Fourier Analysis in Hilbert Spaces

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Why Fourier Belongs in Hilbert Space Fourier Series as an Orthonormal Basis The Riemann-Lebesgue Lemma Plancherel: The Fourier Transform as a Unitary

Why Fourier Belongs in Hilbert Space

Two earlier pages have approached Fourier analysis from very different vantage points. In Fourier Series, we decomposed periodic functions into discrete frequencies and asserted Parseval's identity. In Fourier Transform, we generalized to functions on \(\mathbb{R}\) and stated Plancherel's theorem. In both pages, several foundational results were stated, used, or sketched. Some were explicitly deferred to "the forthcoming page on Fourier analysis in Hilbert spaces." Others, notably the classical Parseval identity for Fourier series, were proved directly but await recognition as instances of a more general Hilbert-space structure. This is that page.

The framework that supplies it is the theory of Hilbert spaces with their orthonormal bases, developed in the functional-analysis block, together with the Riesz-Fischer theorem that places \(L^2\) inside it. With these tools in hand, the classical \(L^2\) Fourier results cease to be a separate subject. They become specific instances of Hilbert-space machinery, recognized retrospectively in the sections that follow. Three constructions do the work: orthonormal-basis decomposition, Bessel's inequality, and bounded extension from a dense subspace.

A different generalization runs in parallel to the one developed here. The complex exponentials are the symmetry functions of the circle, and replacing the circle by a noncommutative compact group leads to a harmonic analysis in which they are superseded by matrix-valued building blocks indexed by the group's symmetries. That theory, the Peter-Weyl theorem, takes the orthonormal-basis decomposition recovered here as its abelian prototype. The present page is the commutative special case of a single general phenomenon.

Convention.
Throughout this page, all Hilbert spaces are taken over \(\mathbb{C}\), with inner product linear in the first slot and conjugate-linear in the second. We inherit the mathematical/PDE convention from the Fourier-series and Fourier-transform pages (positive sign in the coefficient integral and in the forward transform).

Explicitly, the complex Fourier series on \([-L, L]\) takes the form \(f(x) = \sum_n c_n\, e^{-i n \pi x / L}\) with coefficients \(c_n = \tfrac{1}{2L}\int_{-L}^{L} f(x)\, e^{+i n \pi x / L}\, dx\). The Fourier transform on \(\mathbb{R}\) is \(\hat{f}(\xi) = \int_{\mathbb{R}} f(x)\, e^{i\xi x}\, dx\) from the Fourier Transform page. Under this convention, the normalized map \(\mathcal{U}: f \mapsto (2\pi)^{-1/2}\hat{f}\) is the operator we will identify as unitary on \(L^2(\mathbb{R})\).

Fourier Series as an Orthonormal Basis

The Fourier-series page introduced the complex exponential system \(\{e^{-in\pi x/L}\}_{n \in \mathbb{Z}}\) on \([-L, L]\) and showed by direct computation that it is orthogonal with respect to the inner product \(\langle f, g \rangle = \frac{1}{2L}\int_{-L}^{L} f(x)\overline{g(x)}\, dx\). Normalizing, we obtain the orthonormal system \[ e_n(x) := \frac{1}{\sqrt{2L}}\, e^{-i n \pi x / L}, \quad n \in \mathbb{Z}, \] in the Hilbert space \(\mathcal{H} = L^2([-L, L])\) equipped with the unnormalized inner product \(\langle f, g \rangle = \int_{-L}^{L} f(x)\overline{g(x)}\, dx\).

The relationship between this normalization and the normalized inner product used in the Fourier-series page is a uniform rescaling by \(2L\). Either choice produces the same orthonormal system after rescaling the basis vectors. We use the unnormalized inner product here to align with the functional-analysis block.

