Characterization of Absolute and Uniform Continuity

Abstract

In the present paper, by considering nonlinear integral operators and using their approximations via regular summability methods, we obtain characterizations for some function spaces including the space of absolutely continuous functions, the space of uniformly continuous functions, and their other variants. We observe that Bell-type summability methods are quite effective to generalize and improve some related results in the literature. At the end of the paper, we discuss some special cases and applications.

Keywords

• Summability process
• absolute continuity
• uniform continuity
• nonlinear integral operators
• bounded variation

References

1. [1] G.A. Anastassiou and O. Duman, Towards Intelligent Modeling: Statistical Approximation Theory, Intelligent Systems Reference Library, vol. 14, Springer-Verlag, Berlin, 2011.
2. [2] L. Angeloni and G. Vinti, Convergence in variation and rate of approximation for nonlinear integral operators of convolution type, Results Math. 48, 1-23, 2006.
3. [3] L. Angeloni and G. Vinti, Erratum to: Convergence in variation and rate of approximation for nonlinear integral operators of convolution type, Results Math. 57, 387-391, 2010.
4. [4] L. Angeloni and G. Vinti, Variation and approximation in multidimensional setting for Mellin integral operators, New Perspectives on Approximation and Sampling Theory, 299-317, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2014.
5. [5] L. Angeloni and G. Vinti, A characterization of absolute continuity by means of Mellin integral operators, Z. Anal. Anwend 34 (3), 343-356, 2015.
6. [6] L. Angeloni and G. Vinti, Convergence in variation and a characterization of the absolute continuity, Integral Transforms Spec. Funct. 26, 829-844, 2015.
7. [7] L. Angeloni and G. Vinti, A concept of absolute continuity and its characterization in terms of convergence in variation, Math. Nachr. 289 (16), 1986-1994, 2016.
8. [8] I. Aslan and O. Duman, A summability process on Baskakov-type approximation, Period. Math. Hungar. 72 (2), 186-199, 2016.

9. [9] I. Aslan and O. Duman, Summability on Mellin-type nonlinear integral operators, Integral Transforms Spec. Funct. 30 (6), 492-511, 2019.
10. [10] I. Aslan and O. Duman, Approximation by nonlinear integral operators via summability process, Math. Nachr., 2020 (in press) doi: 10.1002/mana.201800187.
11. [11] I. Aslan and O. Duman, Nonlinear approximation in N-dimension with the help of summability methods, submitted for publication.
12. [12] Ö.G. Atlihan and C. Orhan, Summation process of positive linear operators, Comput. Math. Appl. 56, 1188-1195, 2008.
13. [13] C. Bardaro, P.L. Butzer and I. Mantellini, The exponential sampling theorem of signal analysis and the reproducing kernel formula in the Mellin transform setting, Sampl Theory Signal Image Process. 13 (1), 35-66, 2014.
14. [14] C. Bardaro, P.L. Butzer, R.L. Stens and G. Vinti, Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals, Analysis 23, 299-340, 2003.
15. [15] H.T. Bell, A−summability, Dissertation, Lehigh University, Bethlehem., Pa., 1971.
16. [16] H.T. Bell, Order summability and almost convergence, Proc. Amer. Math. Soc. 38, 548-552, 1973.
17. [17] M. Bertero and E.R. Pike, Exponential-sampling method for Laplace and other dilationally invariant transforms. II. Examples in photon correlation spectroscopy and Fraunhofer diffraction, Inverse Probl. 7 (1), 21-41, 1991.
18. [18] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford, 2000.
19. [19] P.L. Butzer and S. Jansche, A direct approach to the Mellin transform, J. Fourier Anal. Appl. 3 (4), 325-376, 1997.
20. [20] P.L. Butzer and S. Jansche, A self-contained approach to Mellin transform analysis for square integrable functions; applications, Integral Transforms Spec. Funct. 8 (3-4), 175-198, 1999.
21. [21] P.L. Butzer and R.J. Nessel, Fourier Analysis and Approximation, Lehrbücher und Monographien aus dem Gebiete der exakten Wissenschaften, Mathematische Reihe 40, Birkhäuser, Basel, Academic Press, New York, 1971.
22. [22] D. Casasent, Optical signal processing, in: Optical Data Processing, 241-282, Springer, Berlin, 1978.
23. [23] A. De Sena and D. Rocchesso, A fast Mellin and scale transform, EURASIP J. Adv. Signal Process, Art. ID 89170, 9 pages, 2007.
24. [24] G.H. Hardy, Divergent Series, Oxford University Press, Oxford, 1949.
25. [25] W.B. Jurkat and A. Peyerimhoff, Fourier effectiveness and order summability, J. Approx. Theory 4, 231-244, 1971.
26. [26] W.B. Jurkat and A. Peyerimhoff, Inclusion theorems and order summability, J. Approx. Theory 4, 245-262, 1971.
27. [27] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80, 167-190, 1948.
28. [28] R.G. Mamedov, The Mellin Transform and Approximation Theory, Elm, Baku, 1991.
29. [29] R.N. Mohapatra, Quantitative results on almost convergence of a sequence of positive linear operators, J. Approx. Theory 20, 239-250, 1977.
30. [30] D.A. Smith and W.F. Ford, Acceleration of linear and logarithmical convergence, Siam J. Numer. Anal. 16, 223-240, 1979.
31. [31] J.J. Swetits, On summability and positive linear operators, J. Approx. Theory 25, 186-188, 1979.
32. [32] L. Tonelli, Su alcuni concetti dell’analisi moderna, Ann. Scuola Norm. Super. Pisa 11 (2), 107-118, 1942.
33. [33] J. Wimp, Sequence Transformations and Their Applications, Academic Press, New York, 1981.