Vector valued Fourier hyperfunctions
Yoshifumi Ito · Kyoto journal of mathematics · 1992
In this paper, we study H-valued F o u rie r h y p e rfu n c tio n s.H ere H i s a complex H ilbert space which is not necessarily separable.W e realize H -valued Fourier hyperfunctions a s elements o f th e dual space of the space of all rapidly decreasing H-valued real analytic functions o r a s -boundary values" of slowly increasing H-valued holo- morphic functions a n d then show that they a r e t h e tw ofold realization o f t h e same H -valued F o u rie r h y p e rfu n c tio n s.W hen we realize H-valued Fourier layperfunctions using H-valued analytic functionals, our treatment is m o re general than other works in th e p o in t th at te s t functions a r e vector valued.T h is idea is also used in Bruning-N agam achi [2], which I knew after submission o f th e present paper.Next, we define th e F o u rie r tra n s fo rm a tio n o f H-valued Fourier hyperfunctions a n d s h o w th a t t h e space o f H-valued Fourier hyperfunctions on the entire space is stable under th e F o u rie r tra n sfo rm a tio n .Further we prove th e Paley-Wiener theorem fo r H-valued Fourier hyperfunctions.Until now, many mathematicians have studied (vector valued) Fourier hyperfunctions : S ato