A SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES

Snježana Maksimović, Svetlana Mincheva-Kamińska, Stevan Pilipović, Petar Sokoloski

Abstract: We introduce and investigate two types of the space ${𝒰}^{*}$ of $s$-ultradistributions meant as equivalence classes of suitably defined fundamental sequences of smooth functions; we prove the existence of an isomorphism between ${𝒰}^{*}$ and the respective space ${𝒟}^{\text{'}*}$ of ultradistributions: of Beurling type if $*=\left(p{!}^t\right)$ and of Roumieu type if $*=\left\{p{!}^t\right\}$. We also study the spaces ${𝒯}^{*}$ and ${\widetilde{𝒯}}^{*}$ of $t$-ultradistributions and $\tilde{t}$-ultradistributions, respectively, and show that these spaces are isomorphic with the space ${𝒮}^{\text{'}*}$ of tempered ultradistributions both in the Beurling and the Roumieu case.

Keywords: fundamental sequences, Hermite expansions

Classification (MSC2000): 46F05; 46F10

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