Abstract
In this paper, we define generalized Fourier-Hermite functionals on a function space <TEX>$C_{a,b}[0,\;T]$</TEX> to obtain a complete orthonormal set in <TEX>$L_2(C_{a,b}[0,\;T])$</TEX> where <TEX>$C_{a,b}[0,\;T]$</TEX> is a very general function space. We then proceed to give a necessary and sufficient condition that a functional F in <TEX>$L_2(C_{a,b}[0,\;T])$</TEX> has a generalized Fourier-Wiener function space transform <TEX>${\cal{F}}_{\sqrt{2},i}(F)$</TEX> also belonging to <TEX>$L_2(C_{a,b}[0,\;T])$</TEX>.
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