The Fourier Transform and Conjugate Symmetry

Introduction

The Fourier transform converts the involution of the functions into a reflection followed by a conjugation. Precisely, $$ \widehat{\bar f}(\xi)=\overline{\hat f(-\xi)}, $$ so that conjugating a function conjugates its transform and reflects the frequency; a real function is therefore characterised by the conjugate symmetry of its transform, $$ f=\bar f\ \Longleftrightarrow\ \hat f(-\xi)=\overline{\hat f(\xi)}, $$ and the even and odd parts of a real function are read off from the real and imaginary parts of its transform. The same identity underlies the conjugate Fourier integral: the conjugate function of a real function has transform $-i\operatorname{sgn}(\xi)$ times the transform of the function, so that the pairing of a function with its conjugate is the decomposition of its spectrum into the positive and negative frequencies, and it is the reason the analytic representation — the function whose spectrum is confined to the positive frequencies — is built from a function and its Hilbert transform. This article develops the conjugate symmetry of the transform, the parity decomposition that comes with it, and the algebra statement that the transform is a star-isomorphism: it carries the involution $f^{*}(x)=\overline{f(-x)}$ of the convolution algebra to the pointwise conjugation of the transforms.

This article is the fourth of the * group of Foundations of Analysis. Its prerequisites are Fourier Analysis on Euclidean Spaces for the transform, the inversion and Plancherel theorems, the reflection $\tilde f(x)=f(-x)$ and the identity $\mathcal F^2=\text{reflection}$, the multiplication formula and the Hilbert transform with symbol $-i\operatorname{sgn}(\xi)$; and Hermitian Measures and Complex Measures, the previous article of this group, for the involution on the measures and its compatibility with the Fourier–Stieltjes transform. The transform is treated in the $L^1\cap L^2$ setting of the Fourier article, the Schwartz class and the tempered distributions being Distributions and Fundamental Solutions, later in this Part. The general singular-integral theory of the conjugate function and its maximal operator belongs to Real Harmonic Analysis, in a later Part, and to Fourier Analysis on Euclidean Spaces; the conjugate Poisson integral belongs to harmonic function theory, later in this Part, and is named only. The Clifford-algebra analogue of the present symmetry is The Clifford–Fourier Transform and Conjugate Symmetry, in the Clifford analysis of a later Part. The convolution operators are the previous group of this category, and the transform acts on them as multiplication by the symbol. No geometry is invoked.

The Involution and the Transform

The Conjugate Symmetry Identity

Theorem (the transform of the conjugate). For $f\in L^1(\mathbb R^n)$, $$ \widehat{\bar f}(\xi)=\overline{\hat f(-\xi)},\qquad \widehat{\bar f}=\widetilde{\overline{\hat f}}, $$ where $\tilde g(x)=g(-x)$ is the reflection; equivalently, in the notation of the Fourier article, $\mathcal F(\bar f)=\overline{\mathcal F f}\circ(-1)$.

Proof. $\widehat{\bar f}(\xi)=\int\overline{f(x)}e^{-2\pi ix\cdot\xi}dx =\overline{\int f(x)e^{2\pi ix\cdot\xi}dx}=\overline{\hat f(-\xi)}$, by the conjugation of the integral and the substitution $x\mapsto-x$. $\blacksquare$

The reflection commutes with the transform, $\widehat{\tilde f}=\widetilde{\hat f}$, as recorded in the Fourier article, where it is also proved that $\mathcal F^2$ is the reflection and $\mathcal F^4=1$; the conjugate symmetry identity is therefore the statement that conjugation is carried by the transform to conjugation after the reflection, and, because $\mathcal F^2$ is the reflection, $$ \mathcal F\circ(\text{conjugation})\circ\mathcal F^{-1}=(\text{conjugation})\circ\mathcal F^2 , $$ so that conjugation and the transform fail to commute by exactly the reflection of order two.

Real Functions and Conjugate-Symmetric Spectra

Theorem (the conjugate-symmetry criterion). A function $f\in L^1\cap L^2$ is real up to equality almost everywhere if and only if its transform obeys the conjugate symmetry $$ \hat f(-\xi)=\overline{\hat f(\xi)} $$ for almost every $\xi$; equivalently, the transform is Hermitian. More generally, $f$ satisfies the shifted Hermitian condition $\bar f=\tau_yf$ for a translation $\tau_y$ if and only if $\hat f$ has the symmetry $\hat f(-\xi)=e^{2\pi iy\cdot\xi}\overline{\hat f(\xi)}$.

