Fourier Series, Fourier Transforms, and Plancherel Theory

Fourier analysis resolves a function into translation modes. With the site’s positive forward phase, a translation becomes a phase factor, differentiation becomes multiplication by momentum, and pointwise multiplication becomes convolution in momentum. The Plancherel theorem makes the same transform a unitary map when momentum space carries the matching measure:

$$\begin{aligned} \widetilde f(p) &=\int_{\mathbb R^n}\mathrm d^n x\, e^{+ip\cdot x}f(x),\\ f(x) &=\int_{\mathbb R^n}\frac{\mathrm d^n p}{(2\pi)^n}\, e^{-ip\cdot x}\widetilde f(p),\\ \int_{\mathbb R^n}\mathrm d^n x\,|f(x)|^2 &=\int_{\mathbb R^n}\frac{\mathrm d^n p}{(2\pi)^n}\, |\widetilde f(p)|^2. \end{aligned}$$

The first integral is literal for $L^1$ functions, and both formulas are pointwise safe on the Schwartz class. For a general $L^2$ function, however, the transform and inverse are defined by norm limits and represent almost-everywhere equivalence classes. Keeping that distinction visible is what makes the familiar position–momentum rules reliable rather than merely formal.

Helpful background. Lebesgue Integration and Convergence Theorems supplies the convergence theorems used in the $L^1$ statements. $L^p$ Spaces, Inequalities, and Weak Convergence supplies density, completion, and almost-everywhere equivalence classes for the $L^2$ extension. Both deepen the justification, but neither is needed to begin.

Function classes and a reliable transform workflow

The analytic sections use $x,p\in\mathbb R^n$ with the Euclidean pairing $p\cdot x$. The QFT application later returns to the site’s $(+---)$ metric and declares its fixed-time spatial phase separately. Throughout,

$$\langle f,g\rangle =\int \mathrm d^n x\,\overline{f(x)}g(x)$$

is conjugate-linear in its first slot.

A dependable Fourier calculation has six steps:

1. Decide whether the domain is periodic or noncompact.
2. Write both the forward and inverse formulas, including every volume or $2\pi$ factor.
3. Name the function class: a trigonometric polynomial, $L^1$, the Schwartz class, or $L^2$.
4. Apply translation, derivative, product, or inversion rules only under hypotheses that justify the change of variables, integration by parts, or interchange of integrals.
5. State whether the answer holds pointwise, almost everywhere, or in norm.
6. Check one known mode or Gaussian and verify Parseval–Plancherel.

The stop rule is equally important: constants on all of $\mathbb R^n$, plane waves, delta functions, and singular kernels are not ordinary $L^1$ or $L^2$ inputs. Their transforms require tempered distributions and Fourier calculus.

Fourier series: periodic space gives discrete momentum

Let

$$\Omega_L=\prod_{j=1}^n[0,L_j), \qquad V=\prod_{j=1}^n L_j, \qquad k_{r,j}=\frac{2\pi r_j}{L_j}, \quad r\in\mathbb Z^n.$$

Opposite faces are identified, so functions on $\Omega_L$ are periodic in each coordinate. Use the Fourier coefficients

$$\widetilde f_r =\int_{\Omega_L}\mathrm d^n x\, e^{+ik_r\cdot x}f(x)$$

and the reconstruction convention

$$f(x) =\frac1V\sum_{r\in\mathbb Z^n} e^{-ik_r\cdot x}\widetilde f_r.$$

The elementary orthogonality calculation is

$$\int_{\Omega_L}\mathrm d^n x\, e^{i(k_r-k_s)\cdot x} =V\,\delta_{rs},$$

where $\delta_{rs}$ is a product of Kronecker deltas. Thus $V^{-1/2}e^{-ik_r\cdot x}$ is an orthonormal family in $L^2(\Omega_L)$.

Completeness, mean-square convergence, and Parseval

Fourier-series Plancherel theorem. The exponential modes form a complete orthonormal basis of $L^2(\Omega_L)$. If

$$S_Nf(x) =\frac1V \sum_{|r_1|,\ldots,|r_n|\leq N} e^{-ik_r\cdot x}\widetilde f_r,$$

then

$$\lim_{N\to\infty}\|S_Nf-f\|_{L^2(\Omega_L)}=0.$$

Moreover, for $f,g\in L^2(\Omega_L)$,

$$\int_{\Omega_L}\mathrm d^n x\, \overline{f(x)}g(x) =\frac1V\sum_{r\in\mathbb Z^n} \overline{\widetilde f_r}\,\widetilde g_r,$$

and hence

$$\|f\|_2^2 =\frac1V\sum_{r\in\mathbb Z^n}|\widetilde f_r|^2.$$

For the one-dimensional circle, completeness, $L^2$ convergence, and Parseval are developed in Axler 2020, Theorems 11.30–11.31 and Example 11.32, pp. 355–356, author-hosted open-access PDF. The rectangular-torus statement follows by taking products of the one-dimensional modes.