The system \(\{e_n\}_{n \in \mathbb{Z}}\) is orthonormal, by the elementary computation carried out in the Fourier-series page and transported through the rescaling. What is not elementary, and what was left open at that stage, is whether this orthonormal system is in fact an orthonormal basis of \(L^2([-L, L])\): that is, whether the closed linear span of \(\{e_n\}\) is all of \(L^2([-L, L])\). Once this is established, Parseval's identity for Fourier series becomes a special case of the abstract Parseval identity in Hilbert spaces. The convergence of partial sums in \(L^2\), invoked but not proved in the Fourier-series page, becomes the expansion statement of that same theorem.

Strategy: from orthogonality to a basis

One characterization of orthonormal bases in Hilbert spaces is decisive here, and it is equivalent to the closed-span definition adopted in Intro to Functional Analysis. An orthonormal set \(\{e_n\}\) in \(\mathcal{H}\) is a basis if and only if the only vector orthogonal to every \(e_n\) is the zero vector. That is, if \(\langle h, e_n \rangle = 0\) for all \(n\) forces \(h = 0\), then the closed span equals \(\mathcal{H}\), and conversely. Each direction is immediate from the orthogonal-decomposition theorem. If the closed span \(M\) is a proper subspace, then the orthogonal complement \(M^\perp\) contains a nonzero vector, contradicting the implication. Conversely, any vector orthogonal to all \(e_n\) lies in \(M^\perp\), which is \(\{0\}\) when \(M = \mathcal{H}\).

For the trigonometric system, this strategy translates into a concrete claim. If \(f \in L^2([-L, L])\) has every Fourier coefficient \(\int_{-L}^{L} f(x)\, e^{+in\pi x/L}\, dx = 0\) for \(n \in \mathbb{Z}\), then \(f = 0\) almost everywhere. The technical core of the argument lies in approximating an arbitrary continuous periodic function by trigonometric polynomials, and then using density of continuous periodic functions in \(L^2\) to conclude.

Trigonometric approximation via the Fejér kernel

For the approximation step we use the classical Fejér kernel construction. We work in the normalization of the Fourier-series page, so that \(c_n\) carries a factor of \(1/(2L)\). The rescaling that relates these \(c_n\) to the abstract inner products \(\langle f, e_n \rangle\) will be reinstated at the end of this section. For a continuous \(2L\)-periodic function \(f\), define the partial sums and Fourier coefficients by \[ \begin{align*} S_N f(x) &= \sum_{|n| \leq N} c_n\, e^{-i n \pi x / L}, \\\\ c_n &= \frac{1}{2L} \int_{-L}^{L} f(y)\, e^{+i n \pi y / L}\, dy, \end{align*} \] and define the Cesàro means \[ \sigma_N f(x) = \frac{S_0 f(x) + S_1 f(x) + \cdots + S_{N-1} f(x)}{N}. \] Substituting the integral expression for \(c_n\) and switching summation with integration (a finite sum, so the interchange is unconditional) gives \[ \sigma_N f(x) = \frac{1}{2L}\int_{-L}^{L} f(y)\, F_N\!\left(\frac{\pi(x-y)}{L}\right) dy, \] where the Fejér kernel \(F_N\) is obtained by tracking the same computation through each partial sum. Writing \(S_n f(x) = \tfrac{1}{2\pi}\int_{-\pi}^{\pi} f(x - Lt/\pi)\, D_n(t)\, dt\) with the Dirichlet kernel \(D_n(t) = \sum_{k=-n}^{n} e^{ikt}\) and averaging over \(n = 0, \ldots, N-1\) gives \(F_N = \tfrac{1}{N}(D_0 + D_1 + \cdots + D_{N-1})\), the Cesàro average of the Dirichlet kernels.

The Fejér kernel admits the closed form \[ F_N(t) = \frac{1}{N}\, \frac{\sin^2(N t / 2)}{\sin^2(t / 2)}, \quad t \in (-\pi, \pi) \setminus \{0\}, \] with the natural limiting value \(F_N(0) = N\), obtained from \(\lim_{t \to 0} \sin^2(Nt/2)/\sin^2(t/2) = N^2\). That value makes \(F_N\) continuous at \(0\) and the closed form valid on all of \([-\pi, \pi]\).