Proof. Apply the conjugate symmetry identity to $\bar f$: if $f=\bar f$ then $\hat f(\xi)=\widehat{\bar f}(\xi)=\overline{\hat f(-\xi)}$. Conversely, if $\hat f(-\xi)=\overline{\hat f(\xi)}$ then $\widehat{\bar f}=\hat f$ by the identity, so $\bar f=f$ by the injectivity of the transform. The translation statement is the same computation with the translation identity $\widehat{(\tau_{-a}f)}(\xi)=e^{-2\pi ia\cdot\xi}\hat f(\xi)$ of the Fourier article. $\blacksquare$

Corollary (the parity dictionary for a real function). Let $f$ be real, and let $f_{\mathrm e}(x)=\frac12(f(x)+f(-x))$ and $f_{\mathrm o}(x)=\frac12(f(x)-f(-x))$ be its even and odd parts. Then $$ \hat f_{\mathrm e}(\xi)=\operatorname{Re}\hat f(\xi),\qquad \hat f_{\mathrm o}(\xi)=i\operatorname{Im}\hat f(\xi), $$ so that the even part of a real function has the real part of the spectrum as its transform and the odd part has $i$ times the imaginary part.

Proof. For real $f$, $\hat f(-\xi)=\overline{\hat f(\xi)}$; the even part has transform $\frac12(\hat f(\xi)+\hat f(-\xi))=\operatorname{Re}\hat f(\xi)$ and the odd part $\frac12(\hat f(\xi)-\hat f(-\xi))=i\operatorname{Im}\hat f(\xi)$. $\blacksquare$

Thus for a real function the parity of $f$ is read from the reality of $\hat f$ and the sign of the reflection, and the transform of a real even function is real and even, of a real odd function purely imaginary and odd. In particular a real even function is determined by the values of its transform on the positive frequencies alone.

The Conjugate Fourier Integral

The Conjugate Function

Definition. Let $f$ be a real function in $L^2(\mathbb R)$, extended to the upper half plane by the Poisson integral. The conjugate function (or harmonic conjugate) of $f$ is the function $$ \tilde f(x)=\frac1\pi\,\mathrm{p.v.}\!\int_{\mathbb R}\frac{f(y)}{x-y}\,dy , $$ the principal-value integral with the conjugate Poisson kernel; it is the unique real function such that $f+i\tilde f$ is the boundary value of a function holomorphic in the upper half plane, up to an additive constant, and it is the value at the boundary of the conjugate Poisson integral $$ Q_y(x)=\frac1\pi\frac{x}{x^2+y^2},\qquad \tilde f(x)=\lim_{y\to0}\!\int Q_y(x-t)f(t)\,dt . $$

Theorem (the symbol of the conjugate function). On $L^2(\mathbb R)$ the conjugate function is the Hilbert transform $H$, the bounded operator with the multiplier $$ \widehat{(Hf)}(\xi)=-i\operatorname{sgn}(\xi)\,\hat f(\xi) , $$ and it is skew-adjoint and involutive in the sense $H^2=-\mathrm{id}$; it commutes with the translations and with the dilations.

Proof. The Fourier transform of the principal value of $1/x$ is $-i\pi\operatorname{sgn}(\xi)$ and the conjugate Poisson kernel multiplies this by $e^{-2\pi y\lvert\xi\rvert}$, so the multiplier at $y=0$ is $-i\operatorname{sgn}(\xi)$, which is bounded; the skew-adjointness is that of a multiplication by a purely imaginary odd function, $H^\dagger=-H$, and $H^2$ has multiplier $(-i\operatorname{sgn})^2=-1$. The commutation with the translations and the dilations is the corresponding statement for the multiplier, from the convolution and dilation identities of the Fourier article. $\blacksquare$

The Hilbert transform and its singular-integral theory are Fourier Analysis on Euclidean Spaces and Real Harmonic Analysis; the present Article records only its relation to the conjugate symmetry.

The Analytic Representation

Definition. For a real function $f\in L^2(\mathbb R)$ the analytic representation (or Hardy extension on the line) is $$ f_{\mathrm a}=f+iHf . $$

Theorem. The analytic representation has spectrum confined to the positive frequencies, $$ \widehat{f_{\mathrm a}}(\xi)=\bigl(1+\operatorname{sgn}(\xi)\bigr)\hat f(\xi) =\begin{cases}2\hat f(\xi),&\xi>0,\\ 0,&\xi<0,\end{cases} $$ and the real function and its conjugate are recovered from it by $f=\operatorname{Re}f_{\mathrm a}$ and $Hf=\operatorname{Im}f_{\mathrm a}$; the conjugate symmetry of the spectrum of $f$ is exactly the statement that the two halves of the spectrum of $f_{\mathrm a}$ are redundant.

