A Theory of Summability on a Space of Generalized Functions

Abstract

A theory of summability of orthonormal sets is introduced in multinormed spaces. The approach which is presented caters for infinite sets , where the index set may be uncountable, and is applied to obtain convergence results in appropriate spaces of test functions and corresponding spaces of generalized functions. These spaces are constructed in a systematic manner that relies heavily on properties of orthonormal bases in Hilbert spaces. A space of almost-periodic generalized functions, in which each generalized function can be expanded in terms of an uncountable basis of exponential functions, is obtained as a special case of our theory.