To see how this closed form arises, recall the standard geometric-series evaluation \(D_n(t) = \frac{\sin((n + \tfrac{1}{2}) t)}{\sin(t/2)}\), valid for \(t \not\equiv 0 \pmod{2\pi}\). Substituting this into \(F_N = \tfrac{1}{N}(D_0 + D_1 + \cdots + D_{N-1})\), \[ F_N(t) = \frac{1}{N\, \sin(t/2)} \sum_{n=0}^{N-1} \sin\!\bigl((n + \tfrac{1}{2})\, t\bigr). \] The inner sum is a sum of sines in arithmetic progression. Using the product-to-sum identity \(2 \sin(\alpha) \sin(\beta) = \cos(\alpha - \beta) - \cos(\alpha + \beta)\) with \(\alpha = (n + \tfrac{1}{2}) t\) and \(\beta = t/2\), each term becomes a telescoping difference of cosines: \[ 2 \sin(t/2) \sin\!\bigl((n + \tfrac{1}{2}) t\bigr) = \cos(n t) - \cos\!\bigl((n + 1) t\bigr). \] Summing from \(n = 0\) to \(N - 1\) telescopes to \(1 - \cos(N t) = 2 \sin^2(N t / 2)\). Dividing by \(2 \sin(t/2)\) gives \(\sum_{n=0}^{N-1} \sin((n + \tfrac{1}{2}) t) = \tfrac{\sin^2(N t / 2)}{\sin(t / 2)}\). Substituting back into the expression for \(F_N\) yields the displayed closed form.

Three properties of \(F_N\) drive the approximation argument:

These three properties characterize what is classically called a good kernel or approximate identity. From them follows the central approximation result:

Theorem: Fejér's Theorem (uniform version)

If \(f\) is continuous and \(2L\)-periodic on \(\mathbb{R}\), then the Cesàro means \(\sigma_N f\) converge to \(f\) uniformly on \(\mathbb{R}\) as \(N \to \infty\). In particular, every continuous \(2L\)-periodic function is the uniform limit of trigonometric polynomials.

Proof:

Fix \(\varepsilon \gt 0\). Because \(f\) is continuous and periodic, it is uniformly continuous on \(\mathbb{R}\). Choose \(\delta \in (0, \pi)\) such that \(|f(x - s) - f(x)| \lt \varepsilon\) for all \(x\) whenever \(|s| \lt \delta L / \pi\). Since \(f\) is \(2L\)-periodic and \(F_N\) is \(2\pi\)-periodic, the integrand in the convolution representation of \(\sigma_N f\) is \(2\pi\)-periodic in the variable \(t = \pi(x-y)/L\), so the integration window may be taken to be \([-\pi, \pi]\). Using the unit-mass property of \(F_N\) to rewrite \(f(x)\) as \(\tfrac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\, F_N(t)\, dt\), we obtain \[ \sigma_N f(x) - f(x) = \frac{1}{2\pi}\int_{-\pi}^{\pi} \bigl[ f(x - L t / \pi) - f(x) \bigr]\, F_N(t)\, dt. \]

Split the integral into \(|t| \lt \delta\) and \(\delta \leq |t| \leq \pi\). On the first region, \(|f(x - Lt/\pi) - f(x)| \lt \varepsilon\) for every \(x\) by uniform continuity, so positivity of \(F_N\) and unit mass give the bound, uniform in \(x\), \[ \left| \frac{1}{2\pi}\int_{|t| \lt \delta} \bigl[f(x - Lt/\pi) - f(x)\bigr] F_N(t)\, dt \right| \leq \varepsilon. \]

