Linear prediction and simultaneous approximation by m-th order Kantorovich type sampling series
Tóm tắt
In the present paper, a new family of sampling type operators is introduced and studied. By the composition of the well-known generalized sampling operators of P.L. Butzer with the usual differential and anti-differential operators of order m, we obtain the so-called m-th order Kantorovich type sampling series. This family of approximation operators are very general and include, as special cases, the well-known sampling Kantorovich and the finite-differences operators. Here, we discuss about pointwise and uniform convergence of the m-th order Kantorovich type sampling series; further, quantitative estimates for the order of approximation have been established together with asymptotic formulas and Voronovskaja type theorems. In the latter results, a crucial role is played by certain algebraic moments of the involved kernels, that can be computed by resorting to the their Fourier transform and to the well-known Poisson’s summation formula. By means of the above results we become able to solve the problems of the simultaneous approximation of a function and its derivatives, both from a qualitative and a quantitative point of view, and of the linear prediction by samples from the past.
Tài liệu tham khảo
Acar, T., Aral, A., Raşa, I.: Approximation by k-th order modifications of Sz’asz–Mirakyan operators. Stud. Sci. Math. Hungar. 53(3), 379–398 (2016)
Aldroubi, A.: Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces. Appl. Comput. Harmon. Anal. 13(2), 151–161 (2002)
Allasia, G., Cavoretto, R., De Rossi, A.: A class of spline functions for landmark-based image registration. Math. Methods Appl. Sci. 35, 923–934 (2012)
Allasia, G., Cavoretto, R., De Rossi, A.: Lobachevsky spline functions and interpolation to scattered data. Comput. Appl. Math. 32, 71–87 (2013)
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)
Angeloni, L., Costarelli, D., Vinti, G.: A characterization of the absolute continuity in terms of convergence in variation for the sampling Kantorovich operators. Med J Math 16(2), 44 (2019)
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)
Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Kantorovich-type generalized sampling series in the setting of Orlicz spaces. Sampl Theor Signal Image Process 6, 29–52 (2007)
Bardaro, C., Mantellini, I.: Approximation properties for linear combinations of moment type operators. Comput. Math. Appl. 62(5), 2304–2313 (2011)
Bardaro, C., Mantellini, I.: Asymptotic formulae for linear combinations of generalized sampling operators. Z. Anal. Anwend. 32(3), 279–298 (2013)
Bartoccini, B., Costarelli, D., Vinti, G.: Extension of saturation theorems for the sampling Kantorovich operators. Complex Anal. Oper. Theory 13(3), 1161–1175 (2019)
Bowman, F.: Introduction to Bessel Function. Dover Publications Inc., New York (1958)
Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation I. Academic Press, New York-London (1971)
Butzer, P.L., Ries, S., Stens, R.L.: Approximation of continuous and discontinuous functions by generalized sampling series. J. Approx. Theory 50, 25–39 (1987)
Butzer, P.L., Schmeisser, G., Stens, R.L.: Basic relations valid for the Bernstein space \(B^p_{ }\) and their extensions to functions from larger spaces with error estimates in term of their distances from \(B^p_{ }\). J. Fourier Anal. Appl. 19, 333–375 (2013)
Butzer, P.L., Splerrstosser, W.: A sampling theorem for duration limited functions with error estimates. Inform. Contr. 34, 55–65 (1977)
Butzer, P.L., Splettstö ßer, W., Stens, R.L.: The sampling theorem and linear prediction in signal analysis. Jahresber. Deutsch. Math.-Verein. 90, 1–70 (1988)
Butzer, P.L., Stens, R.L.: Linear prediction by samples from the past. In: Advanced Topics in Shannon Sampling and Interpolation Theory, pp. 157–183 (1993)
Cantarini, M., Costarelli, D., Vinti, G.: A solution of the problem of inverse approximation for the sampling Kantorovich operators in case of Lipschitz functions. Dol. Res. Notes Approx. 13, 30–35 (2020)
