Entropy Number Of Generalized Sobolev Space In Different Setting

Sobolev space is a kind of important function space,and its approximation properties,such as width,entropy number and term approximation,have been deeply studied.In the paper,we define the generalized Sobolev space Spr by Fourier coefficient sequences in the classical Lebesgue spaces L1,we discuss the entropy number of the generalized Sobolev space in worst-case setting and probabilistic setting,and the exact asymptotic order is obtained which is based on distribution method.That is:1.if 1?q<p<?,r>1/q-1/p,then(?)2.if 1?q?2,r>1/q-1/2,??(0,1/2],then(?)Where,BSpr is a unit ball of Spr.