The Entropy Numbers Of Diagonal Operators In Different Setting

The width and entropy numbers are first proposed for the problem of the approximation in the 20 th century by Kolmogorov,and immediately become hot spot in application area as well as in the field of theory.As we know that the width is considered the approximation problem from algebra,whereas the entropy numbers is concerned with the approximation problem from the perspective of geometry.So entropy numbers together with the width comes into being two aspects of the approximation problem,and widely used in engineering field such as in signal processing,physical prospecting,artificial intelligence,cybernetics,scientific calculation and so on.Therefore,the entropy numbers of important function classes is widely and deeply studied.Researchers find that the basic method for investigate entropy numbers is to transform the entropy numbers of function class into the entropy numbers of the sequence space.So it's obvious that the entropy numbers of the identity operator has become indispensable and necessary.In this paper,we study the entropy numbers of the identity operator for infinite dimensional space in the worst setting and probabilistic setting,and estimate its exact asymptotic order.That is:Where,I_p,q denotes the identity operator from a subspace l_p,r of l_p to l_q.