Approximation properties of interpolation and quasi-interpolation operators
1 November 2018 — 31 October 2020
Description of the project:
Interpolation and quasi-interpolation are among the most important mathematical methods used in many branches of science and engineering. They play a crucial role as a connecting link between continuous-time and discrete-time signals. For proper application of interpolation and quasi-interpolation operators, it is very important to know the quality of approximation of functions by such operators in various settings. The main goal of this project is to study approximation properties of several classes of interpolation and quasi-interpolation operators in various function spaces including weighted Lp spaces, Sobolev spaces, Lipschitz spaces, and other important spaces of functions defined on the multivariate Euclidean space, torus, and hypercube. In particular, we plan to obtain a series of new error estimates for interpolation and quasi-interpolation operators by developing a unified approach based on Fourier transform techniques. The main attention in our research will be drawn to the development of various measures of smoothness that depending on the tasks considered (type of the operator and the function space) will provide full and adequate information about the quality of approximation of a given function by the corresponding operator. In particular, we are interested in studying properties of such objects of harmonic analysis and approximation theory as the Lebesgue constants of interpolation processes, the Fourier transform, different measures of smoothness (special moduli of smoothness and K-functionals). Special attention will be paid in our research to the anisotropic nature of the studied objects.
Publications:
- Yu. Kolomoitsev Approximation by quasi-interpolation operators and Smolyak's algorithm submitted arXiv
- Yu. Kolomoitsev, T. Lomako Asymptotics of the Lebesgue constants for bivariate approximation processes Appl. Math. Comput. 403 (2021), 126192 DOI arXiv
- Yu. Kolomoitsev, M. Skopina Uniform approximation by multivariate quasi-projection operators submitted arXiv
- Yu. Kolomoitsev, M. Skopina Approximation by multivariate quasi-projection operators and Fourier multipliers Appl. Math. Comput. 400 (2021), 125955 DOI arXiv
- Yu. Kolomoitsev, E. Liflyand Asymptotics of the Lebesgue constants for a d-dimensional simplex to appear in Proc. Amer. Math. Soc. DOI (arXiv:2004.12236)
- Yu. Kolomoitsev, J. Prestin Approximation properties of periodic multivariate quasi-interpolation operators submitted (arXiv:2002.04247)
- Yu. Kolomoitsev, S. Tikhonov Smoothness of functions vs. smoothness of approximation processes Bull. Math. Sci. 10 (2020), no. 3, 2030002 DOI (arXiv:1903.00229)
- Yu. Kolomoitsev, A. Krivoshein, M. Skopina Approximation by periodic multivariate quasi-projection operators J. Math. Anal. Appl. 489 (2020), no. 2, 124192 DOI
- Yu. Kolomoitsev, S. Tikhonov Properties of moduli of smoothness in Lp(Rd) J. Approx. Theory 257 (2020), 105423 DOI (arXiv:1907.12788)
- Yu. Kolomoitsev, M. Skopina Quasi-projection operators in the weighted Lp spaces Appl. Comput. Harmon. Anal. 52 (2021), 165-197 DOI (arXiv:1805.10536)
- Yu. Kolomoitsev, M. Skopina Approximation by sampling-type operators in Lp-spaces Math. Methods Appl. Sciences 43 (2020), no. 16, 9358-9374, DOI (arXiv:2001.08796)