A discretization of the continuous fourier transform

Summary

A finite representation of the continuous Fourier transform is obtained in a linear space of dimensionN for the Hermite functions by using some properties of the Hermite polynomialsH _N (x) and its zeros. TheN-dimensional vectors representing a Hermite function and its Fourier-trnasformed function converge to their exact continuous values whenN→∞. The genericN×N matrix representing the kernel of the continuous transform exp [−ipx], preserves the conspicuous properties of this function, and their entries approach the values that this kernel takes atp=x _j andx=x _k , wherex _j andx _k are two zeros ofH _N (x) whenN→∞. These properties lead to propose this kind of matrices as finite representations of the kernel of the continuous Fourier transform for functions other than the Hermite ones, yielding certain quadrature formula for this transform. Some numerical examples of this application are shown and they are compared with those obtained through standard procedures.