Approximation of Gaussian by Scaling Functions and Biorthogonal Scaling Polynomials

The derivatives of the Gaussian function, G(x) = 1 √ 2π e−x 2/2, produce the Hermite polynomials by the relation, (−1)mG(m)(x) = Hm(x)G(x), m = 0, 1, . . . , where Hm(x) are Hermite polynomials of degree m. The orthonormal property of the Hermite polynomials, 1 m! ∫∞ −∞Hm(x)Hn(x)G(x)dx = δmn, can be considered as a biorthogonal relation between the derivatives of the Gaussian, {(−1)nG(n) : n = 0, 1, . . .}, and the Hermite polynomials, {Hm m! : m = 0, 1, . . .}. These relationships between the Gaussian and the Hermite polynomials are useful in linear scale-space analysis and applications to human and machine vision and image processing. The main objective of this paper is to extend these properties to a family of scaling functions that approximate the Gaussian function and to construct a family of Appell sequences of “scaling biorthogonal polynomials” that approximate the Hermite polynomials.