A Quantitative Voronovskaya Formula for Mellin Convolution Operators

Mediterranean Journal of Mathematics

Volumn 7, Issue 4, 2010, Pages 483-501

  Bardaro, Carlo ^a   Mantellini, Ilaria ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

Author keywords

K functional; Mellin convolution operators; Mellin derivatives; Moments; Voronovskaya formula

Indexed keywords

EID: 78649328795     PISSN: 16605446     EISSN: 16605454     Source Type: Journal    
DOI: 10.1007/s00009-010-0062-z     Document Type: Article

References (23)

1. Altomare F.: Limit semigroups of Bernstein-Schnabl operators associated with positive projection. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16(4), 259-279 (1989).
2. Altomare F., Campiti M.: Korovkin-type approximation theory and its applications. Walter de Gruyter, Berlin-New York (1994)
3. Anastassiou G., Gal S.: Approximation theory. Moduli of continuity and global smoothness preservation. Birkhauser, Boston (2000).
4. Barbieri F.: Approssimazione mediante nuclei momento. Atti Sem. Mat. Fis. Univ. Modena 32, 308-328 (1983).
5. C. Bardaro and I. Mantellini, Linear integral operators with homogeneous kernel: approximation properties in modular spaces. Applications to Mellin-type convolution operators and to some classes of fractional operators. Applied Math. Rev. vol I, World Scientific Publ., Ed. G. Anastassiou (2000), 45-67.
6. Bardaro C., Mantellini I.: Pointwise convergence theorems for nonlinear Mellin convolution operators. Int. J. Pure Appl. Math. 27(4), 431-447 (2006).
7. Bardaro C., Mantellini I.: Voronovskaya-type estimates for Mellin convolution operators. Result Math. 50, 1-16 (2007).
8. P. L. Butzer and H. Berens, Semi-groups of operators and approximation. Die Grundlehren der mathematischen Wissenschaften, 145, Springer-Verlag, Berlin-Heidelberg-New York, 1967.
9. Butzer P. L., Nessel R. J.: Fourier Analysis and Approximation I. Academic Press, New York-London (1971).
10. Butzer P. L., Jansche S.: A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3, 325-375 (1997).
11. P. L. Butzer and S. Jansche, Mellin transform, the Mellin-Poisson summation formula and the exponential sampling theorem. Atti Sem. Mat. Fis. Univ. Modena, Suppl. Vol. 46, A special volume dedicated to Professor Calogero Vinti, (1998), 99-122.
12. DeVore R. A., Lorentz G. G.: Constructive Approximation. Springer-Verlag, Berlin, Heidelberg (1993).
13. Ditzian Z., Totik V.: Moduli of smoothness. Springer Series in Computational Mathematics, 9. Springer-Verlag, New York (1987).
14. H. Gonska, P. Pitul and I. Rasa, On Peano's form of the Taylor remainder, Voronovskaja's theorem and the commutator of positive linear operators. Proc. Int. Conf. on Numerical Analysis and Approximation Theory, Cluj-Napoca, Romania, July 5-8, 2006, 55-80.
15. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series and products. sixth edition, Academic Press, 2000.
16. Lorentz G. G.: Approximation of functions. Chelsea Publ. Co., New York (1986).
17. Maligranda L.: The K-functional for symmetric spaces. Lecture Notes in Math. 1070, 169-182 (1984).
18. L. Maligranda, Interpolation spaces in the theory of approximation. in: Methods of functional analysis in approximation theory, Proc. Int. Conf. Bombay, Dec. 16-20 1985, ISNM 76, Birkhä user (1986), 263-279.
19. Mitjagin B. S., Semenov E. M.: Lack of interpolation of linear operators in spaces of smooth functions. Math. USSR-Izv. 11(6), 1229-1266 (1977).
20. J. Peetre, A theory of interpolation of normed spaces. Lecture Notes, Brasilia, 1963 (Notas de Matematica, 39, 1968).
21. Peetre J.: Exact interpolation theorems for Lipschitz continuous functions. Ricerche Mat. 18, 239-259 (1969).
22. Sikkema P. C.: Approximation formulae of Voronovskaya type for certain convolution operators. J. Approx. Theory 26, 26-45 (1979).
23. Voronovskaya E. V.: Determination of the asymptotic form of approximation of functions by the polynomials of S. N. Bernstein. Dokl. Akad. Nauk SSSR A, 79-85 (1932).