A Theory of Summability on a Space of Generalized Functions

Abstract EN

A theory of summability of orthonormal sets is introduced in multinormed spaces.

The approach which is presented caters for infinite sets {ϕj}j∈J, where the index set J 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.