A characterization of the absolute continuity in terms of convergence in variation for the sampling Kantorovich operators. (English) Zbl 1414.41009
The convergence in variation for sampling Kantorovich operators is studied for averaged-type kernels and classical band-limited kernels, respectively. In the case of averaged-type kernels, a characterization of the space of the absolutely continuous functions in terms of the convergence in variation is obtained.
Reviewer: Zoltán Finta (Cluj-Napoca)
MSC:
| 41A25 | Rate of convergence, degree of approximation |
| 41A05 | Interpolation in approximation theory |
| 41A30 | Approximation by other special function classes |
| 47A58 | Linear operator approximation theory |
Keywords:
convergence in variation; sampling Kantorovich series; averaged kernel; variation diminishing-type property; generalized sampling seriesSoftware:
SplinePakReferences:
| [1] | Abel, U., Agratini, O.: On the variation detracting property of operators of Balazas and Szabados. Acta Math. Hungar. 150(2), 383-395 (2016) · Zbl 1399.41037 · doi:10.1007/s10474-016-0642-x |
| [2] | Agratini, O.: An approximation process of Kantorovich type. Math. Notes Miskolc 2(1), 3-10 (2001) · Zbl 0981.41015 · doi:10.18514/MMN.2001.31 |
| [3] | Agratini, O.: On the variation detracting property of a class of operators. Appl. Math. Lett. 19(11), 1261-1264 (2006) · Zbl 1176.41027 · doi:10.1016/j.aml.2005.12.007 |
| [4] | Agrawal, P.N., Baxhaku, B.: Degree of approximation for bivariate extension of Chlodowsky-type q-Bernstein-Stancu-Kantorovich operators. Appl. Math. Comput. 306, 56-72 (2017) · Zbl 1411.41007 |
| [5] | Allasia, G., Cavoretto, R., De Rossi, A.: Lobachevsky spline functions and interpolation to scattered data. Comput. Appl. Math. 32, 71-87 (2013) · Zbl 1273.65015 · doi:10.1007/s40314-013-0011-0 |
| [6] | Allasia, G., Cavoretto, R., De Rossi, A.: Numerical integration on multivariate scattered data by Lobachevsky splines. Int. J. Comput. Math. 90, 2003-2018 (2013) · Zbl 1291.65035 · doi:10.1080/00207160.2013.772144 |
| [7] | Angeloni, L.: Approximation results with respect to multidimensional \[\varphi\] φ-variation for nonlinear integral operators. Z. Anal. Anwend. 32(1), 103-128 (2013) · Zbl 1263.26023 · doi:10.4171/ZAA/1476 |
| [8] | Angeloni, L., Vinti, G.: Approximation with respect to Goffman-Serrin variation by means of non-convolution type integral operators. Numer. Funct. Anal. Optim. 31, 519-548 (2010) · Zbl 1200.41015 · doi:10.1080/01630563.2010.490549 |
| [9] | Angeloni, L., Vinti, G.: A sufficient condition for the convergence of a certain modulus of smoothness in multidimensional setting. Commun. Appl. Nonlinear Anal. 20(1), 1-20 (2013) · Zbl 1279.26028 |
| [10] | Angeloni, L., Vinti, G.: Approximation in variation by homothetic operators in multidimensional setting. Differ. Integral Equ. 26(5-6), 655-674 (2013) · Zbl 1299.41026 |
| [11] | Angeloni, L., Vinti, G.: Convergence and rate of approximation in \[BV^{\varphi }({\mathbb{R}}_+^N)\] BVφ(R+N) for a class of Mellin integral operators. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. (9) Appl. 25(3), 217-232 (2014) · Zbl 1301.41015 · doi:10.4171/RLM/675 |
| [12] | Angeloni, L., Vinti, G.: Convergence in variation and a characterization of the absolute continuity. Integral Transforms Spec. Funct. 26(10), 829-844 (2015) · Zbl 1331.41018 · doi:10.1080/10652469.2015.1062375 |
| [13] | Angeloni, L., Vinti, G.: Discrete operators of sampling type and approximation in \[\varphi\] φ-variation. Math. Nachr. 291, 546-555 (2018) · Zbl 1388.41011 · doi:10.1002/mana.201600508 |
| [14] | Angeloni, L., Costarelli, D., Vinti, G.: A characterization of the convergence in variation for the generalized sampling series. Ann. Acad. Sci. Fenn. Math. 43, 755-767 (2018) · Zbl 1405.41010 · doi:10.5186/aasfm.2018.4343 |
