Interpolation Theorems Of Higher Order Derivatives On Multivariate Function Spaces

Whittaker-Shannon-Kotelnikov Sampling theorem as one of the important basic theories of communication engineering and data processing is widely used to describe the approximation of bandlimited functions. It states that every function which is bandlimited to [-σ,σ] can be reconstructed from its values at that equidistant discrete set of points.Since its introduction in communication engineering by Shannon in 1948,this theorem is widely used in communication engineering.It has received the attention and research of many scholars at home and abroad.They are basically in two directions of the pure mathematics and Applied Mathematics to expand the study of the theorem, and thus created a large number of branches of this theory.One direction is that when the function of the study is not bandlimited, the study found that we can use the bandlimited functions to approximate with the Whittaker series, and the approximation error is called the aliasing error. Another direction is to increase the First order derivative at the sample point[1],and make a further discussion of Hermite type sample theorem.Based on the [1], [2] is extended the theorem to the multi-dimensional space from one dimensional space, and further improved the multivariate Hermite type sample theorem. In this paper, the three-order derivative of the sample points is increased, and the function reconstruction of the two-element integrable-band finite-function set is discussed.Functions defined on the R, if its Fourier transform has a finite compact support, it is called bandlimited.B4σ,ρ(R2)(1<P<+∞),σ={σ1,σ2}∈R2) denotes the set of all those functions from LP[R2), whose Fourier transformation are bandlimited to [-σ, σ]:=[-σ1,σ1]×[-σ2,σ2].That is:(?)f(x)B4σ,ρ(R2),f(x)is a P-integrable function, f(x) is supported in the interval [-σ,σ], where f(x) is the Fourier transform of f(x).In this paper, we prove that a function in B4σ,ρ(R2) can be reconstructed in Lp[R2)by its Hermite cardinal interpolation at sequences{f(kπ/σ)}, {fj’(kπ/σ)},{fij"(kπ/σ)},{fjj"(kπ/σ)},{fjjj’’’(kπ/σ)},and {fjji’’’(kπ/σ)},k∈Z2 with the approach of Harmonic analysis.