A fast converging sampling operator

Year 2022, Volume: 5 Issue: 4, 190 - 201, 01.12.2022

Borislav Draganov

https://doi.org/10.33205/cma.1172005

Cited By: 3

Abstract

We construct a sampling operator with the property that the smoother a function is, the faster its approximation is. We establish a direct estimate and a weak converse estimate of its rate of approximation in the uniform norm by means of a modulus of smoothness and a $K$-functional. The case of weighted approximation is also considered. The weights are positive and power-type with non-positive exponents at infinity. This sampling operator preserves every algebraic polynomial.

Keywords

Sampling operator, sampling series, weighted approximation, direct estimate, weak converse estimate, modulus of smoothness, $K$-functional

References

• T. Acar, O. Alagöz, A. Aral, D. Costarelli, M. Turgay and G. Vinti: Convergence of generalized sampling series in weighted spaces, Demonstr. Math., 55 (2022), 153–162.
• T. Acar, O. Alagöz, A. Aral, D. Costarelli, M. Turgay and G. Vinti: Approximation by sampling Kantorovich series in weighted spaces of functions, Turkish J. Math., 46 (7), Article 7.
• O. Alagöz, M. Turgay, T. Acar and M. Parlak: Approximation by sampling Durrmeyer operators in weighted space of functions, Numer. Funct. Anal. Optim., 43 (2022), 1223-1239.
• A. Aral, T. Acar and S. Kursun: Generalized Kantorovich forms of exponential sampling series, Anal. Math. Phys., 12 (2022), 50.
• S. Artamonov, K. V. Runovski and H. J. Schmeisser: Approximation by families of generalized sampling series, realizations of generalized K-functionals and generalized moduli of smoothness, J. Math. Anal. Appl., 489 (2020), 124138.
• C. Bardaro,I. Mantellini: Asymptotic formulae for linear combinations of generalized sampling operators, Z. Anal. Anwend., 32 (2013), 279-298.
• H. Berens, G. G. Lorentz: Inverse theorems for Bernstein polynomials, Indiana Univ. Math. J., 21 (1972), 693-708.
• P. L. Butzer, R. J. Nessel: Fourier Analysis and Approximation, Birkhäser Verlag, Basel (1971).
• P. L. Butzer, S. Ries and R. L. Stens: Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx. Theory, 50 (1987), 25–39.
• P. L. Butzer, R. L. Stens: Linear prediction by samples from the past, Advanced Topics in Shannon Sampling and Interpolation Theory (1993), 157-183.
• R. A. DeVore, G. G. Lorentz: Constructive Approximation, Springer-Verlag, Berlin (1993).
• Z. Ditzian, V. Totik: Moduli of Smoothness, Springer-Verlag, New York (1987).
• Yu. Kolomoitsev, M. Skopina: Quasi-projection operators in weighted $L_p$ spaces, Appl. Comput. Harmon. Anal., 52 (2021), 165-197.
• Yu. Kolomoitsev, M. Skopina: Approximation by multivariate quasi-projection operators and Fourier multipliers, Appl. Math. Comput., 400 (2021), 125955.
• Yu. Kolomoitsev, M. Skopina: Uniform approximation by multivariate quasi-projection operators, Anal. Math. Phys., 12 (2022), 68.
• M. A. Pinsky: Introduction to Fourier Analysis and Wavelets, American Mathematical Society (2009).
• S. Ries, R. L. Stens: Approximation by generalized sampling series, Procedings of the International Conference “Constructive Theory of Functions”, Varna (Bulgaria) (1984) (B. Sendov, P. Petrushev, R. Maleev, S. Tashev, Eds.), Bulgarian Academy of Sciences, Sofia (1984), 746-756.