Notes on generalized Fourier series with application to gravitational field determination

Abstract

Let{(}φn(x)} be an orthonormal system in the set of Lebesgue square integrable functions L². Let f𝜖L². The generalized Fourier series of f with respect to {(}φn(x)} is the series ∑n=0∞ (f, φn) φn(x), where (f, φn) is the inner product of the functions f an φn. The e existence of a complete orthonormal system in L² is proven. Conditions for convergence of the generalized Fourier series are presented. A discussion of orthogonal polynomials with special emphasis on the Jacobi polynomial systems is presented. A least squares, differential correction, discrete observation procedure is employed to solve the potential equation with boundary conditions in tenns of three special Jacobi systems.