Error Analysis For Shannon Series Approximation With Commonly Smooth Sobolev Classes

The famous Shannon sampling theorem shows that every signal function f=B_?,2 could be perfectly reconstructed from the sampling value of its finte points. In practical application, the signal may be non-band-limited, the property of measuring apparatus and precision of limit,we get sampling values which are often not accurate at this point but its average values near by. Thus all sorts of errors appear when Shannon sampling series being used to reconstruct a signal. Since 1948, Shannon proposed classic Whittakier-Kotelnikov-Shannon theorem, a lot of authors have studied this problem in the field of math and engineering.We study errors for commonly smooth function classes on space W_P^A?R^d? by means of the average sampling under decay condition in this paper. Firstly, we got the uniform bound of aliasing errors where ?=?₁...?_d Secondly, generally we can only get a finite number of sampling values,so we further analyze the uniform boundary of truncation errors in this paper. Finally, we consider that sampling values is a linear functionals and its integer translation, also we study its error...