| [15] | Asdrubali, F., Baldinelli, G., Bianchi, F., Costarelli, D., Rotili, A., Seracini, M., Vinti, G.: Detection of thermal bridges from thermographic images by means of image processing approximation algorithms. Appl. Math. Comput. 317, 160-171 (2018) · Zbl 1426.94008 |
| [16] | Asdrubali, F., Baldinelli, G., Bianchi, F., Costarelli, D., Evangelisti, L., Rotili, A., Seracini, M., Vinti, G.: A model for the improvement of thermal bridges quantitative assessment by infrared thermography. Appl. Energy 211, 854-864 (2018) · doi:10.1016/j.apenergy.2017.11.091 |
| [17] | Bardaro, C., Vinti, G.: General convergence theorem with respect to Cesari variation and applications. Nonlinear Anal. 22(4), 505-518 (1994) · Zbl 0798.49015 · doi:10.1016/0362-546X(94)90171-6 |
| [18] | Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Convergence in variation and rates of approximation for Bernstein-type polynomials and singular convolution integrals. Analysis 23, 299-340 (2003) · Zbl 1049.41015 · doi:10.1524/anly.2003.23.4.299 |
| [19] | Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Sampl. Theory Signal Image Process. 6, 29-52 (2007) · Zbl 1156.41307 |
| [20] | Bardaro, C., Karsli, H., Vinti, G.: Nonlinear integral operators with homogeneous kernels: pointwise approximation theorems. Appl. Anal. 90(3-4), 463-474 (2011) · Zbl 1214.41006 · doi:10.1080/00036811.2010.499506 |
| [21] | Bartoccini, B., Costarelli, D., Vinti, G.: Extension of saturation theorems for the sampling Kantorovich operators. Complex Anal. Oper. Theory (2018). https://doi.org/10.1007/s11785-018-0852-z · Zbl 1416.41017 |
| [22] | Bieniek, M.: Variation diminishing property of densities of uniform generalized order statistics. Metrika 65(3), 297-309 (2007) · Zbl 1433.62120 · doi:10.1007/s00184-006-0077-4 |
| [23] | Boccuto, A., Candeloro, D., Sambucini, A.R.: \[L^p\] Lp spaces in vector lattices and applications. Math. Slov. 67(6), 1409-1426 (2017). https://doi.org/10.1515/ms-2017-0060 · Zbl 1505.28018 · doi:10.1515/ms-2017-0060 |
| [24] | Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation I. Academic Press, New York (1971) · Zbl 1515.42001 · doi:10.1007/978-3-0348-7448-9 |
| [25] | Butzer, P.L., Ries, S., Stens, R.L.: Approximation of continuous and discontinuous functions by generalized sampling series. J. Approx. Theory 50(1), 25-39 (1987) · Zbl 0654.41004 · doi:10.1016/0021-9045(87)90063-3 |
| [26] | Butzer, P.L., Fisher, A., Stens, R.L.: Generalized sampling approximation of multivariate signals: theory and applications. Note Mat. 1(10), 173-191 (1990) · Zbl 0768.42013 |
| [27] | Carnicer, J.M., Goodman, T.N., Pena, J.M.: A generalization of the variation diminishing property. Adv. Comput. Math. 3(4), 375-394 (1995) · Zbl 0837.15019 · doi:10.1007/BF02432004 |
| [28] | Chang, G.Z., Hoschek, J.: Convexity and variation diminishing property of Bernstein polynomials over triangles. Int. Ser. Numer. Math. 75, 61-70 (1985) · Zbl 0563.41008 |
| [29] | Cluni, F., Costarelli, D., Minotti, A.M., Vinti, G.: Applications of sampling Kantorovich operators to thermographic images for seismic engineering. J. Comput. Anal. Appl. 19(4), 602-617 (2015) · Zbl 1369.94021 |
| [30] | Coroianu, L., Gal, S.G.: \[L^p\] Lp-approximation by truncated max-product sampling operators of Kantorovich-type based on Fejer kernel. J. Integral Equ. Appl. 29(2), 349-364 (2017) · Zbl 1371.41016 · doi:10.1216/JIE-2017-29-2-349 |
| [31] | Costarelli, D., Vinti, G.: Pointwise and uniform approximation by multivariate neural network operators of the max-product type. Neural Netw. 81, 81-90 (2016) · Zbl 1439.41009 · doi:10.1016/j.neunet.2016.06.002 |
