Remark

The elementary formulas concerning the Fourier transform of measures remain valid for the Fourier transform of tempered distributions.

Thus, if $\alpha \in \mathbf{N}^n$ and $f\in \mathscr{S}'(\mathbf{R}^n)$, one has

Let $y\in \mathbf{R}^n$. Let us denote by $\boldsymbol{\gamma }(y)$ the endomorphism of $\mathscr{S}'(\mathbf{R}^n)$ defined by

for $f\in \mathscr{S}'(\mathbf{R}^n)$ and $\varphi \in \mathscr{S}(\mathbf{R}^n)$. Let $e_y$ be the character of $\mathbf{R}^n$ such that $e_y(x) =$ exp(2$i\pi x\cdot y$). Then $e_y\in \mathscr{S}'(\mathbf{R}^n)$. We have $\mathscr{F}(e_y) =\varepsilon_y$, and more generally

for every $f\in \mathscr{S}'(\mathbf{R}^n).*$

Let $n\geqslant 1$ be an integer, and let $G =\mathbf{T}^n$, endowed with the normalized Haar measure. The dual group of G is identified with $\mathbf{Z}^n$ by the mapping $h\mapsto \chi_h$, where $\chi_h$ is the unitary character of $\mathbf{T}^n$ obtained by passing to the quotient from the character $x\mapsto$ exp(2$i\pi h\cdot x$) of $\mathbf{R}^n$ (Corollary 3 of II, p. 236). The Fourier transform of a measure $\mu$ on $\mathbf{T}^n$ is identified with the family $(\widehat{\mu}(h))_{h\in\mathbf{Z}^n}$ where

The series

is called the Fourier series of $\mu$.

If $f\in L^1(\mathbf{T}^n)$ is such that its Fourier series converges absolutely in $L^1(\mathbf{Z}^n)$, then $f\in \mathscr{C}(\mathbf{T}^n)$ and

for every $x\in \mathbf{T}^n$ (Theorem 3 of II, p. 222), where

For $f\in L^2(\mathbf{T}^n)$, the fourier inversion formula (Proposition 12 of II, p. 219) says that, if the series with general term $\widehat{f}(h)$ converges absolutely, one has

for almost every $x$ in $\mathbf{T}^n$.

However, even if $f$ is continuous, the Fourier series of $f$ does not generally converge to $f(x)$ for every $x$ (Exercise 30 of II, p. 274). The following result is all the more useful.