The Fourier Operator

Introduction

The Fourier transform is not only a formula for a function but an operator, and as an operator it is unitary. Written $$ (\mathcal Ff)(\xi)=\int_{\mathbb R^n}f(x)e^{-2\pi ix\cdot\xi}\,dx , $$ it extends from the Schwartz class to an isometry of $L^2(\mathbb R^n)$ onto itself, so that it preserves the inner product and the norm; its inverse is the transform with the conjugate character. The operator has a second structural feature, its order: applied twice it reflects the argument, $\mathcal F^2f(x)=f(-x)$, and applied four times it returns $f$, so that $\mathcal F^4=\mathrm{id}$ and its only possible eigenvalues are the fourth roots of unity. In the Schwartz class that possibility is realised: the Hermite functions are eigenvectors with eigenvalues $(-i)^k$, and they form an orthonormal basis of $L^2$ in which $\mathcal F$ is diagonal. This article develops the transform in that operator-theoretic form: the unitary operator, its order-four structure, the eigenbasis that diagonalises it, and the unitary equivalence by which it turns differentiation into multiplication and every translation-invariant operator into a multiplier.

The prerequisites are Fourier Analysis on Euclidean Spaces for the transform on $L^1$ and $\mathcal S(\mathbb R^n)$, the multiplication formula, the inversion theorem, the convolution theorem, the Plancherel theorem and the Hermite functions, Measure Theory and Integration for the $L^2$ space and its inner product, and Convolution Operators, the previous article of this group, for the multiplier operators. The spectral vocabulary — unitary, self-adjoint, eigenbasis, unitary equivalence, the spectral theorem — is quoted as it is used from Banach and Hilbert Spaces, later in this Part, where it is developed; the concrete diagonalisation of $\mathcal F$ by the Hermite functions is proved here, and nothing in the article depends on the general spectral theorem. The adjoint of the Fourier operator is the subject of The Adjoint of the Fourier Operator, in the * Operator group of this category, and the present Article stops at the unitarity from which that adjoint is read. No geometry is invoked; the transform acts on functions and the operator is studied through the integral and the inner product alone.

The Transform as an Operator

Definition on the Schwartz Class

Definition. The Fourier operator on $\mathbb R^n$ is the map $$ (\mathcal Ff)(\xi)=\int_{\mathbb R^n}f(x)e^{-2\pi ix\cdot\xi}\,dx , $$ under the normalisation of Fourier Analysis on Euclidean Spaces; the conjugate transform is $(\overline{\mathcal F}f)(\xi)=\int f(x)e^{2\pi ix\cdot\xi}dx$, and the inverse transform is $\check g(x)=\int g(\xi)e^{2\pi ix\cdot\xi}d\xi$.

On the Schwartz class $\mathcal S(\mathbb R^n)$ the transform is a linear homeomorphism onto itself, of order four; this is the content of the invariance of $\mathcal S$ and of the inversion theorem of the Fourier article. The two identities this article uses throughout are those of that article as well: $\mathcal F(\partial_jf)=2\pi i\xi_j\mathcal Ff$ and $\mathcal F(x_jf)=-\frac{1}{2\pi i}\partial_j\mathcal Ff$, the second from differentiating under the integral.

Extension to $L^2$ and Unitarity

Theorem (Plancherel). The transform extends from $\mathcal S(\mathbb R^n)\cap L^2$ to a unique bounded linear operator $\mathcal F:L^2(\mathbb R^n)\to L^2(\mathbb R^n)$, which is an isometry: $$ \lVert\mathcal Ff\rVert_2=\lVert f\rVert_2,\qquad \langle\mathcal Ff,\mathcal Fg\rangle=\langle f,g\rangle , $$ the inner product being $\langle f,g\rangle=\int f\bar g$. It is onto, hence unitary, and its inverse is the conjugate transform, $\mathcal F^{-1}=\overline{\mathcal F}$.

