Approximation properties of interpolation and quasi-interpolation operators

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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 multipliers and 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 Smolyak algorithm, sparse grids, Littlewood–Paley-type decompositions, 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.