Smooth linear functionals on the white-noise topology

Fix the real orthonormal wavelet basis and level weights $a_l$ of Logarithmic negative Sobolev spaces for white noise. For $0\le s<S$, define the basis Sobolev space $H_2^s=\{g\in L^2([0,1]):{\sum }_{l,k}{a}_l^{2s}\mathrm{\mid }⟨g,{\psi }_{lk}⟩{\mathrm{\mid }}^2<\mathrm{\infty }\}\mathrm{.}$ For $g\in H_2^s$, $s>1\mathrm{/}2$, and $f\in L^2([0,1])$, define $L_g(f)=⟨f,g{⟩}_{L^2}$. For $x=(x_{lk})\in H(\delta )$, the intended extension is the coefficient pairing $L_g(x)={\sum }_{l,k}{x}_{lk}⟨g,{\psi }_{lk}⟩$; Smooth functionals extend continuously to the white-noise topology ↗ proves that this series is absolutely convergent and defines a continuous linear functional on $H(\delta )$ for every $\delta >0$.

In the white-noise experiment $X^{(n)}=f_0+n^{-1\mathrm{/}2}W$, fix $\delta >1\mathrm{/}2$ so that the observation is $H(\delta )$-valued; $L_g(X^{(n)})$ means this continuous pairing with the $H(\delta )$-valued observation. It does not assert that $X^{(n)}$ is an $L^2$ function. The representer is nonzero when $g\mathrm{\ne }0$; the corresponding white-noise variance is $\mathrm{\parallel }g{\mathrm{\parallel }}_2^2$.