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An Inverse Result of Approximation by Sampling Kantorovich Series

Part of: General theory of linear operators Approximations and expansions

Published online by Cambridge University Press:  16 October 2018

Danilo Costarelli Open the ORCID record for Danilo Costarelli [Opens in a new window]  and
Gianluca Vinti
Danilo Costarelli*
Affiliation:
Department of Mathematics and Computer Science, University of Perugia, 1 Via Vanvitelli, 06123 Perugia, Italy (danilo.costarelli@unipg.it; gianluca.vinti@unipg.it)
Gianluca Vinti
Affiliation:
Department of Mathematics and Computer Science, University of Perugia, 1 Via Vanvitelli, 06123 Perugia, Italy (danilo.costarelli@unipg.it; gianluca.vinti@unipg.it)
*
*Corresponding author.
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Abstract

In the present paper, an inverse result of approximation, i.e. a saturation theorem for the sampling Kantorovich operators, is derived in the case of uniform approximation for uniformly continuous and bounded functions on the whole real line. In particular, we prove that the best possible order of approximation that can be achieved by the above sampling series is the order one, otherwise the function being approximated turns out to be a constant. The above result is proved by exploiting a suitable representation formula which relates the sampling Kantorovich series with the well-known generalized sampling operators introduced by Butzer. At the end, some other applications of such representation formulas are presented, together with a discussion concerning the kernels of the above operators for which such an inverse result occurs.

Keywords

inverse resultssampling Kantorovich seriesorder of approximationcentral B-splinesgeneralized sampling operatorssaturation theorem