On the second region, \(|f(x - Lt/\pi) - f(x)| \leq 2 \sup_{\mathbb{R}} |f|\), where \(\sup_{\mathbb{R}} |f| \lt \infty\) by continuity on a compact period. Concentration of \(F_N\) gives \[ \begin{align*} \left| \frac{1}{2\pi}\int_{\delta \leq |t| \leq \pi} \bigl[f(x - Lt/\pi) - f(x)\bigr] F_N(t)\, dt \right| &\leq \frac{2 \sup_{\mathbb{R}} |f|}{2\pi} \int_{\delta \leq |t| \leq \pi} F_N(t)\, dt \\\\ &\longrightarrow 0 \end{align*} \] as \(N \to \infty\), uniformly in \(x\). Combining the two estimates, \(\sup_{\mathbb{R}} |\sigma_N f - f| \leq \varepsilon + \eta_N\), where \(\eta_N \to 0\) as \(N \to \infty\). Since \(\varepsilon\) was arbitrary, \(\sigma_N f \to f\) uniformly. Each \(\sigma_N f\) is a finite linear combination of the \(e^{-in\pi x/L}\), so it is a trigonometric polynomial.

Fejér's theorem is the analytic ingredient behind the following determinacy statement. A continuous periodic function is determined by its full set of Fourier coefficients, because if all of them were zero the Cesàro means would be identically zero, and uniform convergence would then force \(f\) to be zero. This observation, combined with the density of continuous periodic functions in \(L^2\), yields the main theorem of this section.

The trigonometric system is a basis of \(L^2([-L, L])\)

Theorem: Completeness of the Trigonometric System

The orthonormal system \(\{e_n\}_{n \in \mathbb{Z}}\), with \(e_n(x) = (2L)^{-1/2} e^{-in\pi x/L}\), is an orthonormal basis of \(L^2([-L, L])\). Equivalently, the only function \(f \in L^2([-L, L])\) with \(\int_{-L}^{L} f(x)\, e^{+in\pi x/L}\, dx = 0\) for every \(n \in \mathbb{Z}\) is the zero function (almost everywhere).

Proof:

Two preliminaries.
First, the Hilbert space \(L^2([-L, L])\) is separable. The density argument below shows that the trigonometric polynomials are dense in \(L^2([-L, L])\), and replacing each coefficient of such a polynomial by a nearby rational-complex number moves it by an arbitrarily small amount in \(L^2\) norm. The polynomials with rational-complex coefficients therefore form a countable dense subset, so the abstract orthonormal-basis machinery applies. Second, the abstract definition of orthonormal basis and the abstract Parseval identity were stated for \(\mathbb{N}\)-indexed sequences \(\{e_n\}_{n=1}^\infty\), whereas our trigonometric system is indexed by \(\mathbb{Z}\). Any bijection \(\phi: \mathbb{Z} \to \mathbb{N}\) transfers \(\{e_n\}_{n \in \mathbb{Z}}\) to a sequence \(\{e_{\phi^{-1}(k)}\}_{k=1}^\infty\) of the abstract form. The relevant Parseval series \(\sum_k |\langle f, e_{\phi^{-1}(k)} \rangle|^2\) consists of non-negative terms, so it converges absolutely and its value is independent of \(\phi\). We may therefore write the result in the original notation \(\sum_{n \in \mathbb{Z}}\).

By the characterization recalled above, it suffices to show that \(f = 0\) almost everywhere whenever \(f \in L^2([-L, L])\) satisfies \(\langle f, e_n \rangle = 0\) for every \(n \in \mathbb{Z}\). We extend \(f\) to a \(2L\)-periodic function on \(\mathbb{R}\) by periodicity. The orthogonality condition states that all Fourier coefficients of the extension vanish.

Reduction to continuous test functions.
The continuous \(2L\)-periodic functions form a dense subspace of \(L^2([-L, L])\). We sketch this in three steps.

(i) Simple functions are dense in \(L^2([-L, L])\). Here a simple function is a finite linear combination of indicator functions of measurable sets of finite measure. Write \(g \in L^2\) as a signed combination of four non-negative measurable parts, \[ g = (\Re g)^+ - (\Re g)^- + i\,(\Im g)^+ - i\,(\Im g)^- \] each dominated pointwise by \(|g|\). Each part admits a monotone increasing sequence of non-negative simple functions converging to it pointwise, by the standard construction in the theory of the Lebesgue integral, and the resulting complex simple functions \(s_n\) satisfy \(|s_n| \leq |g|\) pointwise. Since \(|s_n - g|^2 \to 0\) pointwise and is dominated by \(4|g|^2 \in L^1\) (because \(g \in L^2\)), the dominated convergence theorem gives \(\|s_n - g\|_2 \to 0\).

