Approximation properties of interpolation and quasi-interpolation operators

插值和准插值是在科学和工程的许多分支中使用的最重要的数学方法。它们作为连续时间和离散时间信号之间的连接纽带起着至关重要的作用。为了正确地应用插值算子和拟插值算子，了解这些算子在各种情况下对函数的逼近质量是非常重要的。本课题的主要目的是研究几种插值算子和拟插值算子在各种函数空间中的逼近性质，这些函数空间包括加权Lp空间、Sobolev空间、Lipschitz空间以及其他在多元欧几里德空间、环面和超立方体上定义的函数的重要空间。特别是，我们计划通过开发基于傅里叶乘法器和傅里叶变换技术的统一方法来获得一系列新的插值算子和准插值算子的误差估计。在我们的研究中，主要的注意力将被吸引到各种平滑度量的发展，这取决于所考虑的任务（算子的类型和函数空间），将提供关于相应算子对给定函数的近似质量的充分和充分的信息。特别是，我们感兴趣的是研究谐波分析和近似理论对象的性质，如Smolyak算法，稀疏网格，littlewood - paly型分解，插值过程的Lebesgue常数，傅里叶变换，不同的平滑度量（平滑的特殊模和k泛函）。在我们的研究中将特别注意研究对象的各向异性。

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.