Multivariate Sample Theorem And N-width

Whittaker-Shannon-Kotelnikov Sampling theorem is one of the important results of numerical analysis,data processing and communication theory,one of the theorem and the sampling theorem in the theory of interpolation is the most basic.It is widely used to describe bandlimited function approximation.Since 1948,Shannon introduced the sampling theorem in communication engineering,it is widely used in communication engineering,and by many scholars in the field of electronic engineering and Mathematics.It is shown that every finite information function in the upper band can be completely reconstructed from the function value of the finite information function.Many scholars try to extend this theorem from two directions of pure mathematics and applied mathematics,and create a large number of branches of this theory.For example:the function of the study is not limited,can be used with Whittaker series with finite function approximation,in the electronic engineering,said its approximation error for confusion error.Another direction is to increase the sample point at the first derivative value,to discuss the Hermite type sample theorem.Functions defined on the R,if its Fourier transform has a finite compact support,it is called bandlimited?B4?,p(Rn)(1<P<+?),?={?1,?2}?Rn)denotes the bandlimited class.In this paper,we prove that a function in B4?,p(Rn)can be reconstructed in Lp(Rn)by its Hermite cardinal interpolation at sequences{f{k?/?)},{f1'(k?/?)},{f2'{k?/?)},�,{fn'(k?/?)},{f11"(k?/?)},{f12"(k?/?)},�{fnn"(k?/?)},{f112"(k?/?)}�,{f122"(k?/?)}�,{f111"'(k?/?)}{f222"'{f?/?)},�,{fnnn""(k?/?)}k?Zn with the approach of Harmonic analysis.