• 대한수학회보
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• 제38권1호
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• pp.7-16
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• 2001

We study the Fourier transformation on the Gelfand-Bruhat space of type S and characterize this space by means of Fourier transform on a locally compact abelian group G. Also, we extend Bochner-Schwartz theorem to the dual space of the Gelfand-Bruhat space and the space of Fourier hyperfunctions on G. respectively.

• 대한수학회논문집
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• 제14권2호
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• pp.329-336
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• 1999

We construct the Gelfand triple based on the space \ulcorner, introduced by Sato and di Silva, of analytic and exponentially decreasing function. This space denoted by(\ulcorner) of white noise test functionals are defined by the operator cosh \ulcorner, A=-(\ulcorner)\ulcorner+x\ulcorner+1. We also note that many properties like generalizations of the Paley-Wiener theorem and the Bochner-Schwartz theorem hold in this space as in the space of Hida distributions.

• 대한수학회보
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• 제33권4호
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• pp.545-551
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• 1996

It is well known in the theory of distributions and proved in [GS, S] that $$ (i) (Bochner-Schwartz) Every positive definite (tempered) distribution is the Fourier transform of a positive tempered measure \mu. $$ $$ (ii) (Schwartz kernel theorem) Let B(\varphi, \psi) be a bilinear distribution. Then for some u \in D'(R^n \times R^n) B(\varphi, \psi) = u(\varphi(x)\bar{\psi}(y)) for every \varphi, \psi \in C_c^\infty. $$ $$ (iii) A translation invariant positive definite bilinear distribution B(\varphi, \psi) is of the form B(\varphi, \psi) = \smallint \varphi(x)\psi(x) d\mu(x) for every \varphi, \psi \in C_c^\infty (R^n), where \mu is a positive tempered measure.