Applications of Hermite Function

In mathematics, the Hermite polynomials are classical orthogonal polynomials sequence that arise in probability, numerical analysis as Gaussian quadrature. They are also used in the systems of theory in connection with non-linear operations on Gaussian noise and other sciences. Our aim in this paper is to estimated the incurred error when hn(x) is replaced by an asymptotic formula, then found for the involved error. This bound is then used to study the accuracy of certain approximation to Hermite expansions and Fourier transforms. Relevant applications of the scheme in different contexts are also included. We successively estimate |fn(x), Gn(x,y)|, which have applications of the later estimation to Hermite expansions. We briefly discuss density estimations and Fourier transforms. The results of thenumerical examples are presented, and compared with analytical solutions to confirm the accuracy of presented scheme.