Approximation properties of interpolation and quasi-interpolation operators

插值和拟插值是科学和工程中最重要的数学方法之一。它们作为连续时间和离散时间信号之间的连接纽带发挥着至关重要的作用。为了更好地应用插值和拟插值算子，了解这类算子在各种情况下对函数的逼近性质是非常重要的，本项目的主要目的是研究几类插值和拟插值算子在各种函数空间中的逼近性质，包括加权Lp空间，Sobolev空间，Lipschitz空间，以及其他重要的定义在多元欧几里得空间、环面和超立方体上的函数空间。特别是，我们计划获得一系列新的误差估计的插值和拟插值算子通过开发一个统一的方法的基础上傅立叶乘法器和傅立叶变换技术。在我们的研究的主要注意力将提请发展的各种措施的顺利，这取决于所考虑的任务（类型的运营商和功能空间）将提供充分和足够的信息的质量近似一个给定的功能由相应的运营商。特别是，我们有兴趣研究的性质等对象的谐波分析和逼近理论的Schwarak算法，稀疏网格，Littlewood-Paley型分解，勒贝格常数的插值过程中，傅立叶变换，不同的措施顺利（特别模的顺利和K-泛函）。在我们的研究中将特别注意研究对象的各向异性性质。

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.