MSC classification

Primary: 41A25: Rate of convergence, degree of approximation
Secondary: 41A05: Interpolation 41A30: Approximation by other special function classes 47A58: Operator approximation theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 265 - 280
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Alavi, J. and Aminikhah, H., Applying cubic B-Spline quasi-interpolation to solve 1D wave equations in polar coordinates, ISRN Comput. Math. 2013 (2013), 710529.Google Scholar
2.Asdrubali, F., Baldinelli, G., Bianchi, F., Costarelli, D., Rotili, A., Seracini, M. and Vinti, G., Detection of thermal bridges from thermographic images by means of image processing approximation algorithms, Appl. Math. Comput. 317 (2018), 160–171.Google Scholar
3.Bardaro, C. and Mantellini, I., Voronovskaja formulae for Kantorovich generalized sampling series, Int. J. Pure Appl. Math 62(3) (2010), 247–262.Google Scholar
4.Bardaro, C. and Mantellini, I., Asymptotic formulae for multivariate Kantorovich type generalized sampling series, Acta. Math. Sin. Engl. Ser. 27(7) (2011), 1247–1258.Google Scholar
5.Bardaro, C. and Mantellini, I., On convergence properties for a class of Kantorovich discrete operators, Numer. Funct. Anal. Optim. 33(4) (2012), 374–396.Google Scholar
6.Bardaro, C., Butzer, P. L., Stens, R. L. and Vinti, G., Kantorovich-type generalized sampling series in the setting of Orlicz spaces, Sampl. Theory Signal Image Process. 6(1) (2007), 29–52.Google Scholar
7.Bardaro, C., Butzer, P. L., Stens, R. L. and Vinti, G., Prediction by samples from the past with error estimates covering discontinuous signals, IEEE Trans. Inf. Theory 56(1) (2010), 614–633.Google Scholar
8.Bardaro, C., Karsli, H. and Vinti, G., Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems, Appl. Anal. 90(3–4) (2011), 463–474.Google Scholar
9.Bede, B., Coroianu, L. and Gal, S. G., Approximation by max-product type operators (Springer International Publishing, 2016).Google Scholar
10.Brezis, H., Functional analysis, Sobolev spaces and partial differential equations (Springer, New York–Dordrecht–Heidelberg–London, 2010).Google Scholar
11.Butzer, P. L. and Nessel, R. J., Fourier analysis and approximation I (Academic Press, New York–London, 1971).Google Scholar
12.Butzer, P. L. and Stens, R. L., Linear prediction by samples from the past, in Advanced topics in Shannon sampling and interpolation theory, pp. 157–183 (Springer, 1993).Google Scholar
13.Cluni, F., Costarelli, D., Minotti, A. M. and Vinti, G., Applications of sampling Kantorovich operators to thermographic images for seismic engineering, J. Comput. Anal. Appl. 19(4) (2015), 602–617.Google Scholar
14.Cluni, F., Costarelli, D., Minotti, A. M. and Vinti, G., Enhancement of thermographic images as tool for structural analysis in earthquake engineering, NDT & E Int. 70 (2015), 60–72.Google Scholar
15.Coroianu, L. and Gal, S. G., Approximation by nonlinear generalized sampling operators of max-product kind, Sampl. Theory Signal Image Process. 9(1–3) (2010), 59–75.Google Scholar
16.Coroianu, L. and Gal, S. G., Approximation by max-product sampling operators based on sinc-type kernels, Sampl. Theory Signal Image Process. 10(3) (2011), 211–230.Google Scholar
17.Coroianu, L. and Gal, S. G., Saturation results for the truncated max-product sampling operators based on sinc and Fejér-type kernels, Sampl. Theory Signal Image Process. 11(1) (2012), 113–132.Google Scholar
18.Costarelli, D. and Spigler, R., Convergence of a family of neural network operators of the Kantorovich type, J. Approx. Theory 185 (2014), 80–90.Google Scholar
19.Costarelli, D. and Vinti, G., Approximation by multivariate generalized sampling Kantorovich operators in the setting of Orlicz spaces, Boll. Unione Mat. Ital., Special 9(IV) (2011), 445–468 (special volume dedicated to Professor Giovanni Prodi).Google Scholar
20.Costarelli, D. and Vinti, G., Approximation by nonlinear multivariate sampling-Kantorovich type operators and applications to image processing, Numer. Funct. Anal. Optim. 34(8) (2013), 819–844.Google Scholar
21.Costarelli, D. and Vinti, G., Order of approximation for sampling Kantorovich operators, J. Integr. Equ. Appl. 26(3) (2014), 345–368.Google Scholar
22.Costarelli, D. and Vinti, G., Rate of approximation for multivariate sampling Kantorovich operators on some functions spaces, J. Integr. Equ. Appl. 26(4) (2014), 455–481.Google Scholar
23.Costarelli, D. and Vinti, G., Degree of approximation for nonlinear multivariate sampling Kantorovich operators on some functions spaces, Numer. Funct. Anal. Optim. 36(8) (2015), 964–990.Google Scholar
24.Costarelli, D. and Vinti, G., Approximation by max-product neural network operators of Kantorovich type, Results Math. 69(3) (2016), 505–519.Google Scholar
25.Costarelli, D. and Vinti, G., Max-product neural network and quasi-interpolation operators activated by sigmoidal functions, J. Approx. Theory 209 (2016), 1–22.Google Scholar
26.Costarelli, D., Minotti, A. M. and Vinti, G., Approximation of discontinuous signals by sampling Kantorovich series, J. Math. Anal. Appl. 450(2) (2017), 1083–1103.Google Scholar
27.Do, M. N. and Lu, Y. M., A theory for sampling signals from a union of subspaces, IEEE Trans. Signal. Process. 56(6) (2008), 2334–2345.Google Scholar
28.Fix, G. and Strang, G., Fourier analysis of the finite element method in Ritz–Galerkin theory, Stud. Appl. Math. 48 (1969), 268–273.Google Scholar
29.Gao, W. and Wu, Z., Quasi-interpolation for linear functional data, J. Comput. Appl. Math. 236(13) (2012), 3256–3264.Google Scholar
30.Higgins, J. R., Sampling theory in Fourier and signal analysis: foundations (Oxford University Press, Oxford, 1996).Google Scholar
31.Kivinukk, A. and Tamberg, G., Interpolating generalized Shannon sampling operators, their norms and approximation properties, Sampl. Theory Signal Image Process. 8 (2009), 77–95.Google Scholar
32.Kivinukk, A. and Tamberg, G., On approximation properties of sampling operators by dilated kernels, 8th International Conference on Sampling Theory and Applications (SAMPTA'09) Marseille, France, May 18–22, 2009 (Société Mathématique de France, Marseille, 2009).Google Scholar
33.Liu, S. and Wang, C. C., Quasi-interpolation for surface reconstruction from scattered data with radial basis function, Comput. Aided. Geom. Des. 29(7) (2012), 435–447.Google Scholar
34.Marsden, M. J., An identity for spline functions with applications to variation-diminishing spline approximation, J. Approx. Theory 3(1) (1970), 7–49.Google Scholar
35.Monaghan, J. J., Extrapolating B splines for interpolation, J. Comput. Phys. 60(2) (1985), 253–262.Google Scholar
36.Neuman, E., Moments and Fourier transforms of B-splines, J. Comput. Appl. Math. 7(1) (1981), 51–62.Google Scholar
37.Orlova, O. and Tamberg, G., On approximation properties of generalized Kantorovich-type sampling operators, J. Approx. Theory 201 (2016), 73–86.Google Scholar
38.Ries, S. and Stens, R. L., Approximation by generalized sampling series, in Constructive Theory of Functions, pp. 746–756 (Sofia, 1984).Google Scholar
39.Speleers, H., Multivariate normalized Powell-Sabin B-splines and quasi-interpolants, Comput. Aided. Geom. Des. 30(1) (2013), 2–19.Google Scholar
40.Tamberg, G., Approximation by generalized Shannon sampling operators generated by band-limited kernels, Proc. Appl. Math. Mech. 8(1) (2008), 10937–10940.Google Scholar
41.Tamberg, G., On truncation errors of some generalized Shannon sampling operators, Numer. Algorithms 55(2) (2010), 367–382.Google Scholar
42.Unser, M. A., Ten good reasons for using spline wavelets, in Optical Science, Engineering and Instrumentation 1997, pp. 422–431 (International Society for Optics and Photonics, 1997).Google Scholar
43.Vinti, G. and Zampogni, L., A general approximation approach for the simultaneous treatment of integral and discrete operators, Adv. Nonlinear Stud. (2017), DOI: 10.1515/ans-2017-6038.Google Scholar
44.Wang, R. H., Yu, R. G. and Zhu, C. G., A numerical method for solving KdV equation with multilevel B-spline quasi-interpolation, Appl. Anal. 92(8) (2013), 1682–1690.Google Scholar