. Anal. Optim. 33 (4) (2012), 374-396.
• [6] C. Bardaro, J. Musielak, G.Vinti, Modular estimates and modular convergence for a class of nonlinear operators, Math. Japon. 39 (1994), 7-14.
• [7] C. Bardaro, J. Musielak, G.Vinti, On absolute continuity of a modular connected with strong summability, Comment. Math. Prace Mat. 34 (1994), 21-33.
• [8] C. Bardaro, J. Musielak, G.Vinti, Approximation by nonlinear integral operators in some modular function spaces, Ann. Polon. Math. 63 (2) (1996), 173-182.
• [9] C. Bardaro, J. Musielak, G. Vinti, Nonlinear operators of integral type in some function spaces. Fourth International Conference on Function Spaces (Zielona Góra, 1995) Collect. Math. 48 (1997), 409-422.
• [10] C. Bardaro, J. Musielak, G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, New York, Berlin, 9, 2003.
• [11] C. Bardaro, G. Vinti, Some Inclusion Theorems for Orlicz and Musielak-Orlicz Type Spaces, Annali di Matematica Pura e Applicata 168 (1995), 189-203.
• [12] C. Bardaro, G. Vinti, Modular approximation by nonlinear integral operators on locally compact groups, Comment. Math. Prace Mat. 35 (1995), 25-47.
• [13] C. Bardaro, G. Vinti, A general approach to the convergence theorems of generalized sampling series, Applicable Analysis 64 (1997), 203-217.
• [14] C. Bardaro, G. Vinti, Uniform convergence and rate of approximation for a nonlinear version of the generalized sampling operator, Results in Mathematics, special volume dedicated to Professor P.L. Butzer, 34 (3/4) (1998), 224-240.
• [15] C. Bardaro, G.Vinti, On the order of modular approximation for nets of integral operators in modular Lipschitz classes, Funct. Approx. Comment. Math., special issue dedicated to Prof. Julian Musielak, 26 (1998), 139-154.
• [16] C. Bardaro, G. Vinti, Nonlinear sampling type operators: approximation properties and regular methods of summability, Nonlinear Anal. Forum 6 (1) (2001), 15-26.
• [17] C. Bardaro, G. Vinti, An Abstract Approach to Sampling Type Operators Inspired by the Work of P.L. Butzer - Part II - Nonlinear Operators, Sampling Theory in Signal and Image Processing 3 (1) (2004), 29-44.
• [18] 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.
• [19] P.L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition 3 (1983), 185-212.
• [20] 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.
• [21] P.L. Butzer, G. Hinsen, Reconstruction of bounded signal from pseudo-periodic, irregularly spaced samples, Signal Processing 17 (1989), 1-17.
• [22] P.L. Butzer, R.J. Nessel, Fourier Analysis and Approximation I, Academic Press, New York-London, 1971.
• [23] 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.
• [24] 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.
• [25] P.L. Butzer, W. Splettstößer, R.L. Stens, The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein 90 (1988), 1-70.
• [26] P.L. Butzer, R.L. Stens, Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Review 34 (1) (1992), 40-53.
• [27] 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.
• [28] F. Cluni, D. Costarelli, A.M. Minotti, G. Vinti, Multivariate sampling Kantorovich operators: approximation and applications to civil engineering, to appear in: EURASIP. Proceedings of SampTA 2013, 10th International Conference on Sampling Theory and Applications, July 1st - July 5th, 2013, Jacobs University, Bremen.
• [29] D. Costarelli, G. Vinti, Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces, Bollettino U.M.I., Special volume dedicated to Prof. Giovanni Prodi 9 (IV) (2011), 445-468.
• [30] D. Costarelli, G. Vinti, Approximation by Nonlinear Multivariate Sampling Kantorovich Type Operators and Applications to Image Processing, Numerical Functional Analysis and Optimization 34 (8) (2013), 819-844.
