Improved Sampling Theorem Of Hermite Type And The Estimation Of Multidimensional Class

Whittaker-Shannon-Kotelnikov sampling theorem is widely used to describe bandlimited function approximation, which has been a very important topic in many fields. For several years, it has expected to some aspects, such as choosing different measures of convergence, discussing the convergence of functions with derivations at the sampling sequences, researching the convergence problem in multidimensional space. A bandlimited function can be reconstructed by its sampling sequences with the first order derivatives, which is Hermite sampling theorem. In this paper, we improve the Hermite sampling theorem and further discuss reconstruction and convergence of the class with the second order derivatives at the sampling sequences in one dimensional function class space and multidimensional function class space.In the first part, let B3σ,ρ(R)(1<p<∞,σ>0) denote the bandlimited clas: that is, f(x) in BB3σ,ρ(R) is a p-integrable function and f(x) is supported in the interval [-σ,σ], where f(x) is the Fourier transform of f(x). In this part, we prove that a function in B3σ,ρ(R)(1<p<∞,σ>0) can be reconstructed in Lp(R) by its Hermite cardinal interpolation at sequences {f(kπ/σ)},{f’(kπ/σ)} and {f"(kπ/σ)} with the approach of Harmonic analysis, and then compute the order of approximation. In the second part, we extend the theorem above to multidimensional bandlimited class. Let B3σ,ρ(Rn)(1<p<∞,σ={σ1,σ2,...σn}∈R+n) denote the multidimensional bandlimited class:that is,f(x) in B3σ,ρ(Rn)(1<p<∞,σo={σ1,σ2,...σn}∈R+n) is a p-integrable function and f(x)is supported in the interval [-σ,σ]:=[-σ1,σ1]×[-σ2,σ2]×...×[-σn.σn],where f(x) is the Fourier transforrm of f(x).It shpws that a multidimensional bandlimited function in B3σ,ρ(Rn)(1<p<∞,σo={σ1,σ2,...σn}∈R+n) can be completely reconstructed in Lp(Rn))(1<p<∞)by its sampling sequences{f(kπ/σ)},{fj’(kπ/σ)},{fjj"(kπ/σ)},and {fij"(kπ/σ)},and then research its approximation,for i,j=1,2,...,,n,i≠j and kπ/σ=(k1π/σ1,k1π/σ1,...,knπ/σn),k=(k1,k2,...,kn).