Proof sketch. Orthogonality makes $S_N$ an orthogonal projection and gives Bessel’s inequality. Trigonometric polynomials separate points of the torus, are closed under conjugation, and contain the constants, so they are uniformly dense in continuous periodic functions. Continuous functions are dense in $L^2(\Omega_L)$; therefore the union of the finite mode spaces is dense. Hilbert-space projection then gives $S_Nf\to f$ in $L^2$, and the inner-product identity follows by continuity.

This theorem asserts mean-square convergence. It does not assert uniform convergence, and it does not select the value of an $L^2$ representative at a particular point. Changing one point leaves every Fourier coefficient unchanged.

A sawtooth check

Take $f(x)=x$ on $(-\pi,\pi]$ and regard it as a $2\pi$-periodic $L^2$ function. Its coefficients in the convention above are

$$\widetilde f_0=0, \qquad \widetilde f_r =\int_{-\pi}^{\pi}x e^{irx}\,\mathrm dx =-\frac{2\pi i(-1)^r}{r}, \quad r\neq0.$$

Thus, in the $L^2$ sense,

$$x =2\sum_{r=1}^{\infty} \frac{(-1)^{r+1}}{r}\sin(rx).$$

The periodic representative jumps at the identified endpoint, so continuous partial sums cannot converge to it uniformly. Parseval gives an independent normalization check:

$$\frac1{2\pi}\int_{-\pi}^{\pi}x^2\,\mathrm dx =2\sum_{r=1}^{\infty}\frac1{r^2} =\frac{\pi^2}{3},$$

and therefore $\sum_{r=1}^{\infty}r^{-2}=\pi^2/6$. This calculation needs no pointwise convergence claim.

Operations on periodic functions

For a periodic $C^1$ function, integration by parts has no boundary term:

$$\begin{aligned} \widetilde{\partial_j f}_r &=\int_{\Omega_L}\mathrm d^n x\, e^{ik_r\cdot x}\partial_jf(x)\\ &=-ik_{r,j}\widetilde f_r. \end{aligned}$$

Likewise, if $(T_af)(x)=f(x-a)$ with subtraction understood modulo the periods, then

$$\widetilde{T_af}_r =e^{+ik_r\cdot a}\widetilde f_r.$$

For trigonometric polynomials—and more generally when the coefficient sums permit rearrangement—a pointwise product gives a discrete convolution:

$$\widetilde{fg}_r =\frac1V\sum_{s\in\mathbb Z^n} \widetilde f_s\widetilde g_{r-s}.$$

The volume factor follows from the chosen unnormalized forward coefficient. Convolution estimates and broader convergence hypotheses are developed on the next page.

Fourier transforms on noncompact space

For $f\in L^1(\mathbb R^n)$, the forward integral exists for every $p$ and satisfies

$$|\widetilde f(p)|\leq\|f\|_1.$$

In fact, $\widetilde f$ is uniformly continuous and tends to zero as $|p|\to\infty$; this is the Riemann–Lebesgue lemma. These properties, along with the derivative and translation rules, appear in Axler 2020, Definition 11.47 and Theorems 11.49–11.55, pp. 363–367, author-hosted open-access PDF. The cited source is one-dimensional; the same estimates and dominated-convergence arguments work componentwise on $\mathbb R^n$.

The bound does not by itself provide inversion. One useful theorem is: if $f\in L^1$, $\widetilde f\in L^1$, and $f$ is represented by a continuous function, then

$$f(x) =\int_{\mathbb R^n}\frac{\mathrm d^n p}{(2\pi)^n}\, e^{-ip\cdot x}\widetilde f(p)$$

for every $x$. Without the continuity assumption, the equality holds almost everywhere after choosing the inverse-transform representative. A $d$-dimensional proof under these hypotheses, using Gaussian regularization, is given in Speck 2011, Theorem 4.1, pp. 9–10, MIT OpenCourseWare PDF.

