Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces

Danilo Costarelli; Gianluca Vinti

Bollettino dell'Unione Matematica Italiana (2011)

  • Volume: 4, Issue: 3, page 445-468
  • ISSN: 0392-4041

Abstract

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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 φ ⁢ ( ℝ n ) that allows us to deal the case of not necessarily continuous signals/images. The convergence theorems in L p ⁢ ( ℝ n ) - spaces, L α ⁢ log β ⁡ L ⁢ ( ℝ n ) -spaces and exponential spaces follow as particular cases. Several graphical representations, for the various examples and Image Processing applications are included.

How to cite

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Costarelli, Danilo, and Vinti, Gianluca. "Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces." Bollettino dell'Unione Matematica Italiana 4.3 (2011): 445-468. <http://eudml.org/doc/290713>.

@article{Costarelli2011,
abstract = {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.},
author = {Costarelli, Danilo, Vinti, Gianluca},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {445-468},
publisher = {Unione Matematica Italiana},
title = {Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces},
url = {http://eudml.org/doc/290713},
volume = {4},
year = {2011},
}

TY - JOUR
AU - Costarelli, Danilo
AU - Vinti, Gianluca
TI - Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2011/10//
PB - Unione Matematica Italiana
VL - 4
IS - 3
SP - 445
EP - 468
AB - 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.
LA - eng
UR - http://eudml.org/doc/290713
ER -

