Fourier transform with different measure I am interested in the following question: can we generalize Fourier theory to different $L^2(\mu)$ spaces, not just the Lebesgue measure?
For example on $\mathbb{R}$, defining the Fourier transform of $f$ as
$$ \mathcal{F}[f](\omega)=\int_\mathbb{R}f(x)e^{-i\omega x}d\mu(x). $$
Is there literature on this subject? Do any properties hold with this (in particular Plancherel’s theorem)?  I have tried to look this idea up online but found nothing.
I am a beginner when it comes to Fourier theory, I apologize if it is a simple question.
 A: Yes, in fact one can define it on measures on topological groups (besides integrable functions with respect to Haar measures). In any event, lets just focus on $\mathbb{R}^n$. There, the Fourier transform is well define on the space of complex (or finite real valued) Borel measures $\mathcal{M}(\mathbb{R}^n)$. There is also an inversion formula that shows that the Fourier transform a one to one function on $\mathcal{M}$. The Fourier transform in this setting  is often used in probability theory under the name of characteristic function.
Definition:
Let $\mu$ be a Complex measure on
$(\mathbb{R}^d,\mathscr{B}(\mathbb{R}^d))$. The Fourier transform  or characteristic function  of $\mu$ is the function defined as
$$\widehat{\mu}({\bf t})=\int\exp(i{\bf t}\cdot{\bf x})\mu(d{\bf x})$$

When $\mu\ll\lambda_d$ and $\mu=f\cdot\lambda_d$, then the Fourier transform of $\mu$ is closely related to the Fourier transform of $f$. and $\hat{\mu}(-2\pi i t)=\hat{f}(t)$.

Here are a few important results on uniqueness:
Theorem: Suppose that   $\mu$ and $\nu$ are complex measures (measures of finite variation) on $\mathscr{B}(\mathbb{R}^d)$. Then, $\mu=\nu$ iff
$\widehat{\mu}=\widehat{\nu}$.
Theorem: Let $\mu$ be a complex Borel measure on $\mathbb{R}$ and let
$\widehat{\mu}$ be its characteristic function. Then

*

*For any $-\infty<a<b <\infty$,
$$
\begin{align}
\mu((a,b))+\frac12\mu(\{a,b\})=\frac{1}{2\pi}
\lim_{T\rightarrow\infty}\int^T_{-T}\int^b_a
e^{-iyt}\widehat{\mu}(t)\,dy\,dt \tag{1}\label{inversion}
\end{align}
$$
$$
\mu(\{a\})=\lim_{T\rightarrow\infty}\frac{1}{2T}\int^T_{-T}
e^{-iat}\widehat{\mu}(t)\,\tag{2}\label{two}
$$
2. If $f\in\mathcal{L}_1(\mathbb{R},\lambda)$ then
$$\begin{align}
f(x)=\frac{1}{2\pi}\lim_{T\rightarrow\infty}\int^T_{-T}e^{-iyt}\widehat{f}(-t/2\pi)\,dt\tag{3}\label{inversion3}
\qquad\text{a.s.}
\end{align}
$$
3. If $\widehat{\mu}\in\mathcal{L}_1$, then $\mu\ll\lambda$ and
$$\begin{align}
\frac{d\mu}{d\lambda}(y)=\frac{1}{2\pi}\int
e^{-ity}\widehat{\mu}(t)\,dt\qquad\text{a.s.}\tag{4}\label{density-prob}
\end{align}
$$
With regards smoothness:
Theorem: Let $\mu$ be a finite positive measure on $(\mathbb{R}^n,\mathscr{B}(\mathbb{R}^n))$. If $\partial^\alpha\widehat{\mu}(0)$ exits and is finite for all $|\alpha|=2m$ then,  $\widehat{\mu}\in\mathcal{C}^{2m}(\mathbb{R}^n)$; furthermore, for all  $\alpha\in\mathbb{Z}^n_+$ with $|\alpha|=2m$,
$\int |x^\alpha|\,\mu(dx)<\infty$ and $\partial^\alpha\widehat{\mu}(t)=i^{|\alpha|}\int x^\alpha e^{ix\cdot t}\,\mu(dx)$.
Theorem: Suppose that $\mu$ is a complex measure on $(\mathbb{R}^n,\mathscr{B}(\mathbb{R}^n)$. If
$$
  \int_{\mathbb{R}^n}|x_j|^m|\mu|(dx)<\infty,
  $$
then the partial derivative
$\partial^m_j\widehat{\mu}$ exists,  is uniformly continuous, and
$$\begin{align}
\partial^k_j \widehat{\mu}(t)=i^k\int_{\mathbb{R}^n} x^k_j e^{i x\cdot t}\mu(dx),\quad 0\leq k\leq m.
\end{align}\tag{1}\label{deriv-m-charac}
$$
Moreover, if $|x|^m=\Big(\sum^n_{j=1}x^2_j\Big)^{\frac{m}{2}}\in\mathcal{L}_1(|\mu|)$, then $\widehat{\mu}\in\mathcal{C}^m(\mathbb{R}^n)$, and
$$\begin{align}
\widehat{\mu}(t)= \sum_{0\leq|\alpha|\leq m}\frac{i^{|\alpha|}}{\alpha!}t^\alpha \int x^\alpha\,\mu(dx) + o(|t|^m)\tag{2}\label{otaylor}
\end{align}$$
Theorem: (Bochner--Herglotz)
$\varphi:\mathbb{R}^d\rightarrow\mathbb{C}$ is the characteristic
function of a finite nonnegative measure $\mu$ in  $\mathscr{B}(\mathbb{R}^d)$
iff $\varphi$ is a bounded positive definite  continuous function.