The Schwartz class as a safe computational core

The Schwartz class $\mathcal S(\mathbb R^n)$ consists of smooth functions for which

$$\sup_{x\in\mathbb R^n} |x^\alpha\partial^\beta f(x)|<\infty$$

for every pair of multi-indices $\alpha,\beta$. Such functions and all their derivatives decay faster than every inverse power. Consequently every integration by parts and every interchange of integrals used below is absolutely controlled.

Fourier transformation maps $\mathcal S$ bijectively to itself, and the displayed inverse formula holds pointwise. A standard proof inserts a Gaussian factor in momentum space, uses Fubini at positive regulator, and then removes the regulator by dominated convergence; see Dyatlov 2022, Theorem 11.6, pp. 121–122, and Theorem 11.15, pp. 124–125, official MIT lecture-note PDF, translated by $p\mapsto-p$. The general smoothing mechanism is developed in Convolution, Approximate Identities, and Poisson Summation; here the Schwartz theorem supplies a clean domain for the operational rules.

Translating common source conventions

Axler and Speck use

$$\mathcal F_-f(\xi) =\int\mathrm d^n x\,e^{-2\pi i\xi\cdot x}f(x).$$

Their frequency and the site’s frequency are related by

$$\widetilde f(p) =\mathcal F_-f\!\left(-\frac{p}{2\pi}\right), \qquad \mathrm d^n\xi =\frac{\mathrm d^n p}{(2\pi)^n}.$$

This change simultaneously reverses the phase, rescales frequency, and turns their unit $L^2(\mathrm d^n\xi)$ normalization into the site’s $L^2(\mathrm d^n p/(2\pi)^n)$ normalization. Translating only the exponential but not the measure would break inversion and Plancherel.

Dyatlov instead uses the unscaled negative-phase transform

$$\mathcal F_Df(\xi) =\int\mathrm d^n x\,e^{-i\xi\cdot x}f(x),$$

so only a sign reversal is needed:

$$\widetilde f(p)=\mathcal F_Df(-p).$$

Thus a source formula may require a sign reversal, a frequency rescaling, or both. The exponent and the momentum measure must always be translated together.

The operational dictionary and why it works

For $f,g\in\mathcal S(\mathbb R^n)$, the site convention gives

Position-space operations and their momentum-space images in the positive-forward convention

Operation in position space	Result in momentum space
f(x − a)	exp(+i p·a) f̃(p)
exp(−i b·x) f(x)	f̃(p − b)
∂ⱼf(x)	−i pⱼ f̃(p)
xⱼf(x)	−i ∂f̃(p)/∂pⱼ
Complex conjugate of f(x)	Complex conjugate of f̃(−p)
(f ∗ g)(x)	f̃(p) g̃(p)
f(x) g(x)	∫ dⁿq/(2π)ⁿ f̃(q) g̃(p − q)

Here

$$(f*g)(x) =\int_{\mathbb R^n}\mathrm d^n y\,f(y)g(x-y).$$

For a real-valued function, the conjugation row becomes the reality condition

$$\widetilde f(-p)=\overline{\widetilde f(p)}.$$

Two short derivations fix the signs. For translation, set $y=x-a$:

$$\begin{aligned} \widetilde{f(\,\cdot-a\,)}(p) &=\int\mathrm d^n x\,e^{ip\cdot x}f(x-a)\\ &=e^{ip\cdot a}\widetilde f(p). \end{aligned}$$

For differentiation, decay removes the boundary term:

$$\begin{aligned} \widetilde{\partial_jf}(p) &=\int\mathrm d^n x\,e^{ip\cdot x}\partial_jf(x)\\ &=-ip_j\widetilde f(p). \end{aligned}$$

Differentiating the transform integral instead gives

$$\partial_{p_j}\widetilde f(p) =i\widetilde{x_jf}(p),$$

which is the coordinate-multiplication row. Hence any polynomial $P(\partial)$ with constant coefficients is diagonalized:

$$\widetilde{P(\partial)f}(p) =P(-ip)\widetilde f(p), \qquad \widetilde{-\Delta f}(p)=|p|^2\widetilde f(p).$$

For the product formula, insert the inverse transform of $g$ and use Fubini:

$$\begin{aligned} \widetilde{fg}(p) &=\int\mathrm d^n x\,e^{ip\cdot x}f(x) \int\frac{\mathrm d^n q}{(2\pi)^n} e^{-iq\cdot x}\widetilde g(q)\\ &=\int\frac{\mathrm d^n q}{(2\pi)^n} \widetilde f(p-q)\widetilde g(q). \end{aligned}$$

The convolution formula follows similarly after setting $z=x-y$ in the double integral. Schwartz decay makes both Fubini steps legal. Outside this safe class, the identities need separate integrability hypotheses; they are not automatic algebra on arbitrary functions.

