| | | |

## Characterization of Absolute and Uniform Continuity

#### İsmail ASLAN [1] , Oktay DUMAN [2]

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.
Summability process, absolute continuity, uniform continuity, nonlinear integral operators, bounded variation
• [1] G.A. Anastassiou and O. Duman, Towards Intelligent Modeling: Statistical Approximation Theory, Intelligent Systems Reference Library, vol. 14, Springer-Verlag, Berlin, 2011.
• [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] 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] 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] 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] L. Angeloni and G. Vinti, Convergence in variation and a characterization of the absolute continuity, Integral Transforms Spec. Funct. 26, 829-844, 2015.
• [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] I. Aslan and O. Duman, A summability process on Baskakov-type approximation, Period. Math. Hungar. 72 (2), 186-199, 2016.
• [9] I. Aslan and O. Duman, Summability on Mellin-type nonlinear integral operators, Integral Transforms Spec. Funct. 30 (6), 492-511, 2019.
• [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] I. Aslan and O. Duman, Nonlinear approximation in N-dimension with the help of summability methods, submitted for publication.
• [12] Ö.G. Atlihan and C. Orhan, Summation process of positive linear operators, Comput. Math. Appl. 56, 1188-1195, 2008.
• [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] 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] H.T. Bell, A−summability, Dissertation, Lehigh University, Bethlehem., Pa., 1971.
• [16] H.T. Bell, Order summability and almost convergence, Proc. Amer. Math. Soc. 38, 548-552, 1973.
• [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] J. Boos, Classical and Modern Methods in Summability, Oxford University Press, Oxford, 2000.
• [19] P.L. Butzer and S. Jansche, A direct approach to the Mellin transform, J. Fourier Anal. Appl. 3 (4), 325-376, 1997.
• [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] 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] D. Casasent, Optical signal processing, in: Optical Data Processing, 241-282, Springer, Berlin, 1978.
• [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] G.H. Hardy, Divergent Series, Oxford University Press, Oxford, 1949.
• [25] W.B. Jurkat and A. Peyerimhoff, Fourier effectiveness and order summability, J. Approx. Theory 4, 231-244, 1971.
• [26] W.B. Jurkat and A. Peyerimhoff, Inclusion theorems and order summability, J. Approx. Theory 4, 245-262, 1971.
• [27] G.G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80, 167-190, 1948.
• [28] R.G. Mamedov, The Mellin Transform and Approximation Theory, Elm, Baku, 1991.
• [29] R.N. Mohapatra, Quantitative results on almost convergence of a sequence of positive linear operators, J. Approx. Theory 20, 239-250, 1977.
• [30] D.A. Smith and W.F. Ford, Acceleration of linear and logarithmical convergence, Siam J. Numer. Anal. 16, 223-240, 1979.
• [31] J.J. Swetits, On summability and positive linear operators, J. Approx. Theory 25, 186-188, 1979.
• [32] L. Tonelli, Su alcuni concetti dell’analisi moderna, Ann. Scuola Norm. Super. Pisa 11 (2), 107-118, 1942.
• [33] J. Wimp, Sequence Transformations and Their Applications, Academic Press, New York, 1981.
Primary Language en Mathematics Mathematics Orcid: 0000-0001-9753-6757Author: İsmail ASLAN (Primary Author)Institution: Hacettepe UniversityCountry: Turkey Orcid: 0000-0001-7779-6877Author: Oktay DUMAN Institution: TOBB UNIVERSITY OF ECONOMICS AND TECHNOLOGYCountry: Turkey Publication Date : October 6, 2020
 Bibtex @research article { hujms585581, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1550 - 1565}, doi = {10.15672/hujms.585581}, title = {Characterization of Absolute and Uniform Continuity}, key = {cite}, author = {Aslan, İsmail and Duman, Oktay} } APA Aslan, İ , Duman, O . (2020). Characterization of Absolute and Uniform Continuity . Hacettepe Journal of Mathematics and Statistics , 49 (5) , 1550-1565 . DOI: 10.15672/hujms.585581 MLA Aslan, İ , Duman, O . "Characterization of Absolute and Uniform Continuity" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1550-1565 Chicago Aslan, İ , Duman, O . "Characterization of Absolute and Uniform Continuity". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1550-1565 RIS TY - JOUR T1 - Characterization of Absolute and Uniform Continuity AU - İsmail Aslan , Oktay Duman Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.585581 DO - 10.15672/hujms.585581 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1550 EP - 1565 VL - 49 IS - 5 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.585581 UR - https://doi.org/10.15672/hujms.585581 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Characterization of Absolute and Uniform Continuity %A İsmail Aslan , Oktay Duman %T Characterization of Absolute and Uniform Continuity %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 5 %R doi: 10.15672/hujms.585581 %U 10.15672/hujms.585581 ISNAD Aslan, İsmail , Duman, Oktay . "Characterization of Absolute and Uniform Continuity". Hacettepe Journal of Mathematics and Statistics 49 / 5 (October 2020): 1550-1565 . https://doi.org/10.15672/hujms.585581 AMA Aslan İ , Duman O . Characterization of Absolute and Uniform Continuity. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1550-1565. Vancouver Aslan İ , Duman O . Characterization of Absolute and Uniform Continuity. Hacettepe Journal of Mathematics and Statistics. 2020; 49(5): 1550-1565.

Authors of the Article