Proof. The isometry on $\mathcal S$ is Plancherel's theorem of Fourier Analysis on Euclidean Spaces; $\mathcal S$ is dense in $L^2$, so the isometry extends uniquely to a bounded isometry of $L^2$, and the extension is given by the $L^2$ limit of the transforms of an approximating sequence. The inner product identity is the polarisation of the norm identity over $\mathbb C$ (the article works over $\mathbb K=\mathbb R$ or $\mathbb C$; over $\mathbb R$ the norm identity is equivalent to the inner product identity by the polarisation formula). The inverse is the conjugate transform, since the inversion theorem gives $\check{\hat f}=f$ on $\mathcal S$ and hence, by density and continuity, on $L^2$; surjectivity follows. $\blacksquare$

The word unitary for $\mathcal F$ is used in the sense of Banach and Hilbert Spaces, later in this Part: a bounded operator with $\mathcal F^\dagger\mathcal F=\mathcal F\mathcal F^\dagger=\mathrm{id}$. The dagger is the adjoint, in the sense of Conventions in Mathematics; the star, by contrast, names the involution of the elements, $\bar f$. The unitarity of $\mathcal F$ is the statement that $\overline{\mathcal F}=\mathcal F^{-1}$.

The Order-Four Structure

The Square of the Transform Is Parity

Definition. The parity operator $P$ is the reflection $(Pf)(x)=f(-x)$. It is unitary and self-adjoint, and $P^2=\mathrm{id}$; hence $P$ has the two eigenvalues $+1$ and $-1$, with the even functions as the $+1$ eigenspace and the odd functions as the $-1$ eigenspace.

Theorem. On $L^2(\mathbb R^n)$ one has $\mathcal F^2=P$ and consequently $\mathcal F^4=\mathrm{id}$, $\mathcal F^{-1}=\mathcal F^3$, and $\mathcal F^3=\mathcal FP=P\mathcal F$.

Proof. On $\mathcal S$, inserting the definition and exchanging the order of integration, which the rapid decrease justifies, $$ \mathcal F^2f(x)=\int \mathcal Ff(\xi)e^{-2\pi ix\cdot\xi}\,d\xi =\iint f(y)e^{-2\pi iy\cdot\xi}e^{-2\pi ix\cdot\xi}\,dy\,d\xi =\int f(y)\Bigl(\int e^{-2\pi i(x+y)\cdot\xi}d\xi\Bigr)dy , $$ and the inner integral is the delta distribution at $x+y$, so it selects $y=-x$ and $\mathcal F^2f=f(-x)$. Both sides extend to $L^2$ by continuity and density. Then $\mathcal F^4=(\mathcal F^2)^2=P^2=\mathrm{id}$, and $\mathcal F^{-1}=\mathcal F^3$ follows by multiplying by $\mathcal F$; the last identity is $\mathcal F\mathcal F^2=\mathcal F^2\mathcal F$, that is, parity commutes with $\mathcal F$, which holds because $\mathcal F^2$ is a function of $\mathcal F$. $\blacksquare$

Corollary (the eigenvalues). A unitary operator satisfying $\mathcal F^4=\mathrm{id}$ has spectrum contained in the fourth roots of unity $\{1,i,-1,-i\}$. In finite dimension it would be diagonalisable with these eigenvalues; on the infinite-dimensional $L^2$ the same eigenvalues occur, but the operator has no eigenvectors in $L^2$ beyond the eigenbasis below, and no other spectral values.

Proof. If $\mathcal F^4=\mathrm{id}$ and $\lambda$ is an eigenvalue, then $\lambda^4=1$. The spectrum statement is the spectral mapping theorem for the polynomial $z^4-1$, applied in Banach and Hilbert Spaces, later in this Part. $\blacksquare$

The Hermite Eigenbasis

The Hermite Functions

Definition. On $\mathbb R$ the Hermite functions are obtained from the Gaussian by the operators of multiplication by $x$ and of differentiation, $$ h_k(x)=\frac{(-1)^k}{(2^kk!\sqrt\pi)^{1/2}}e^{\pi x^2}\frac{d^k}{dx^k}\bigl(e^{-2\pi x^2}\bigr), \qquad k=0,1,2,\dots, $$ and on $\mathbb R^n$ the product functions $h_\alpha(x)=h_{\alpha_1}(x_1)\cdots h_{\alpha_n}(x_n)$, for multi-indices $\alpha$, are the Hermite functions of several variables. They lie in the Schwartz class and form an orthonormal basis of $L^2(\mathbb R^n)$.