Parseval and the Plancherel extension

For $f,g\in\mathcal S$, insert the inverse transform of $g$ into the position inner product:

$$\begin{aligned} \int\mathrm d^n x\,\overline{f(x)}g(x) &=\int\mathrm d^n x\,\overline{f(x)} \int\frac{\mathrm d^n p}{(2\pi)^n} e^{-ip\cdot x}\widetilde g(p)\\ &=\int\frac{\mathrm d^n p}{(2\pi)^n} \overline{\widetilde f(p)}\widetilde g(p). \end{aligned}$$

This is Parseval’s identity on the dense test class. Setting $g=f$ gives the norm identity. The proof uses inversion and Fubini, not a formal integral of a plane wave asserted to be a delta function.

Constructing the transform on all of L²

Let $f\in L^2(\mathbb R^n)$ and choose $f_j\in\mathcal S$ such that $f_j\to f$ in $L^2(\mathrm d^n x)$. Parseval gives

$$\int\frac{\mathrm d^n p}{(2\pi)^n} |\widetilde f_j(p)-\widetilde f_k(p)|^2 =\|f_j-f_k\|_2^2.$$

Thus $(\widetilde f_j)$ is Cauchy in $L^2(\mathrm d^n p/(2\pi)^n)$. Define $\widetilde f$ to be its norm limit. If a second approximating sequence is used, the same identity applied to the difference shows that it has the same limit. The inverse transform extends in the same way, so

$$\mathcal F: L^2(\mathbb R^n,\mathrm d^n x) \longrightarrow L^2\!\left(\mathbb R^n, \frac{\mathrm d^n p}{(2\pi)^n}\right)$$

is unitary. With ordinary Lebesgue measure $\mathrm d^n p$ on the target, the equivalent norm statement is

$$\|\widetilde f\|_{L^2(\mathrm d^n p)} =(2\pi)^{n/2}\|f\|_{L^2(\mathrm d^n x)}.$$

For $f\in L^1\cap L^2$, the integral transform and the Plancherel extension agree almost everywhere. The density construction, together with the warning that a general $L^2$ transform is not a pointwise integral, is given in Axler 2020, Theorem 11.82 and Definitions 11.86–11.87, pp. 375–377, author-hosted open-access PDF. The full $d$-dimensional extension and inner-product identity appear in Dyatlov 2022, Theorem 11.29 and Remark 11.30, pp. 131–132, official MIT lecture-note PDF; the norm factor there becomes the target measure $\mathrm d^n p/(2\pi)^n$ used here.

Plancherel also transports strong convergence exactly:

$$f_j\to f\text{ in }L^2(\mathrm d^n x) \quad\Longleftrightarrow\quad \widetilde f_j\to\widetilde f \text{ in }L^2\!\left(\frac{\mathrm d^n p}{(2\pi)^n}\right).$$

It does not turn norm convergence into pointwise convergence.

What each function class permits

Guarantees supplied by each Fourier-analysis function class

Input class	What is guaranteed
Schwartz class 𝒮	Pointwise forward and inverse transforms; all displayed operational identities.
L¹	A bounded uniformly continuous transform that vanishes at infinity; inversion needs extra hypotheses.
L¹ ∩ L²	The integral transform agrees almost everywhere with the Plancherel transform.
L²	A forward transform and inverse defined in norm as equivalence classes; no general pointwise integral.
Plane waves, constants, deltas, and singular kernels	Stop ordinary-function calculus and use tempered distributions.

