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Functions with given moments and weight functions for orthogonal polynomials. (English) Zbl 0777.44003

The author gives a constructive technique to find smooth functions with given moments. Earlier in the well-known moment problem it was guaranteed that for every sequence of complex numbers there is a function of bounded variation with the given moments in these numbers.

The author shows how these weight functions are found. He uses the technique of Fourier and Hankel transform in Schwartz spaces to establish the respective functions with the given moments. Namely, he finds the functions for the classical orthogonal polynomials: the Bessel polynomials, the Hermite polynomials, the Laguerre and generalized Laguerre polynomials, and the Jacobi polynomials.

44A60Moment problems (integral transforms)
33C45Orthogonal polynomials and functions of hypergeometric type