Fourier Analysis — Foundations¶

Job: Define Fourier series and the Fourier transform; state Plancherel and convolution theorems; track function-space hypotheses; connect to PDE multipliers and distributions.

Core¶

Fourier series on the torus¶

Fourier series on $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$ (or $[0,2\pi]$): $$ \hat f(n)=\frac{1}{2\pi}\int_0^{2\pi}f(\theta)e^{-in\theta}\,d\theta,\qquad S_Nf=\sum_{|n|\le N}\hat f(n)e^{in\theta}. $$ For $f\in L^2(\mathbb{T})$: $\|f\|_2^2=2\pi\sum|\hat f(n)|^2$ (Parseval; ONB $\{e^{in\theta}/\sqrt{2\pi}\}$). Pointwise: Carleson ($L^2$ a.e.); classical Dini / Dirichlet–Jordan for continuous BV. Gibbs phenomenon at jumps: overshoot $\to$ fixed fraction of jump size. Fejér means: Cesàro sums restore uniform convergence for continuous $f$ (positive kernel).

Fourier transform on $\mathbb{R}^n$¶

One common normalization: $$ \hat f(\xi)=\int_{\mathbb{R}^n}f(x)e^{-2\pi i x\cdot\xi}\,dx,\qquad f(x)=\int_{\mathbb{R}^n}\hat f(\xi)e^{2\pi i x\cdot\xi}\,d\xi $$ for $f,\hat f\in L^1$. Riemann–Lebesgue: $\hat f\in C_0$. Extends by density/unitarity to $L^2$:

Plancherel. Unique unitary $\mathcal{F}:L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$ extending the $L^1\cap L^2$ transform, $\|\hat f\|_2=\|f\|_2$, inversion in $L^2$ sense.

Convolution theorem: $\widehat{f*g}=\hat f\hat g$ (e.g. $L^1$). Differentiation: $\widehat{\partial^\alpha f}=(2\pi i\xi)^\alpha\hat f$ for Schwartz $f$. Translation / modulation dualities: $\widehat{f(\cdot-x_0)}=e^{-2\pi i x_0\cdot\xi}\hat f$, etc.

Schwartz space and tempered distributions¶

Schwartz $\mathcal{S}$: rapidly decreasing smooth functions; $\mathcal{F}:\mathcal{S}\to\mathcal{S}$ isomorphism. Dual tempered distributions $\mathcal{S}'$: $\langle\hat u,\varphi\rangle=\langle u,\hat\varphi\rangle$ (convention-dependent conjugate). Constant-coeff PDE: $P(D)u=f\mapsto P(2\pi i\xi)\hat u=\hat f$ as multipliers on $\mathcal{S}'$.

Inequalities and uncertainty¶

Hausdorff–Young: $\|\hat f\|_{p'}\le C_p\|f\|_p$ for $1\le p\le2$, $1/p+1/p'=1$ (Beckner sharp constants). Young for convolutions: $\|f*g\|_r\le\|f\|_p\|g\|_q$ when $1+1/r=1/p+1/q$.

Uncertainty (heuristic form): $\|xf\|_2\|\xi\hat f\|_2\ge\frac{n}{4\pi}\|f\|_2^2$ (normalization variants). Equality for Gaussians.

Poisson summation (Schwartz): $\sum_{k\in\mathbb{Z}^n}f(k)=\sum_{\xi\in\mathbb{Z}^n}\hat f(\xi)$ — bridges periodization and sampling; dual to DFT aliasing.

Multipliers and applications (pointer)¶

$m(\xi)$ defines $T_m f=\mathcal{F}^{-1}(m\hat f)$. Mihlin/Hörmander multiplier theorems give $L^p$ bounds from derivative decay of $m$. Spectral multipliers for $-\Delta$ and wave/Schrödinger propagators $e^{it\sqrt{-\Delta}}$, $e^{it\Delta}$.

Object	Fourier side
$\partial^\alpha$	polynomial multiplier
Heat $e^{t\Delta}$	$e^{-4\pi^2 t
Convolution op.	multiplication by $\hat K$
Periodization	sampling of $\hat f$ (Poisson)

See it¶

Heat equation $u_t=\Delta u$ on $\mathbb{R}^n$: $\partial_t\hat u=-4\pi^2|\xi|^2\hat u$, so $\hat u(\xi,t)=e^{-4\pi^2 t|\xi|^2}\hat u_0$. Inverse transform: convolution with the Gaussian kernel — prototype spectral solution.

Sampling (Shannon, idealized). Bandlimited $f$ with $\operatorname{supp}\hat f\subset[-1/2,1/2]$ recovered from samples $f(n)$ via $\operatorname{sinc}$ — Poisson + periodization of $\hat f$ without overlap.

Check¶

Why is $L^1$ Fourier transform not surjective onto $C_0$?

$\hat f\in C_0$ by Riemann–Lebesgue, but not every $C_0$ function is a Fourier transform of an $L^1$ function (constraints from the Banach algebra structure of $L^1$ under convolution).

State Plancherel carefully for $\hat f(\xi)=\int f(x)e^{-ix\cdot\xi}\,dx$.

Unitary form requires a $(2\pi)^{-n/2}$ factor (or Plancherel measure $(2\pi)^{-n}d\xi$). Always check convention when quoting $\|\hat f\|_2=\|f\|_2$.

Why do Fejér means of a continuous $f$ converge uniformly while $S_Nf$ may not?

Fejér kernel is a positive approximate identity; Dirichlet kernel is not positive and $\|D_N\|_1\sim\log N$ $\Rightarrow$ possible divergence at points (du Bois-Reymond).

Pitfalls¶

Normalization and $2\pi$ placement differ by community (probability, HA, signal processing).
Pointwise a.e. Fourier inversion needs more than $L^1$ membership of $\hat f$ alone.
Discrete FFT periodizes and samples — aliasing is Poisson dual.
Not every distribution is tempered; exponential growth blocks $\mathcal{S}'$.
Multiplier boundedness on $L^p$ fails for rough $m$ (e.g. ball multiplier for $p\neq2$, Fefferman).

Refs¶

Stein–Shakarchi IV; Folland, Real Analysis / Harmonic Analysis; Katznelson; Grafakos I–II; Hörmander ALPDO I; Stein, Singular Integrals.

Next¶

Index · PDE · Functional analysis · Numerical spectral methods · Microlocal analysis · Dispersive PDE — Strichartz estimates via Fourier restriction · Harmonic analysis — LCA groups, Pontryagin duality, CZ singular integrals · Restriction & Strichartz · Oscillatory & Carleson · Hardy & BMO · Abstract Harmonic & Pontryagin