A Gaussian normalization and sign check

For $a>0$, let

$$g_a(x)=e^{-a|x|^2/2}.$$

Because $x_jg_a=-a^{-1}\partial_jg_a$, the operational rules give

$$\partial_{p_j}\widetilde g_a(p) =i\widetilde{x_jg_a}(p) =-\frac{p_j}{a}\widetilde g_a(p).$$

Therefore the transform is a Gaussian in $p$. Evaluating it at $p=0$ fixes the constant:

$$\widetilde g_a(p) =\left(\frac{2\pi}{a}\right)^{n/2} e^{-|p|^2/(2a)}.$$

This is the scaled $n$-dimensional version of Axler 2020, Example 11.51, pp. 365–366, author-hosted open-access PDF. Substitution in the inverse formula returns $g_a$ with no residual factor. Plancherel gives the independent check

$$\begin{aligned} \int\mathrm d^n x\,|g_a(x)|^2 &=\left(\frac{\pi}{a}\right)^{n/2},\\ \int\frac{\mathrm d^n p}{(2\pi)^n} |\widetilde g_a(p)|^2 &=\left(\frac{\pi}{a}\right)^{n/2}. \end{aligned}$$

Finally, $g_a(x-b)$ has transform $e^{+ip\cdot b}\widetilde g_a(p)$. The Gaussian therefore checks the phase, the inverse measure, the $2\pi$ power, and the translation sign at once.

From a periodic box to continuous momentum

The momentum spacing in direction $j$ is $\Delta k_j=2\pi/L_j$. Hence, for $H\in\mathcal S(\mathbb R^n)$,

$$\begin{aligned} \frac1V\sum_{r\in\mathbb Z^n}H(k_r) &=\sum_{r\in\mathbb Z^n} \left[\prod_{j=1}^n\frac{\Delta k_j}{2\pi}\right]H(k_r)\\ &\longrightarrow \int_{\mathbb R^n}\frac{\mathrm d^n k}{(2\pi)^n}H(k) \end{aligned}$$

as every $L_j\to\infty$. This Riemann-sum limit explains the matching between $V^{-1}\sum_r$ and $\int\mathrm d^n k/(2\pi)^n$; it does not authorize replacing every sum by an integral. Boundary effects, zero modes, and insufficient decay can obstruct the limit. Poisson summation on the next page makes the discrete–continuous comparison more precise.

Position–momentum representations and free modes

Consider a classical real scalar field in $(1+n)$-dimensional Minkowski spacetime with the site metric $(+---)$ and periodic spatial box $\Omega_L$. At fixed time, use

$$\begin{aligned} \widehat\phi_r(t) &=\int_{\Omega_L}\mathrm d^n x\, e^{-ik_r\cdot x}\phi(t,x),\\ \phi(t,x) &=\frac1V\sum_{r\in\mathbb Z^n} e^{+ik_r\cdot x}\widehat\phi_r(t). \end{aligned}$$

The spatial phase is deliberately opposite to the analytic convention used above. It is the spatial part of the site’s spacetime phase because

$$p\cdot x=p^0t-\mathbf p\cdot\mathbf x, \qquad e^{+ip\cdot x} =e^{+ip^0t}e^{-i\mathbf p\cdot\mathbf x}.$$

Equivalently, $\widehat\phi_r=\widetilde\phi_{-r}$ relative to the earlier periodic notation. Thus the fixed-time rules are

$$\widehat{\partial_j\phi}_r =+ik_{r,j}\widehat\phi_r, \qquad \widehat{-\nabla^2\phi}_r =|k_r|^2\widehat\phi_r.$$

The Klein–Gordon equation

$$(\partial_t^2-\nabla^2+m^2)\phi=0$$

therefore becomes one ordinary oscillator equation per momentum:

$$\left(\partial_t^2+\omega_r^2\right)\widehat\phi_r(t)=0, \qquad \omega_r^2=|k_r|^2+m^2.$$

When $\omega_r>0$, each coefficient has the form

$$\widehat\phi_r(t) =A_r e^{-i\omega_rt}+B_r e^{+i\omega_rt}.$$

There is one endpoint exception. If the field is massless and $r=0$, then $\omega_0=0$ and the mode equation is $\partial_t^2\widehat\phi_0=0$, so

$$\widehat\phi_0(t)=A_0+B_0t.$$

The massless zero mode must therefore be handled separately; two coincident constant exponentials do not span its solution space.