| [32] | Costarelli, D., Vinti, G.: Convergence for a family of neural network operators in Orlicz spaces. Math. Nachr. 290(2-3), 226-235 (2017) · Zbl 1373.47010 · doi:10.1002/mana.201600006 |
| [33] | Costarelli, D., Vinti, G.: Estimates for the neural network operators of the max-product type with continuous and p-integrable functions. Results Math. 73(1), 12 (2018). https://doi.org/10.1007/s00025-018-0790-0 · Zbl 1390.41020 · doi:10.1007/s00025-018-0790-0 |
| [34] | Costarelli, D., Vinti, G.: An inverse result of approximation by sampling Kantorovich series. Proc. Edinb. Math. Soc. 62(1), 265-280 (2019) · Zbl 1428.41019 |
| [35] | Costarelli, D., Minotti, A.M., Vinti, G.: Approximation of discontinuous signals by sampling Kantorovich series. J. Math. Anal. Appl. 450(2), 1083-1103 (2017) · Zbl 1373.41018 · doi:10.1016/j.jmaa.2017.01.066 |
| [36] | Cottin, C., Gavrea, I., Gonska, H.H., Kacsó, D.P., Zhou, D.X.: Global smoothness preservation and the variation-diminishing property. J. Inequal. Appl. 4(2), 91-114 (1999). https://doi.org/10.1155/S1025583499000314 · Zbl 0970.41015 · doi:10.1155/S1025583499000314 |
| [37] | DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993) · Zbl 0797.41016 · doi:10.1007/978-3-662-02888-9 |
| [38] | Ferracina, L., Spijker, M.N.: Stepsize restrictions for the total-variation-diminishing property in general Runge-Kutta methods. SIAM J. Numer. Anal. 42(3), 1073-1093 (2004) · Zbl 1080.65087 · doi:10.1137/S0036142902415584 |
| [39] | Heilmann, M., Rasa, I.: A nice representation for a link between Bernstein-Durrmeyer and Kantorovich operators. Commun. Comput. Inf. Sci. 655, 312-320 (2017) · Zbl 1514.41018 |
| [40] | Karsli, H.: On convergence of Chlodovsky and Chlodovsky-Kantorovich polynomials in the variation seminorm. Mediterr. J. Math. 10(1), 41-56 (2013) · Zbl 1263.41007 · doi:10.1007/s00009-012-0186-4 |
| [41] | Karsli, H., Oksuzer Yilik, O., Yesildal, F.T.: Convergence of the Bernstein-Durrmeyer operators in variation seminorm. Results Math. 72(3), 1257-1270 (2017) · Zbl 1376.41020 · doi:10.1007/s00025-017-0653-0 |
| [42] | Kivinukk, A., Metsmagi, T.: The variation detracting property of some Shannon sampling series and their derivatives. Sampl. Theory Signal Image Process. 13, 189-206 (2014) · Zbl 1346.94080 |
| [43] | Krejci, P., Roche, T.: Lipschitz continuous data dependence of sweeping processes in BV spaces. Discrete Contin. Dyn. Syst. Ser. B 15, 637-650 (2011) · Zbl 1214.49022 · doi:10.3934/dcdsb.2011.15.637 |
| [44] | Laczkovich, M., Sós, V.T.: Functions of bounded variation. In: Real Analysis. Springer, New York, pp. 399-406 (2015) · Zbl 1325.26001 |
| [45] | Orlova, O., Tamberg, G.: On approximation properties of generalized Kantorovich-type sampling operators. J. Approx. Theory 201, 73-86 (2016) · Zbl 1329.41030 · doi:10.1016/j.jat.2015.10.001 |
| [46] | Sancetta, A.: Nonparametric estimation of distributions with given marginals via Bernstein-Kantorovich polynomials: L1 and pointwise convergence theory. J. Multivariate Anal. 98(7), 1376-1390 (2007) · Zbl 1116.62039 · doi:10.1016/j.jmva.2007.02.004 |
| [47] | Schumaker, L.L.: Spline Functions—Computational Methods. SIAM, Philadelphia (2015) · Zbl 1333.65018 · doi:10.1137/1.9781611973907 |
| [48] | Speleers, H.: Multivariate normalized Powell-Sabin B-splines and quasi-interpolants. Comput. Aided Geom. Design 30(1), 2-19 (2013) · Zbl 1255.65038 · doi:10.1016/j.cagd.2012.07.005 |
| [49] | Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005) · Zbl 1075.65021 |
| [50] | Ziemer, W.P.: Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation, vol. 120. Springer Science and Business Media, New York (2012) · Zbl 0692.46022 |
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