Pseudo Width And Entropy Number Of Multivariate Sobolev Space With Mixed Derivatives In Probabilistic Setting

Sobolev space is an important function space which is used widely in physics,mechanics,computational mathematics and partial differential equations.The pseudo-width and entropy number can be good methods to measure the uncertain characteristics of the information flow in the system.In particular,the entropy number,which is closely related to the information-based radius in the system,so that the pseudo-width and entropy number are widely used in information theory and learning theory.Therefore,studying the pseudo-width and entropy of Sobolev space has important practical background and theoretical significance.In this thesis,we study the pseudo-width and entropy number of the multivariate Sobolev space with mixed derivatives in probabilistic setting.By using the method of discretization,we first consider discretization theorems to estimate the upper and lower bounds of the pseudo-width and entropy,which are crucial to convert them into the corresponding pseudo-width and entropy number in the finite-dimensional space.And the exact asymptotic orders of the pseudo-width and entropy number are estimated,i.e.(?)...