What is the Fourier transform of a functional?

Question

Generally Fourier transforms are defined as follows:

$$ F(s)=\int dxf(x)e^{-isx}.$$

Similarly, is it possible to define Fourier transform for a functional $\psi[\phi(x)]$ ?

Answer 1

The Fourier transform can be defined for all linear and continuous functionals on the Schwartz space $\mathscr{S}(G)$, whenever $G$ is a locally compact and abelian group. The definition is by duality: $$\hat{T}(f)=T(\hat{f})\; ,\; f\in\mathscr{S}(G)$$ where $T\in \mathscr{S}'(G)$, and $$\hat{f}(\chi)=\int_{G} \chi(g) f(g)\mathrm{d}h(g)\; ,$$ $\chi\in \hat{G}$ a character and $h$ the Haar measure.

Another (related) notion of Fourier transform can be given for finite countably additive measures in the algebra of cylinders $C(X,\Gamma)$ (for a space $X$ with cylinders generated by the vector space $\Gamma\subset \mathbb{R}^X$) that are countably additive when restricted to the $\sigma$-algebra generated by any finite dimensional subspace of generators (let us call the set of them $M_{\mathrm{cyl}}(X,\Gamma)$). Essentially, $$\hat{\mu}(\gamma)=\int_{\mathbb{R}}e^{it}\mathrm{d}\mu_{\gamma}(t)\;,\; \gamma\in \Gamma\; ,$$ where $\mu_{\gamma}=\gamma\, _* \, \mu$ is the finite Radon measure on $\mathbb{R}$ obtained by the pushforward of the finitely additive measure $\mu$ by the function $\gamma$. In particular, the Fourier transform can be defined for any Radon probability measure on $X$, if $X$ is a topological vector space.

In both cases the Fourier transform is a bijection. In the first case, it is a bijection of $\mathscr{S}'(G)$ onto $\mathscr{S}'(\hat{G})$. In the second case, it is a bijection from $M_{\mathrm{cyl}}(X,\Gamma)$ onto $ACP(\Gamma)$, where the latter is the space of functions from $\Gamma$ to $\mathbb{C}$ that are of positive type ($\sum_{i,j\in \mathrm{Fin}}f(\gamma_{i}-\gamma{j})\bar{z}_{j}z_i \geq 0$ for any finite combination with complex coefficients $\{z_{i}\}$) and almost continuous (i.e. continuous when restricted to any finite dimensional subspace).

It is not possible however to define a Fourier transform and a Fourier space that generalize the standard concepts for more general set of functionals (as far as I know).