Fourier and Fourier-Mehler Transforms

Abstract

Recall that in Section 4.A we have the following J- and f-transformations defined on the space (f)* of generalized white noise functionals: for Ф∈(f)* and ξ∈f(ℝ),

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeqabiqaaa % qaamaabmaabaGaamivaiabfA6agbGaayjkaiaawMcaamaabmaabaGa % eqOVdGhacaGLOaGaayzkaaGaeyypa0JaeSOEIaNafuOPdyKbaebaca % GGSaGaamyzamaaCaaaleqabaGaamyAaiablQNiWjabgwSixlaacYca % cqaH+oaEcqWI7jIzaaGccqWI7jIzcaGGSaaabaWaaeWaaeaacaWGSb % GaeuOPdyeacaGLOaGaayzkaaWaaeWaaeaacqaH+oaEaiaawIcacaGL % PaaacqGH9aqpcqWI6jcCcuqHMoGrgaqeaiaacQdacaWGLbWaaWbaaS % qabeaacqWI6jcCcqGHflY1caGGSaGaeqOVdGNaeS4EIygaaOGaaiOo % aiablUNiMjabg2da9iaadwgadaahaaWcbeqaaiabgkHiTmaalaaaba % GaaGymaaqaaiaaikdaaaGaaiiFaiabe67a4jaacYhadaqhaaadbaGa % aGOmaaqaaiaaikdaaaaaaOGaeSOEIaNafuOPdyKbaebacaGGSaGaam % yzamaaCaaaleqabaGaeSOEIaNaeyyXICTaaiilaiabe67a4jablUNi % MbaakiablUNiMbaaaaa!7E69! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\begin{array}{*{20}{c}} {\left( {T\Phi } \right)\left( \xi \right) = \prec \bar \Phi ,{e^{i \prec \cdot ,\xi \succ }} \succ ,} \\ {\left( {l\Phi } \right)\left( \xi \right) = \prec \bar \Phi :{e^{ \prec \cdot ,\xi \succ }}: \succ = {e^{ - \frac{1}{2}|\xi |_2^2}} \prec \bar \Phi ,{e^{ \prec \cdot ,\xi \succ }} \succ } \end{array}$$