Hermitean Ultradistributions

Luís Camilo do Canto de Loura and Francisco Caetano di Sigmaringen dos Santos Viegas

Abstract: In this paper we introduce the hermitean ultradistributions by a duality argument. The method we use is exactly the same introduced by Schwartz for the distribution theory. Our space of hermitean ultradistributions contains all the Schwartz tempered distributions but is not related with the space $\calc{D}'$ of all distributions. In our space $\calc{G}'$ of hermitean ultradistributions the derivatives are linear continuous operators and the Fourier transform is a vectorial and topological isomorphism. The construction of $\calc{G}'$ is based on the construction of a space of functions $\calc{G}$, strictly included in the Schwartz space $\calc{S}$, but still dense in $\calc{S}$. This space $\calc{G}$ is an inductive limit of finite-dimensional vector spaces. Finally we give a sequential representation of our hermitean ultradistributions and we apply the theory to the series of multipoles used by physicists.

Keywords: Distributions; ultradistributions; Fourier transform; multipole series.

Classification (MSC2000): 46

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