References

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  1. BARDARO, C. - BUTZER, P. L. - STENS, R. L. - VINTI, G., Kantorovich-Type Generalized Sampling Series in the Setting of Orlicz Spaces, Sampling Theory in Signal and Image Processing, 6, No. 1 (2007), 29-52. Zbl1156.41307MR2296881
  2. BARDARO, C. - MANTELLINI, I., 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. MR1645731
  3. BARDARO, C. - MUSIELAK, J. - VINTI, G., Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, New York, Berlin, 9, 2003. MR1994699DOI10.1515/9783110199277
  4. BARDARO, C. - VINTI, G., Modular convergence in generalized Orlicz spaces for moment type operators, Applicable Analysis, 32 (1989), 265-276. Zbl0668.42009MR1030099DOI10.1080/00036818908839853
  5. BARDARO, C. - VINTI, G., A general approach to the convergence theorems of generalized sampling series, Applicable Analysis, 64 (1997), 203-217. Zbl0878.47016MR1460079DOI10.1080/00036819708840531
  6. BARDARO, C. - VINTI, G., 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. Zbl1137.41334MR2090110
  7. BEZUGLAYA, L. - KATSNELSON, V., The sampling theorem for functions with limited multi-band spectrum I, Zeitschrift für Analysis und ihre Anwendungen, 12 (1993), 511-534. Zbl0786.30019MR1245936DOI10.4171/ZAA/550
  8. BUTZER, P. L., A survey of the Whittaker-Shannon sampling theorem and some of its extensions, J. Math. Res. Exposition, 3 (1983), 185-212. Zbl0523.94003MR724869
  9. BUTZER, P. L. - ENGELS, W. - RIES, S. - STENS, R. L., 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. Zbl0617.41020MR833479DOI10.1137/0146020
  10. BUTZER, P. L. - FISHER, A. - STENS, R. L., Generalized sampling approximation of multivariate signals: theory and applications, Note di Matematica, 10, Suppl. n. 1 (1990), 173-191. Zbl0768.42013MR1193522
  11. BUTZER, P. L. - HINSEN, G., Reconstruction of bounded signal from pseudo-periodic, irregularly spaced samples, Signal Processing, 17 (1989), 1-17. MR995998DOI10.1016/0165-1684(89)90068-6
  12. BUTZER, P. L. - NESSEL, R. J., Fourier Analysis and Approximation, I, Academic Press, New York-London, 1971. Zbl0217.42603MR510857
  13. BUTZER, P. L. - RIES, S. - STENS, R. L., 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. MR820537
  14. BUTZER, P. L. - RIES, S. - STENS, R. L., Approximation of continuous and discountinuous functions by generalized sampling series, J. Approx. Theory, 50 (1987), 25-39. Zbl0654.41004MR888050DOI10.1016/0021-9045(87)90063-3
  15. BUTZER, P. L. - SPLETTSTOßER, W. - STENS, R. L., The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.-Verein, 90 (1988), 1-70. Zbl0633.94002MR928745
  16. BUTZER, P. L. - STENS, R. L., Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Review, 34 (1) (1992), 40-53. Zbl0746.94002MR1156288DOI10.1137/1034002
  17. BUTZER, P. L. - STENS, R. L., 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. MR1221748
  18. DODSON, M. M. - SILVA, A. M., Fourier Analysis and the Sampling Theorem, Proc. Ir. Acad., 86, A (1985), 81-108. Zbl0583.42003MR821425
  19. FOLLAND, G. B., Real Analysis: Modern techniques and their applications, Wiley and Sons, 1984. Zbl0549.28001MR767633
  20. HIGGINS, J. R., Five short stories about the cardinal series, Bull. Amer. Math. Soc., 12 (1985), 45-89. Zbl0562.42002MR766960DOI10.1090/S0273-0979-1985-15293-0
  21. HIGGINS, J. R., Sampling Theory in Fourier and Signal Analysis: Foundations, Oxford Univ. Press, Oxford, 1996. Zbl0872.94010
  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. JERRY, A. J., The Shannon sampling-its various extensions and applications: a tutorial review, Proc. IEEE, 65 (1977), 1565-1596. 
  24. KOZLOWSKI, W. M., Modular Function Spaces, (Pure Appl. Math.) Marcel Dekker, New York and Basel, 1988. Zbl0661.46023MR1474499
  25. KRASNOSEL'SKǏI, M. A. - RUTICKǏI, YA. B., Convex Functions and Orlicz Spaces, P. Noordhoff Ltd. - Groningen - The Netherlands, 1961. MR126722
  26. MALIGRANDA, L., Orlicz Spaces and Interpolation, Seminarios de Matematica, IMECC, Campinas, 1989. Zbl0874.46022MR2264389
  27. MANTELLINI, I. - VINTI, G., Approximation results for nonlinear integral operators in modular spaces and applications, Ann. Polon. Math., 81 (1) (2003), 55-71. Zbl1019.41013MR1977761DOI10.4064/ap81-1-5
  28. MUSIELAK, J., Orlicz Spaces and Modular Spaces, Springer-Verlag, Lecture Notes in Math., 1034, 1983. Zbl0557.46020MR724434DOI10.1007/BFb0072210
  29. MUSIELAK, J. - ORLICZ, W., On modular spaces, Studia Math., 28 (1959), 49-65. Zbl0086.08901MR101487DOI10.4064/sm-18-1-49-65
  30. RAO, M. M. - REN, Z. D., Theory of Orlicz Spaces, Pure and Appl. Math., Marcel Dekker Inc. New York-Basel-Hong Kong, 1991. MR1113700
  31. RAO, M. M. - REN, Z. D., Applications of Orlicz Spaces, Monographs and Textbooks in Pure and applied Mathematics, vol. 250, Marcel Dekker Inc., New York, 2002. Zbl0997.46027MR1890178DOI10.1201/9780203910863
  32. RIES, S. - STENS, R. L., Approximation by generalized sampling series, Constructive Theory of Functions '84, Sofia (1984), 746-756. 
  33. SHANNON, C. E., Communication in the presence of noise, Proc. I.R.E., 37 (1949), 10-21. MR28549
  34. VINTI, C., 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. Zbl1010.01021MR1635582DOI10.1007/BF03322036
  35. VINTI, G., Approximation in Orlicz spaces for linear integral operators and applications, Rendiconti del Circolo Matematico di Palermo, Serie II, N. 76 (2005), 103-127. Zbl1136.41306MR2175550
  36. VINTI, G. - ZAMPOGNI, L., 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. Zbl1223.41014MR2816117

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