Nonlinear Approximation Characteristic Of Sobolev Space With Bounded Mixed Derivative In Different Settings

The width theory, as an important research direction in the development of modern mathematics, has a close contact with the computational complexity. It can translate the question of computational complexity and optimal error bounds into calculate the width of the corresponding classes of function in different computational case settings. We investigate nonlinear approximation characteristic of multivariate Sobolev class with mixed derivative MWrd2（T）in different settings, such as worst case, average case, and probabilistic case setting. We present sharp bounds on the Kolmogorov, probabilistic andp average N widths ofMWrd2（r1T）,2, that isWhere is a subspace ofL1（Td）, in which the Fourier series is absolutely convergent in qsense, and is the Gaussian measure in theS（Tdq）, in addition, is depending only on the eigenvalues of the correlation operator of the measure (see^[22]).