On Equivalence of Certain Types of Series of Orthonormal Functions

One approach to the problem lies in showing that expansion (1) is equivalent to another expansion, usually a Fourier series, whose behavior is known. In the case of orthogonal polynomials, where un{x) =x ~, the writer was able to show, under conditions not too restrictive, that the expansions of a function in terms of the two sets of orthogonal polynomials corresponding, respectively, to different weight functions converge to the same value or diverge together. It is the purpose of this note to point out that similar results can be obtained for systems of orthonormal functions constructed from a set [un(x) ] which has the property that the product of any two members