Reality in position space is equivalent to

$$\widehat\phi_{-r}(t) =\overline{\widehat\phi_r(t)}.$$

Parseval independently checks the diagonalization. If $\phi(t,\cdot)\in H^1(\Omega_L)$ and $\partial_t\phi(t,\cdot)\in L^2(\Omega_L)$, the classical quadratic energy is

$$E(t) =\frac12\int_{\Omega_L}\mathrm d^n x\, \left[ (\partial_t\phi)^2+|\nabla\phi|^2+m^2\phi^2 \right],$$

and mode space gives

$$E(t) =\frac1{2V}\sum_{r\in\mathbb Z^n} \left[ |\partial_t\widehat\phi_r|^2 +\omega_r^2|\widehat\phi_r|^2 \right].$$

The reduction of the free Klein–Gordon equation to independent momentum-space oscillators is also given in Tong 2006–2007, § 2.1, p. 21, equations (2.4)–(2.7), official lecture-note PDF. Tong writes the same inverse spatial phase $e^{+i\mathbf p\cdot\mathbf x}$; the boxed convention above is its finite-volume counterpart.

This calculation is deliberately finite-volume and classical. It does not choose positive frequency, fix symplectic normalization, quantize the mode coefficients, or treat continuum plane waves as ordinary $L^2$ functions. The Klein–Gordon Field and Its Modes develops those physical choices, while Foundations places the position–momentum description in the broader free-field construction.

Diagnostics and common pitfalls

Mixing transform conventions. A sign change in the forward exponential must be accompanied by the matching inverse phase. A frequency rescaling by $2\pi$ also changes the momentum measure.

Treating every $L^2$ transform as an integral. An $L^2$ function need not belong to $L^1$. Its Fourier transform is defined by completion and is itself an almost-everywhere class.

Reading mean-square convergence pointwise. Fourier-series convergence in $L^2$ neither chooses the value at a jump nor proves uniform convergence. Always state the convergence sense.

Dropping the boundary term. The derivative rule requires decay, periodicity, compact support, or a weak/distributional formulation that controls integration by parts.

Using product–convolution identities without Fubini. The displayed proof is safe for Schwartz functions. Broader classes need explicit integrability or norm estimates.

Transforming a plane wave as an ordinary function. A continuum plane wave has neither finite $L^1$ nor finite $L^2$ norm. Its delta-function transform is a distributional statement.

Confusing a Fourier series with sampled-data Fourier analysis. A periodic continuum function has infinitely many Fourier coefficients. A discrete Fourier transform additionally samples and truncates, introducing aliasing questions not treated here.

Forgetting the opposite-momentum reality condition. A real field does not have independent complex coefficients at $k$ and $-k$.

Replacing a box sum by an integral automatically. The replacement is a limit with decay and convergence hypotheses, not a typographical rule.

Confusing spatial modes with a quantum particle split. Fourier analysis diagonalizes the classical linear equation. Positive frequency, vacuum, and operator-valued fields require additional physical structure.

What Fourier analysis now controls

Fourier series give a discrete orthogonal decomposition on a periodic domain; Fourier transforms give the continuous counterpart on $\mathbb R^n$. Translations act by phases, constant-coefficient derivatives act by polynomials in momentum, and products are converted into convolutions. Parseval–Plancherel preserves the quadratic inner product exactly when the momentum measure matches the transform convention. Those statements are pointwise on a safe test class and extend to all of $L^2$ by completion.

Useful continuations are:

• Convolution, Approximate Identities, and Poisson Summation develops convolution bounds, smoothing limits, periodization, and controlled discrete–continuous identities.
• Tempered Distributions and Fourier Calculus treats plane waves, deltas, polynomial growth, and singular Fourier transforms.
• Symbols, Characteristics, and PDE Type uses the multiplier $P(-ip)$ to read differential operators in momentum space.
• Sturm–Liouville Problems and Eigenfunction Expansions generalizes the mode-expansion idea to nontrivial domains, weights, and boundary conditions.
• The Klein–Gordon Field and Its Modes supplies the exact physical treatment of the free scalar modes introduced here.

Exercises

Phase and derivative signs. Starting from the positive-forward transform, derive the transforms of $f(x-a)$ and $\partial_jf(x)$. Which hypotheses enter the two derivations?

Solution

For translation, set $y=x-a$:

$$\int\mathrm d^n x\,e^{ip\cdot x}f(x-a) =e^{ip\cdot a}\widetilde f(p).$$

This requires a legitimate translation change of variables, which holds for $L^1$ functions and for the Schwartz class. For the derivative, integration by parts gives

$$\int\mathrm d^n x\,e^{ip\cdot x}\partial_jf(x) =-ip_j\widetilde f(p)$$

when the boundary term vanishes. Schwartz decay is a sufficient hypothesis; periodicity is another in the Fourier-series setting.

