Interpolation Theorem With Three Derivative

The classical Whittaker-Shannon-Kotelnikov sampling theorem is suggested by Shannonin 1948.The approximation problem with finite function is discussed by the theorem,it has been applied in communication. For several years,it has been expected to some aspects by scholars,such as choosing different measures of convergence,discussing the convergence of functions with derivatives at the sampling sequences.In this paper, we discuss the reconstruction and convergence of the class with the third order derivatives at the sampling sequences in one dimensional function class space and it’s inverse theorem.In the first part,let B4σ,p(R)(1<p<+∞,σ>0)denote the bandlimited class,that is,f(x) in B4σ,p(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 paper,we prove that a function in B4σ,p(R)(1<p<+∞,σ>0)can be reconstructed Lp(R) by its Hermite cardinal interpolation at sequences f(kπ/σ)}k∈Z,{f’(kπ/σ)}k∈Z, {f"(kπ/σ)}k∈Zznd {f’’’(kπ/σ)}k∈Z with the approach of Harmonic analysis. In the second part,we discuss the inverse theorem of the above theorem,that is,let y={yk}k∈Z,y’={y’k}k∈Z,y"={y"k}k∈Z,y,y’,y",y’’’∈lp,1<p<∞,then a unique g(x) in B4σ,p(R) is proved satisfying the conditions.If a g(x) satisfying g(kπ/σ)=yk,g’(kπ/σ)=y’k,g"(kπ/σ)=y"k,g’’’(kπ/σ)=y’’’k,then y={yk}k∈Z,y’={y’k}k∈Z,y’’={y’’k}k∈Z,y,y’,y",y’’’∈lp,1<p<∞.In the third part, we compute the order of approximation.