Improved Interpolation Method And Best Approximation

In the field of signal processing,it is a hot research direction to use the Whittaker-Shannon-Kotelnikov Sample Theorem to restore the class of functions with finite signals and to seek the optimal restoration method.In the past 50 years,experts and scholars have tried to generalize and extend the theorem from many angles.Generally speaking,there are three main research directions:the first is to study the optimal recovery problem under different spatial measurement scales;Secondly,the convergence of the sample theorem is studied not only by using the information of the original function,but also by using the information of the first derivative and even the information of the higher derivative.The third is to generalize the theorem in higher dimensional space.In the above research,constructing better and more effective interpolation method is the key to the problem.On the basis of previous researches,this paper constructs a new interpolation method based on sin5x/x2,sin5x/x3,sin5x/x4,sin5x/x5funtion,to solve the problem of restoring interpolated nodes with high-dimensional derivative information.It is proved that any function f(x)with a finite set of functions B5?,p(1<p<?,?>0)can be completely reconstructed at Lp(R)(1<p<?)scale from the values of its original function and the values of its first,second and third derivatives on some sample sequences.The main results are as follows:Setf(x)?B5?,p,1<p<? there is a)f(x)=(?){[1/?5(x-k?/?)5+5/6?3(x-k?/?)3]f(k?/?)+[1/?4(x-k?/?)4+5/6?2(x-k?/?)2]f'(x-k?/?)+1/2?3(x-k?/?)3f"(k?/?)+1/6?2(x-k?/?)2f'"(k?/?)}sin5?(x-k?/?).And the series converges uniformly on the right;(b)?f(x)-(?){[1/?5(x-k?/?)5+5/6?3(x-k?/?)3]f(k?/?)[1/?4(x-k?/?)4+5/6?2(x-k?/?)2]f'(x-k?/?)+1/2?3(x-k?/?)3f"(k?/?)+1/6?2(x-k?/?)2f'"(k?/?)}sin5?(x-k?/?)?p(R)?0,N??.