One periodic mode. On $\Omega_L$, let $f(x)=e^{-ik_s\cdot x}$. Compute every $\widetilde f_r$ and verify Parseval.

Solution

Orthogonality gives

$$\widetilde f_r =\int_{\Omega_L}\mathrm d^n x\, e^{i(k_r-k_s)\cdot x} =V\delta_{rs}.$$

The position-space norm is $\int_{\Omega_L}|f|^2=V$. The coefficient side is

$$\frac1V\sum_r|\widetilde f_r|^2 =\frac1V V^2 =V,$$

so the normalization agrees.

An $L^2$ transform without an $L^1$ integral. Show that $f(x)=(1+|x|)^{-3/4}$ belongs to $L^2(\mathbb R)$ but not to $L^1(\mathbb R)$. In what sense does its Fourier transform exist?

Solution

The function is bounded near the origin. At infinity, $|f(x)|$ is comparable to $|x|^{-3/4}$, whose integral diverges, whereas $|f(x)|^2$ is comparable to $|x|^{-3/2}$, whose integral converges. Thus $f\in L^2\setminus L^1$.

Choose $f_j\in\mathcal S$ with $f_j\to f$ in $L^2$. Plancherel makes $\widetilde f_j$ Cauchy in $L^2(\mathrm dp/(2\pi))$; its limit is the Fourier transform of $f$. The literal absolutely convergent integral is not justified, and the result is an almost-everywhere equivalence class.

Gaussian constants. In one dimension, transform $g_a(x)=e^{-ax^2/2}$ and verify both sides of Plancherel.

Solution

The transform is

$$\widetilde g_a(p) =\sqrt{\frac{2\pi}{a}}\,e^{-p^2/(2a)}.$$

Then

$$\int_{\mathbb R}|g_a(x)|^2\,\mathrm dx =\int_{\mathbb R}e^{-ax^2}\,\mathrm dx =\sqrt{\frac\pi a},$$

while

$$\begin{aligned} \int_{\mathbb R}\frac{\mathrm dp}{2\pi} |\widetilde g_a(p)|^2 &=\frac1a\int_{\mathbb R}e^{-p^2/a}\,\mathrm dp\\ &=\sqrt{\frac\pi a}. \end{aligned}$$

The $2\pi$ factor and target measure are therefore consistent.

Free modes in the spatial convention. Apply the fixed-time transform with phase $e^{-ik_r\cdot x}$ to the Klein–Gordon equation. Derive the oscillator frequency and the reality condition.

Solution

The spatial derivative maps to $+ik_{r,j}$, so $-\nabla^2$ maps to $|k_r|^2$. Therefore

$$\left(\partial_t^2+|k_r|^2+m^2\right) \widehat\phi_r(t)=0,$$

with $\omega_r=\sqrt{|k_r|^2+m^2}$. If $\phi$ is real, then

$$\begin{aligned} \overline{\widehat\phi_r(t)} &=\int_{\Omega_L}\mathrm d^n x\, e^{+ik_r\cdot x}\phi(t,x)\\ &=\widehat\phi_{-r}(t). \end{aligned}$$

Thus the opposite momenta are conjugate rather than independent.

References

• Sheldon Axler, Measure, Integration & Real Analysis, Graduate Texts in Mathematics 282, Springer (2020), Chapter 11, pp. 340–379, author-hosted open-access PDF. Fourier coefficients, $L^2$ Fourier-series convergence, transform rules, inversion, Gaussian normalization, and the Plancherel extension.
• Semyon Dyatlov, Lecture Notes for 18.155: Differential Analysis, MIT (2022), Dyatlov 2022, §§ 11.1–11.2, pp. 119–132, official lecture-note PDF. Schwartz-space calculus, Gaussian-regularized inversion, and the full $L^2$ Plancherel extension in $n$ dimensions.
• Jared Speck, 18.152 Introduction to Partial Differential Equations: Class Meetings 16–18, The Fourier Transform on $\mathbb R^n$, MIT OpenCourseWare (Fall 2011), pp. 1–11, open course-notes PDF. $n$-dimensional Gaussian regularization and Fourier inversion under the stated $L^1$ hypotheses.
• David Tong, Quantum Field Theory, Cambridge Part III lecture notes (2006–2007), Tong 2006–2007, § 2.1, p. 21, official lecture-note PDF. Spatial Fourier decomposition of the free Klein–Gordon equation into oscillator modes.