Proof. The orthonormality and completeness are those of the Hermite basis of Fourier Analysis on Euclidean Spaces; the normalising constant is chosen so that $\lVert h_k\rVert_2=1$, and the products are orthonormal in $L^2(\mathbb R^n)$ by Fubini. $\blacksquare$

The Eigenvalues

Theorem. The Fourier operator is diagonal in the Hermite basis: $$ \mathcal Fh_\alpha=(-i)^{\lvert\alpha\rvert}h_\alpha , \qquad\text{so}\qquad \mathcal F=\sum_\alpha(-i)^{\lvert\alpha\rvert}\langle\cdot,h_\alpha\rangle h_\alpha . $$

Proof. On $\mathbb R$ the two identities $\mathcal F(\partial_xf)=2\pi i\xi\mathcal Ff$ and $\mathcal F(xf)=-\frac1{2\pi i}\partial_x\mathcal Ff$ show that $\mathcal F$ commutes with the creation and annihilation operators $x\pm\frac1{2\pi}\frac{d}{dx}$ which raise and lower $k$, hence that $\mathcal F$ carries $h_k$ to a multiple of $h_k$; the eigenvalue is then fixed on $h_0$ by the Gaussian computation $\mathcal F(e^{-\pi x^2})=e^{-\pi x^2}$ of the Fourier article, giving eigenvalue $1$ for $k=0$, and the commutation with the raising operator multiplies the eigenvalue by $-i$ at each step, giving $\mathcal Fh_k=(-i)^kh_k$. In $n$ variables the eigenvalue of $h_\alpha$ is the product of the one-variable eigenvalues along the coordinates, $(-i)^{\lvert\alpha\rvert}$. $\blacksquare$

Corollary (the four eigenspaces). The eigenvalues $1,i,-1,-i$ correspond to the values $\lvert\alpha\rvert\equiv0,1,2,3\pmod 4$, so $L^2(\mathbb R^n)$ is the orthogonal direct sum of four closed subspaces on which $\mathcal F$ acts as the scalars $1,i,-1,-i$; each is infinite-dimensional for $n\geq1$. This is the diagonalisation of a unitary operator of order four, and the spectral theorem for a unitary operator is the general statement behind it.

The Fourier Operator as a Unitary Equivalence

Differentiation Becomes Multiplication

Theorem. With $D_j=\frac1{2\pi i}\partial_j$ acting on the Schwartz class, one has $$ \mathcal FD_j\mathcal F^{-1}=M_{x_j},\qquad \mathcal FM_{x_j}\mathcal F^{-1}=-D_j , $$ where $M_{x_j}$ is multiplication by the coordinate $x_j$; equivalently $\mathcal F$ is a unitary equivalence carrying the differentiation $\frac1{2\pi i}\partial_j$ into multiplication by the coordinate.

Proof. The first identity is the rearrangement of $\mathcal F(\partial_jf)=2\pi i\xi_j\mathcal Ff$, the second of $\mathcal F(x_jf)=-\frac1{2\pi i}\partial_j\mathcal Ff$; both are the differentiation identities of Fourier Analysis on Euclidean Spaces. On $\mathcal S$ they are identities of operators, and they extend to the closures. $\blacksquare$

The theorem is the reason the transform solves constant-coefficient equations: a polynomial in the derivatives becomes multiplication by the same polynomial in the coordinates, so a differential equation becomes an algebraic one.

Translation and Convolution Become Multiplication

Theorem. For the translation $\tau_y$ and the character $e_y(\xi)=e^{-2\pi iy\cdot\xi}$ one has $$ \mathcal F\tau_y\mathcal F^{-1}=M_{e_y} , $$ and for $k\in L^1(\mathbb R^n)$, with $T_k$ the convolution operator of the previous article, $$ \mathcal FT_k\mathcal F^{-1}=M_{\hat k},\qquad \mathcal F^{-1}T_m\mathcal F=M_m $$ for the multiplier operator $T_m$.

Proof. $\mathcal F(\tau_yf)=\widehat{f(\cdot-y)}=e_y\mathcal Ff$ is the translation identity of the Fourier article and gives the first formula; the convolution theorem $\widehat{k*f}=\hat k\hat f$ gives $T_k=\mathcal F^{-1}M_{\hat k}\mathcal F$, and the multiplier operator is defined by $T_m=\mathcal F^{-1}M_m\mathcal F$. $\blacksquare$

Corollary. Conjugation by $\mathcal F$ is a unitary isomorphism of the algebra of translation-invariant bounded operators onto the algebra of multiplication operators $M_m$ with $m\in L^\infty$; it carries composition to pointwise multiplication, adjunction to complex conjugation of the symbol, and compactness to the vanishing of the symbol. This is the operator form of the Fourier multiplier correspondence of Convolution Operators.