Proof. The multiplier of $iH$ is $i(-i\operatorname{sgn})=\operatorname{sgn}$, so the multiplier of the analytic representation is $1+\operatorname{sgn}$, which is $2$ on the positive frequencies and $0$ on the negative ones; the reconstruction is the definition. $\blacksquare$

This is the analytic form of the conjugate symmetry: a real function is determined by the positive half of its spectrum, and the conjugate function is the price of discarding the negative half. The systematic use of this decomposition in the theory of analytic functions is the theory of the Hardy space and the Plemelj formulae, which belong to the complex analysis of a later Part.

The Transform as a Star-Isomorphism

The Two Involutions

Definition. On the functions on a group two involutions are in play: the value-conjugation $\bar f(x)=\overline{f(x)}$, and the convolution involution $$ f^{*}(x)=\overline{f(-x)}, $$ which combines the conjugation with the reflection and is the adjoint of convolution; the distinction is the one recorded in Hermitian Measures and Complex Measures, the previous article of this group, between the value-conjugation $\bar\mu$ and the adjoint involution $\mu^{*}$ of the group measure algebra.

Theorem (the transform turns the conjugation into the value-conjugation). In the setting of the convolution algebra of a group, $$ \widehat{f^{*}}(\xi)=\overline{\hat f(\xi)}, $$ so that the Fourier transform is a star-isomorphism: it carries the convolution involution $f\mapsto f^{*}$ of the algebra of functions to the pointwise conjugation of the transforms, and the product to the product. For a finite abelian group the transform is an isomorphism of the group algebra with its star onto the algebra of functions on the dual group with pointwise product and pointwise conjugation; for $\mathbb Z$ and $\mathbb R$ it is a star-homomorphism of $\ell^1$ or $L^1$ into the bounded continuous functions, with dense image, whose extension is the Gelfand transform of the algebra.

Proof. $\widehat{f^{*}}(\xi)=\int\overline{f(-x)}e^{-2\pi ix\cdot\xi}dx =\overline{\int f(-x)e^{2\pi ix\cdot\xi}dx}=\overline{\int f(u)e^{-2\pi iu\cdot\xi}du}=\overline{\hat f(\xi)}$, which is the statement; the product statement is the convolution theorem of the Fourier article, and the isomorphism statements are the Fourier inversion and Plancherel theorems. $\blacksquare$

Corollary (the algebraic reading of the conjugate symmetry). The value-conjugation identity $\widehat{\bar f}=\widetilde{\overline{\hat f}}$ and the involution identity $\widehat{f^{*}}=\overline{\hat f}$ differ by the reflection: $\bar f=f^{*}\circ(\text{reflection})$, and the transform commutes with the reflection. The conjugate symmetry of the previous sections is therefore the shadow, at the level of the involution of the algebra, of the elementary identity $\mathcal F^2=\text{reflection}$.

Summary

The Fourier transform carries complex conjugation to conjugation followed by reflection, $\widehat{\bar f}(\xi)=\overline{\hat f(-\xi)}$, so that a function is real exactly when its transform obeys the conjugate symmetry $\hat f(-\xi)=\overline{\hat f(\xi)}$; for a real function the even part has the real part of the spectrum as its transform and the odd part $i$ times the imaginary part, and a real even function has a real even transform. The conjugate function of a real function on the line is the Hilbert transform, with multiplier $-i\operatorname{sgn}(\xi)$, skew-adjoint and involutive with square $-\mathrm{id}$; the analytic representation $f+iHf$ has spectrum confined to the positive frequencies, which is the sharpest form of the redundancy of the conjugate-symmetric spectrum, and its analytic theory is that of the Hardy space and the Plemelj formulae. On the convolution algebra the Fourier transform is a star-isomorphism carrying the involution $f^{*}(x)=\overline{f(-x)}$ to the pointwise conjugation of the transforms, so that the conjugate symmetry of the spectrum and the involution of the algebra are the reflection of order two of the transform and the inversion theorems read in two ways.

Summary of Notation

Symbol	Meaning
$\bar f$, $\bar\mu$	Value-conjugation $f(x)\mapsto\overline{f(x)}$ and $\mu(A)\mapsto\overline{\mu(A)}$
$\tilde f(x)=f(-x)$	Reflection, $\mathcal F^2=\text{reflection}$, $\mathcal F^4=1$
$\widehat{\bar f}(\xi)=\overline{\hat f(-\xi)}$	Conjugate symmetry of the transform
$f_{\mathrm e},f_{\mathrm o}$	Even and odd parts, $\frac12(f(x)\pm f(-x))$
$\tilde f$, $H$	Conjugate function, Hilbert transform, multiplier $-i\operatorname{sgn}$
$Q_y$	Conjugate Poisson kernel $\frac1\pi\frac{x}{x^2+y^2}$
$f_{\mathrm a}=f+iHf$	Analytic representation, spectrum on the positive frequencies
$f^{*}(x)=\overline{f(-x)}$	Convolution involution, $\widehat{f^{*}}=\overline{\hat f}$