Cieri, E., Costarelli, D., Fiorucci, B., Isernia, G., Seracini, M., Simonte, G., Vinti, G.: Computed tomography post-processing for abdominal aortic aneurysm lumen recognition in unenhanced exams. Ann. Vasc. Surg. 60, 407–414 (2019)
Constales, D., De Bie, H., Lian, P.: A new construction of the Clifford–Fourier kernel. J. Fourier Anal. Appl. 23(2), 462–483 (2017)
Coroianu, L., Costarelli, D., Gal, S.G., Vinti, G.: The max-product generalized sampling operators: convergence and quantitative estimates. Appl. Math. Comput. 355, 173–183 (2019)
Costarelli, D., Minotti, A.M., Vinti, G.: Approximation of discontinuous signals by sampling Kantorovich series. J. Math. Anal. Appl. 450(2), 1083–1103 (2017)
Costarelli, D., Seracini, M., Vinti, G.: A comparison between the sampling Kantorovich algorithm for digital image processing with some interpolation and quasi-interpolation methods. Appl. Math. Comput. 374, 125046 (2020)
Costarelli, D., Vinti, G.: Order of approximation for sampling Kantorovich operators. J. Integr. Equ. Appl. 26(3), 345–368 (2014)
Costarelli, D., Vinti, G.: An inverse result of approximation by sampling Kantorovich series. Proc. Edinburgh Math. Soc. 62(1), 265–280 (2019)
Costarelli, D., Vinti, G.: Inverse results of approximation and the saturation order for the sampling Kantorovich series. J. Approx. Theory 242, 64–82 (2019)
Costarelli, D., Vinti, G.: Saturation by the Fourier transform method for the sampling Kantorovich series based on bandlimited kernels. Anal. Math. Phys. 9, 2263–2280 (2019)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation, vol. 303. Springer, Berlin (1993)
Feichtinger, H., Gröchenig, K.: Theory and practice of irregular sampling. In: Benedetto, J., Frazier, M. (eds.) Wavelets: Mathematics and Applications, pp. 305–363. CRC Press Inc., London (1994)
Gonska, H.: Quantitative Korovkin-type theorems on simultaneous approximation. Math. Z. 186, 419–433 (1984)
Gonska, H., Heilmann, M., Raşa, I.: Kantorovich operators of order k. Numer. Funct. Anal. Optim. 32(7), 717–738 (2011)
Gröchenig, K.: Reconstruction algorithms in irregular sampling. Math. Comput. 59, 181–194 (1992)
Heilmann, M., Raşa, I.: \(k\)-th order Kantorovich type modification of the operators \(U_{n}^{\rho }\). J. Appl. Funct. Anal. 9(3–4), 320–334 (2014)
Kacsó, D.: Certain Bernstein–Durrmeyer Operators Preserving Linear Functions. University of Duisburg-Essen, Habilitationsschrift (2008)
Knoop, H.B., Pottinger, P.: Ein Satz vom Korovkin-Typ fur Ck-Raume. Math. Z. 148, 23–32 (1976)
Kolomoitsev, Y.S., Skopina, M.A.: Approximation by multivariate Kantorovich–Kotelnikov operators. J. Math. Anal. Appl. 456(1), 195–213 (2017)
Mastroianni, G., Themistoclakis, W.: Pointwise estimates for polynomial approximation on the semiaxis. J. Approx. Theory 162(11), 2078–2105 (2010)
Orlova, O., Tamberg, G.: On approximation properties of generalized Kantorovich-type sampling operators. J. Approx. Theory 201, 73–86 (2016)
S. Ries, R.L. Stens, Approximation by generalized sampling series. In: Constructive Theory of Functions’84, Sofia, pp. 746–756 (1984)
Sendov, B., Popov, V.: Konvergenz von Ableitungen linearer Operatoren, Handwritten German translation of notes used by the authors at the “Seminar on Interpolation and Convexity,” Cluj-Napoca, September 1–10 (1968)
Sendov, B., Popov, V.: The convergence of the derivatives of positive linear operators. C. R. Acad. Bulgare Sci. (in Russian) 22, 507–509 (1969)
Sendov, B., Popov, V.: Convergence of the derivatives of positive linear operators. B’lgar. Akad. Nauk. Otdel. Mat. Fiz. Nauk. Izv. Mat. Inst. (in Bulgarian) 11, 107–115 (1970)
Strang, G., Fix, G.: A Fourier analysis of the finite element variational method. Constr. Aspects Funct. Anal. 793–840 (1971)
Tamberg, G.: On truncation errors of some generalized Shannon sampling operators. Numer. Algorithms 55(2), 367–382 (2010)
Totik, V.: Problems and solutions concerning Kantorovich operators. J. Approx. Theory 37, 51–68 (1983)
Unser, M.: Ten good reasons for using spline wavelets. Wavel. Appl. Signal Image Process. 3169(5), 422–431 (1997)
Vinti, G., Zampogni, L.: A unifying approach to convergence of linear sampling type operators in Orlicz space. Adv. Differ. Equ. 16, 573–600 (2011)