The Discrete and Finite Models

The Finite Abelian Group

Example (the discrete Fourier transform). Let $G$ be a finite abelian group of order $N$, with the counting measure normalised to total mass one, and let $\widehat G$ be its character group. The discrete Fourier transform is $$ (\mathcal Ff)(\chi)=\frac1{\sqrt N}\sum_{g\in G}f(g)\overline{\chi(g)} , \qquad f\in\ell^2(G), $$ and the characters form an orthonormal basis of $\ell^2(G)$, so $\mathcal F$ is a unitary operator. For $G=\mathbb Z/N\mathbb Z$ it is the matrix with entries $N^{-1/2}\omega^{jk}$, $\omega=e^{-2\pi i/N}$, which is symmetric, and its square is the reversal of the coordinates, so $\mathcal F^2=P$ and $\mathcal F^4=\mathrm{id}$ exactly as on $\mathbb R^n$. This is the finite model of the theory: the four eigenvalues, the order-four structure and the diagonalisation by characters are all visible in an $N$-dimensional unitary matrix.

The finite abelian group transform and the general character theory are the subject of Character Theory and, in its analytic form, of Harmonic Analysis on Groups; the matrix above is recorded here because it is the exact analogue of the integral operator of the present Article and because its four-eigenvalue structure is the one the integral transform shares.

The Group $\mathbb Z$ and the Circle

Example ($\mathbb Z$ and $\mathbb T$). On $\ell^2(\mathbb Z)$ the transform to $L^2(\mathbb T)$ is $(\mathcal Fa)(\theta)=\sum_{n\in\mathbb Z}a_ne^{-2\pi in\theta}$, a unitary isomorphism by Parseval for Fourier series; its inverse is the coefficient map. The shift of Convolution Operators becomes multiplication by $e^{-2\pi i\theta}$, and a convolution operator becomes multiplication by a function on the circle. The same dictionary of this article — order four, diagonalisation, equivalence of translation and multiplication — holds verbatim, with the Hermite basis replaced by the characters.

Summary

The Fourier operator $\mathcal Ff(\xi)=\int f(x)e^{-2\pi ix\cdot\xi}dx$ extends from the Schwartz class to a unitary operator of $L^2(\mathbb R^n)$, isometric for the inner product, with inverse the conjugate transform. It has order four: $\mathcal F^2=P$ is parity and $\mathcal F^4=\mathrm{id}$, so its spectrum lies in the fourth roots of unity, and it is diagonalised in the Hermite basis by $\mathcal Fh_\alpha=(-i)^{\lvert\alpha\rvert}h_\alpha$, which splits $L^2$ into four eigenspaces. As a unitary equivalence it carries differentiation into multiplication by the coordinate and, by the convolution theorem, carries every translation-invariant operator into its multiplier, so that the translation-invariance, the multiplier and the four-eigenvalue structure of the transform are three faces of the same diagonalisation. The finite abelian group $\mathbb Z/N\mathbb Z$ and the pair $\mathbb Z,\mathbb T$ exhibit the identical structure with characters in place of Hermite functions.

Summary of Notation

Symbol	Meaning
$\mathcal F$, $\overline{\mathcal F}$	Fourier operator and its conjugate
$\check g$	Inverse transform, $\check g(x)=\int g(\xi)e^{2\pi ix\cdot\xi}d\xi$
$P$	Parity, $(Pf)(x)=f(-x)$; $\mathcal F^2=P$
$h_\alpha$	Hermite functions, orthonormal basis with $\mathcal Fh_\alpha=(-i)^{\lvert\alpha\rvert}h_\alpha$
$D_j$	$\frac1{2\pi i}\partial_j$; $\mathcal FD_j\mathcal F^{-1}=M_{x_j}$
$M_{x_j}$, $M_m$	Multiplication operators
$\tau_y$, $e_y$	Translation and character, $\mathcal F\tau_y\mathcal F^{-1}=M_{e_y}$
$T_k$, $T_m$	Convolution operator and multiplier operator
$\omega$	$e^{-2\pi i/N}$, the primitive $N$-th root of the finite transform