Frame Expansions of Test Functions, Tempered Distributions, and Ultradistributions

Abstract

The paper is devoted to frame expansions in Fréchet spaces. First we review some results which concern series expansions in general Fréchet spaces via Fréchet and General Fréchet frames. Then we present some new results on series expansions of tempered distributions and ultradistributions, and the corresponding test functions, via localized frames and coefficients in appropriate sequence spaces.

Localization of Fréchet frames and expansion of generalized functions

“…As noticed in [32], having in mind the known expansions of tempered distributions (S(R + )) ′ [35,15] and Beurling ultradistributions (G α α (R + )) ′ [16,23,22], and their test spaces, by the use of the Laguerre orthonormal basis l n , n ∈ N, and validity of the corresponding properties P (ln) , we can transfer the above results to the mentioned classes of distributions and ultradistributions over R + . Remark 6.4.…”