# Fourier Transform of Measures on Hilbert, Banach, or general Topological Vector Space

After reading this question on MathOverflow I began to wonder to what extent Fourier transforms can be defined on Hilbert spaces, Banach spaces or their duals with the weak-* topology (or even a general TVS). In the infinite-dimensional setting we can lose local compactness and so classical harmonic analysis is inadequate.

My guess is that for a sufficiently nice topological vector space $E$, it's Pontryagin dual $\hat E$ (i.e. it's space of characters) is simply the topological dual $E^*$. I also think that for a nice enough measure $\mu$ over $E$ we can define the Fourier transform as $$\hat\mu: E^*\to\mathbb{C} \\ \hat\mu(\varphi) = \int_E e^{i\langle \varphi, x\rangle} \,\mathrm{d}\mu(x)$$ where $\langle\cdot,\cdot\rangle:E^*\times E\to\mathbb{R}$ is the duality pairing. Furthermore, I would expect $\hat\mu=0$ iff $\mu=0$.

I'm particularly interested in the case where $E$ is some dual Banach space with the weak-* topology. For example, $E$ being $M(K)$ with the weak-* topology for some compact Hausdorff $K$ (in this case I'd be looking at Fourier analysis for measures on measures).

Is my intuition above correct? Are there any reference to study the Fourier theory of measures on such infinite-dimensional spaces? Or if the theory falls apart, that would be good to know as well. Thanks!

• There are books that discuss Fourier transforms (a.k.a. characteristic functions) of probability measures on Banach spaces. For example, you can visit statmathbc.wordpress.com and download Volume 2 of Infinitely divisible and stable measures. FT's os measures on TVS's is in research papers and I do not know of any book on this topic. Dec 7 '17 at 7:39
• This en.wikipedia.org/wiki/Minlos%27_theorem might be interesting. Dec 7 '17 at 8:12

The answers to many of the above questions can be found in Measure Theory (Vol. II) by Vladimir Bogachev. Section 7.12 is about measures on linear spaces and section 7.13 is about the Fourier transform of such measures.

Bogachev defines the Fourier transform as:

Here $Cyl(X,G)$ is the $G$-cylindrical $\sigma$-algebra of the topological vector space $X$ for a set of linear functionals $G$. A quasi-measure on $Cyl(X,G)$ is an additive real function $\mu$ on $Cyl(X,G)$ such that all finite dimensional projections $P_*\mu=\mu\circ P^{-1}$ are bounded and countably additive measures. Note that if we do a change of variables we get $$\tilde\mu(f) = \int_E e^{it} \,\mathrm{d}f_*\mu(x) = \int_E e^{if(x)} \,\mathrm{d}\mu(x) = \int_E e^{i\langle f, x\rangle} \,\mathrm{d}\mu(x)$$ So the definition agrees with the question statement.

The following lemma from the book answers the question of injectivity:

Since this formulation applies when $X$ is locally convex and $G=X^*$, we can also apply it to the case where $X=M(K)$ with the weak-* topology (in which case $X^*=C(K)$).