Translation invariant and positive definite bilinear fourier hyperfunctions

  • Jaeyoung Chung (Department of Mathematics, Kunsan National University, Kunsan 573-360) ;
  • Chung, Soon-Yeong (Department of Mathematics, Sogang University, Seoul 121-742) ;
  • Kim, Dohan (Department of Mathematics, Seoul National University, Seoul 151-742)
  • Published : 1996.11.01

Abstract

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.

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