On pointwise convergence of nets of Mellin-Kantorovich convolution operators

• Autorzy: Bardaro C. , Mantellini I.

Identyfikatory

DOI

10.14708/cm.v53i2.782

• Języki publikacji: EN

Abstrakty

EN

Here we study pointwise approximation and asymptotic formulae for a class of Mellin-Kantorovich type integral operators, both in linear and nonlinear form.

Słowa kluczowe

EN

Mellin-Kantorovich convolution operators; Mellin derivatives; moments; Lipschitz conditions; Voronovskaja formula

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Commentationes Mathematicae
• Rocznik: 2013
• Tom: Vol. 53, No. 2
• Strony: 65--77
• Opis fizyczny: Bibliogr. 36 poz.

Twórcy

autor

Bardaro C.

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

autor

Mantellini I.

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

Bibliografia

• [1] L. Angeloni and G. Vinti, Approximation in variation by homothetic operators in multidimensional setting, Differential Integral Equation, 26(5-6) (2013), 655–674.
• [2] L. Angeloni and G. Vinti, Variation and approximation in multidimensional setting for Mellin operators, submitted (2013).
• [3] C. Bardaro, H. Karsli and G. Vinti, Nonlinear integral opereators with homogeneous kernels; pointwise approximation theorems, Appl. Anal.,90(3-4) (2011), 463-474.
• [4] C. Bardaro, H. Karsli and G. Vinti, On pointwise convergence of Mellin type nonlinear m-singular integral operators, Comm. Appl. Nonlinear Anal., 20(2) (2013), 25-39
• [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] C. Bardaro and I. Mantellini, Approximation properties in abstract modular spaces for a class of general sampling-type operators, Appl. Anal. 85 (2006), 383–413.
• [7] C. Bardaro and I. Mantellini, Pointwise convergence theorems for nonlinear Mellin convolution operators. Int. J. Pure Appl. Math, 27(4) (2006), 431–447.
• [8] C. Bardaro and I. Mantellini, Voronovskaya-type estimates for Mellin convolution operators, Results in Math. 50(1-2) (2007), 1-16.
• [9] C. Bardaro and I. Mantellini, A quantitative Voronovskaya formula for Mellin convolution operators, Mediterr. J. Math. 7(4) (2010), 483-501.
• [10] C. Bardaro and I. Mantellini, A note on the Voronovskaya theorem for Mellin-Fejer convolution operators, Appl. Math. Letters 24 (2011), 2064-2067.
• [11] C. Bardaro and I. Mantellini, On Voronovskaja formula for linear combinations of Mellin-Gauss-Weierstrass operators, Appl. Math. Comput. 218 (2012), 10171-10179.
• [12] C. Bardaro and I. Mantellini, Approximation properties for linear combinations of moment type operators, Comput. Math. Appl. 62 (2011), 2304-2313.
• [13] C. Bardaro and I. Mantellini, Asymptotic behaviour of Mellin-Fejer convolution operators, East J. Approx 17(2) (2011), 181-201.
• [14] C. Bardaro and I. Mantellini, On the iterates of Mellin-Fejer convolution operators, Acta Appl. Math. 121 (2012), 2304-2313.
• [15] C. Bardaro, J. Musielak and G. Vinti, Nonlinear integral operators and applications De Gruyter Series in Nonlinear Analysis and Appl. Vol.9, 2003.
• [16] M. Bertero and E.R. Pike, Exponential sampling method for Laplace and other dilationally invariant transforms I. Singular-system analysis. II. Examples in photon correction spectroscopy and Frauenhofer diffraction, Inverse Problems 7 (1991), 1-20; 21-41.
• [17] P.L. Butzer and R.J. Nessel, Fourier Analysis and Approximation I. Academic Press, New York-London, 1971.
• [18] P.L. Butzer and S. Jansche, A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3 (1997), 325-375.
• [19] P.L. Butzer and S. Jansche, The exponential sampling theorem of signal analysis, Atti Sem. mat. Fis. Univ. Modena, Suppl. Vol. 46, special volume dedicated to Professor Calogero Vinti (1998), 99-122.
• [20] R.A. DeVore and G.G. Lorentz, Constructive Approximation, Grundlehren der Mathematischen Wissenschaften 303, Springer Verlag, Berlin, 1993.
• [21] Z. Ditzian and V. Totik, Modulus of smoothness, Springer Series in Computational Mathematics 9, Springer-Verlag, New York, 1987.
• [22] H. Karsli, On approximation properties of non convolution type nonlinear integral operators, Anal. Theory Appl. 26(2), (2010), 140-152.
• [23] H. Karsli, Some convergence results for nonlinear singular integral operators, Demonstratio Math. (2013), to appear.
• [24] W. Kolbe and R.J. Nessel, Saturation theory in connections with the Mellin transform methods, SIAM J. Math. Anal. 3 (1972), 246-262.
• [25] R.G. Mamedov, The Mellin transform and approximation theory, (in Russian), Elm", Baku, 1991.
• [26] I. Mantellini, On the asymptotic behaviour of linear combinations of Mellin-Picard type operators, to appear in Math. Nachr. 286(17-18) (2013), 1820-1832.
• [27] 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.
• [28] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer-Verlag, Berlin, 1983.
• [29] J. Musielak, Approximation by nonlinear singular integral operators in generalized Orlicz spaces, Comment. Math. 31 (1991), 79-88.
• [30] J. Musielak, Nonlinear approximation in some modular function spaces I, Math. Japonica, 38 (1993), 83-90.
• [31] J. Musielak, On approximation by nonlinear integral operators with generalized Lipschitz kernel over a compact abelian group, Comment. Math. 35 (1993), 99-104.
• [32] T. Swiderski and E. Wachnicki, Nonlinear singular integrals depending on two parameters, Comment. Math. 40 (2000), 181-189.
• [33] F. Ventriglia and G. Vinti, Nonlinear Kantorovich-type operators: a unified approach, submitted (2013).
• [34] G. Vinti and L. Zampogni, A unified approach for the convergence of linear Kantorovich-type operators, submitted (2013)
• [35] E.V. Voronovskaja, Determination of the asymptotic form of approximation of functions by the polynomials of S.N. Bernstein, Dokl. Akad. Nauk SSSR, A (1932), 79-85.
• [36] A.H. Zemanian, Generalized Integral Transformation, Interscience, New York, 1968.