Approximation properties of interpolation and quasi-interpolation operators

(2018-2021)

DFG project KO 5804/1

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.

Principal investigator: Yurii Kolomoitsev
Staff: Tetiana Lomako

Corresponding publications

	Sharp Lp-error estimates for sampling operators Yurii Kolomoitsev, Tetiana Lomako J. Approx. Theory, Available online 26 June 2023, 105941, 2023, preprint available at arXiv.
	Sparse grid approximation in weighted Wiener spaces Yurii Kolomoitsev, Tetiana Lomako, Sergey Tikhonov J. Fourier Anal. Appl. 29, 19, 2023, preprint available at arXiv.
	Uniform approximation by multivariate quasi-projection operators Yurii Kolomoitsev, Maria Skopina Anal. Math. Phys. 12, no. 2, Paper No. 68, 2022, preprint available at arXiv.
	Approximation by quasi-interpolation operators and Smolyak's algorithm Yurii Kolomoitsev J. of Complexity 69, 101601, 2022, preprint available at arXiv.
	Approximation properties of periodic multivariate quasi-interpolation operators Yurii Kolomoitsev, Jürgen Prestin J. Approx. Theory 270, 105631, 2021, preprint available at arXiv.
	Asymptotics of the Lebesgue constants for bivariate approximation processes Yurii Kolomoitsev, Tetiana Lomako Appl. Math. Comput. 403, 126192, 2021, preprint available at arXiv.
	Asymptotics of the Lebesgue constants for a d-dimensional simplex Yurii Kolomoitsev, Elijah Liflyand Proc. Amer. Math. Soc. 149, 2911-2926, 2021, preprint available at arXiv.
	Quasi-projection operators in the weighted Lp spaces Yurii Kolomoitsev, Maria Skopina Appl. Comput. Harmon. Anal. 52, 165-197, 2021, preprint available at arXiv.
	Approximation by multivariate quasi-projection operators and Fourier multipliers Yurii Kolomoitsev, Maria Skopina Appl. Math. Comput. 400, 125955, 2021, preprint available at arXiv.
	Smoothness of functions vs. smoothness of approximation processes Yurii Kolomoitsev, Sergey Tikhonov Bull. Math. Sci. 10, no. 3, 2030002, 2020, preprint available at arXiv.
	Approximation by periodic multivariate quasi-projection operators Yurii Kolomoitsev, Aleksandr Krivoshein, Maria Skopina J. Math. Anal. Appl. 489, no. 2, 124192, 2020, preprint available at arXiv.
	Properties of moduli of smoothness in Lp(Rd) Yurii Kolomoitsev, Sergey Tikhonov J. Approx. Theory 257, 105423, 2020, preprint available at arXiv.
	Approximation by sampling-type operators in Lp-spaces Yurii Kolomoitsev, Maria Skopina Math. Methods Appl. Sciences 43, no. 16, 9358-9374, 2020, preprint available at arXiv.