Kantorovich-type generalized sampling series in the setting of Orlicz spaces

Sampling Theory in Signal and Image Processing

Volumn 6, Issue 1, 2007, Pages 29-52

  Bardaro, C ^a   Vinti, G ^a   Butzer, P L ^b   Stens, R L ^b  

^a UNIVERSITY OF PERUGIA   (Italy)

^b RWTH AACHEN UNIVERSITY   (Germany)

Author keywords

Generalized sampling operators; Irregular sampling; Kantorovich type operators; L log L space; Modular convergence; Orlicz spaces

Indexed keywords

EID: 33846326538     PISSN: 15306429     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. Angeloni, L. and Vinti, G., A unified approach to approximation results with applications to nonlinear sampling theory, Int. J. Math. Sci., 3(1), 93-128, 2004.
2. Bardaro, C., Butzer, P. L., Stens, R. L., and Vinti, G., Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals, Analysis (Munich), 23(4), 299-340, 2003.
3. Bardaro, C., Butzer, P. L., Stens, R. L, and Vinti, G., Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals, J. Math. Anal. Appl., 316(1), 269-306, 2006.
4. Bardaro, C. and Mantellini, I., On global approximation properties of discrete operators in Orlicz spaces, J. Inequal. Pure Appl. Math. 6(4), Article 123 (electronic), 2005.
5. Bardaro, C., Musielak, J., and Vinti, G., Nonlinear Integral Operators and Applications, de Gruyter Series in Nonlinear Analysis and Applications, vol. 9, Walter de Gruyter & Co., Berlin, 2003.
6. Bennett, C. and Rudnick, K., On Lorentz-Zygmund Spaces, Dissertationes Math. (Rozprawy Mat.) 175, 1980.
7. Bernstein, S. N., Démonstration du théorème de Weierstraß, fondée sur le calcul des probabilités., Commun. Math. Charkow (2), 13, 1-2, 1912, in French.
8. Butzer, P. L. and Lei, J., Errors in truncated sampling series with measured sampled values for not-necessarily bandlimited functions, Funct. Approx. Comment. Math., 26, 25-39, 1998.
9. Butzer, P. L. and Lei, J., Approximation of signals using measured sampled values and error analysis, Commun. Appl. Anal., 4(2), 245-255, 2000.
10. Butzer, P. L., Mawhin, J., and Vetro, P., eds., Charles-Jean de La Vallée Poussin, Collected works/Oeuvres scientifiques. Vol. II. Integration and measure, probability, ordinary differential equations, mechanics, geometry, Académie Royale de Belgique, Brussels, and Circolo Matematico di Palermo, Palermo, 2001.
11. Butzer, P. L. and Nessel, R. J., Fourier Analysis and Approximation, Birkhäuser Verlag, Basel, and Academic Press, New York, 1971.
12. Butzer, P. L., Schmeisser, G., and Stens, R. L., An introduction to sampling analysis, In: Nonuniform Sampling, Theory and Practice (Marvasti, F., ed.), Information Technology: Transmission, Processing and Storage, 17-121, Kluwer Academic/Plenum Publishers, New York, 2001.
13. Butzer, P. L., Splettstößer, W., and Stens, R. L., The sampling theorem and linear prediction in signal analysis, Jahresber. Deutsch. Math.- Verein., 90, 1-70, 1988.
14. de La Vallée Poussin, Ch.-J., Sur l'intégrale de Lebesgue, Trans. Amer. Math. Soc., 16, 435-501, 1915, in French.
15. Edmunds, D. E. and Krbec, M., Two limiting cases of Sobolev imbeddings, Houston J. Math., 21(1), 119-128, 1995.
16. Fiorenza, A., Some remarks on Stein's L log L result, Differential Integral Equations, 5(6), 1355-1362, 1992.
17. Gröchenig, K., Reconstruction algorithms in irregular sampling, Math. Comp., 59(199), 181-194, 1992.
18. Hencl, S., A sharp form of an embedding into exponential and double exponential spaces, J. Funct. Anal., 204(1), 196-227, 2003.
19. Kantorovich, L. V., Sur certains développements suivant les polynomes de la forme de S. Bernstein. I, C. R. Acad. Sc. URSS, 563-568, 1930, in Russian.
20. Krasnosel'skiǐ, M. A., Topological Methods in the Theory of Nonlinear Integral Equations, The Macmillan Co., New York, 1964.
21. Krasnosel'skiǐ, M. A. and Rutickiǐ Ja. B., Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961.
22. Lindbeck, A., ed., Nobel Lectures in Economic Sciences 1969-1980, World Scientific Publishers, Singapore, 1992.
23. Lorentz, G. G., Zur Theorie der Polynome von S. Bernstein., Mat. Sb. (N. S.), 2, 543-556, 1937, in German.
24. Musielak, J., Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983.
25. Rao, M. M. and Ren, Z. D., Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York, 1991.
26. Rao, M. M. and Ren, Z. D., Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 250, Marcel Dekker Inc., New York, 2002.
27. Ries, S. and Stens, R. L., Approximation by generalized sampling series, In: Constructive Theory of Functions, Proc. Conf., Varna, Bulgaria, 1984 (Sendov, Bl., Petrushev, P., Maleev, R., and Tashev, S., eds.), 746-756, Publishing House of the Bulgarian Academy of Sciences, Sofia, 1984.
28. Schmeisser, H. J. and Sickel, W., Sampling theory and function spaces, In: Applied Mathematics Reviews (Anastassiou, G. A., ed.), vol. 1, 205-284, World Scientific Publishing Co. Inc., River Edge, NJ, 2000.
29. Splettstößer, W., Über die Approximation stetiger Funktionen durch die klassischen und durch die neuen Abtastsummen mit Fehlerabschätzungen, Doctoral Dissertation, RWTH Aachen, Aachen, 1977, in German.
30. Stein, E. M., Note on the class L log L, Studia Math., 32, 305-310, 1969.
31. Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.
32. Stens, R. L., Error estimates for sampling sums based on convolution integrals, Inform. and Control, 45(1), 37-47, 1980.
33. Stens, R. L., Sampling with generalized kernels, In: Sampling Theory and Signal Analysis: Advanced Topics (Higgins, J. R. and Stens, R. L., eds.), 130-157, Oxford Science Publications, Oxford University Press, Oxford, 1999.
34. Theis, M., Über eine Approximationsformel von de la Vallée-Poussin, Math. Z., 3, 93-113, 1919, in German.
35. Triebel, H., The Structure of Functions, Birkhäuser Verlag, Basel, 2001.
36. Vinti, G., A general approximation result for nonlinear integral operators and applications to signal processing, Appl. Anal., 79(1-2), 217-238, 2001.