• [31] D. Costarelli, G. Vinti, Order of approximation for sampling Kantorovich operators, submitted, (2013).
• [32] M.M. Dodson, A.M. Silva, Fourier Analysis and the Sampling Theorem, Proc. Ir. Acad. 86 A (1985), 81-108.
• [33] C. Donnini, G. Vinti, Approximation by Means of Kantorovich Generalized Sampling Operators in Musielak-Orlicz spaces, PanAmerican Mathematical Journal 18 (2) (2008), 1-18.
• [34] J.R. Higgins, Five short stories about the cardinal series, Bull. Amer. Math. Soc. 12 (1985), 45-89.
• [35] J.R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford Univ. Press, Oxford, 1996.
• [36] J.R. Higgins, R.L. Stens, Sampling Theory in Fourier and Signal Analysis: advanced topics, Oxford Science Publications, Oxford Univ. Press, Oxford, 1999.
• [37] A.J. Jerry, The Shannon sampling-its various extensions and applications: a tutorial review, Proc. IEEE 65 (1977), 1565-1596.
• [38] W.M. Kozlowski, Modular Function Spaces, (Pure Appl. Math.) Marcel Dekker, New York and Basel, 1988.
• [39] M.A. Krasnosel’skiˇi, Ya.B. Rutickiˇi, Convex Functions and Orlicz Spaces, P. Noordhoff Ltd. - Groningen - The Netherlands, 1961.
• [40] L. Maligranda, Orlicz Spaces and Interpolation, Seminarios de Matematica, IMECC, Campinas, 1989.
• [41] I. Mantellini, G. Vinti, Approximation results for nonlinear integral operators in modular spaces and applications, Ann. Polon. Math. 81 (1) (2003), 55-71.
• [42] J. Musielak, On some approximation problems in modular spaces, in Constructive Function Theory, Proc. Int.Conf. Varna, June 1-5, 1981, pp. 455-461, Publ. House Bulgarian Acad. Sci., Sofia 1983.
• [43] J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Lecture Notes in Math. (1034), 1983.
• [44] J. Musielak, Approximation by nonlinear singular integral operators in generalized Orlicz spaces, Comment. Math. Prace Mat. 31 (1991), 79-88.
• [45] J. Musielak, Nonlinear approximation in some modular function spaces I, Math. Japon. 38 (1993), 83-90.
• [46] J. Musielak, On the approximation by nonlinear integral operators with generalized Lipschitz kernel over a compact abelian group, Comment. Math. Prace Mat. 33 (1993), 99-104.
• [47] J. Musielak, On nonlinear integral operators, Atti Sem. Mat. Fis. Univ. Modena 47 (1999), 183-188.
• [48] J. Musielak, Approximation by a nonlinear convolution operator in modular function spaces, Rocznik Nauk.-Dydakt. Prace Mat. 17 (2000), 181-190.
• [49] J. Musielak, W. Orlicz, On modular spaces, Studia Math. 28 (1959), 49-65.
• [50] M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces, Pure and Appl. Math., 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, vol. 250, Marcel Dekker Inc., New York, 2002.
• [52] S. Ries, R.L. Stens, Approximation by generalized sampling series, Constructive Theory of Functions’84, Sofia, 1984, 746-756.
• [53] C.E. Shannon, Communication in the presence of noise, Proc. I.R.E. 37 (1949), 10-21.
• [54] 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.
• [55] G. Vinti, A general approximation result for nonlinear integral operators and applications to signal processing, Applicable Analysis 79 (2001), 217-238.
• [56] 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.
• [57] G. Vinti, L. Zampogni, Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces, Journal of Approximation Theory 161 (2009), 511-528.
• [58] G. Vinti, L. Zampogni, A Unifying Approach to Convergence of Linear Sampling Type Operators in Orlicz Spaces, Advances in Differential Equations 16 (5-6) (2011), 573-600.

Identyfikatory

DOI

10.14708/cm.v53i2.792