(ii) Indicators of measurable sets of finite measure are approximated in \(L^2\) by indicators of finite unions of open intervals. By outer regularity of Lebesgue measure, every such set \(E\) is contained in an open set \(U\) with \(|U \setminus E| \lt \varepsilon^2 / 2\). Decomposing \(U\) as a countable disjoint union of open intervals \(\bigsqcup_k I_k\) and truncating to a finite union \(V = I_1 \cup \cdots \cup I_K\) with \(\sum_{k \gt K} |I_k| \lt \varepsilon^2 / 2\) gives \(|V \triangle E| \lt \varepsilon^2\), hence \(\|\mathbf{1}_V - \mathbf{1}_E\|_2 \lt \varepsilon\).

(iii) An indicator of an open interval \((a, b) \subset [-L, L]\) is approximated in \(L^2\) by continuous functions vanishing outside \([a - \varepsilon, b + \varepsilon]\) and equal to \(1\) on \([a, b]\), a trapezoidal cutoff. Choosing \(\varepsilon\) small enough that \([a - \varepsilon, b + \varepsilon] \subset (-L, L)\), each such function extends by zero to a continuous \(2L\)-periodic function on \(\mathbb{R}\).

Chaining the three approximations, every \(g \in L^2([-L, L])\) is the \(L^2\)-limit of continuous \(2L\)-periodic functions. It therefore suffices to show \(\langle f, g \rangle = 0\) for every continuous \(2L\)-periodic \(g\) on \(\mathbb{R}\), since \(f\) is then orthogonal to a dense subset of \(L^2([-L, L])\) and must vanish.

From trigonometric polynomials to continuous \(g\) via Fejér.
Fix a continuous \(2L\)-periodic \(g\). By Fejér's theorem above, there exist trigonometric polynomials \(P_N(x) = \sum_{|n| \leq m_N} a_n^{(N)} e^{-in\pi x / L}\) such that \(P_N \to g\) uniformly on \([-L, L]\). Uniform convergence on a bounded interval implies \(L^2\) convergence, since \[ \begin{align*} \int_{-L}^{L} |P_N(x) - g(x)|^2\, dx &\leq 2L\, \|P_N - g\|_\infty^2 \\\\ &\to 0. \end{align*} \] By the Cauchy-Schwarz inequality, \[ \begin{align*} |\langle f, P_N \rangle - \langle f, g \rangle| &= |\langle f, P_N - g \rangle| \\\\ &\leq \|f\|_2\, \|P_N - g\|_2 \\\\ &\to 0, \end{align*} \] so \(\langle f, P_N \rangle \to \langle f, g \rangle\). But each \(P_N\) is a finite linear combination of the \(e^{-in\pi x/L}\), and \(f\) is assumed orthogonal to every such exponential, so \(\langle f, P_N \rangle = 0\) for every \(N\). Passing to the limit, \(\langle f, g \rangle = 0\).

Since \(g\) was an arbitrary continuous \(2L\)-periodic function, the density argument forces \(f = 0\) in \(L^2([-L, L])\), which is the same as \(f = 0\) almost everywhere.

Retrospective recognition: classical Parseval as a Hilbert-space corollary

With the orthonormal-basis property established, the Hilbert-space Parseval identity applies directly to the trigonometric system. Specializing the abstract statement \(\|h\|^2 = \sum_n |\langle h, e_n \rangle|^2\) to \(\mathcal{H} = L^2([-L, L])\) and \(e_n(x) = (2L)^{-1/2} e^{-in\pi x/L}\), the inner product becomes \[ \begin{align*} \langle f, e_n \rangle &= \int_{-L}^{L} f(x)\, \overline{e_n(x)}\, dx \\\\ &= \frac{1}{\sqrt{2L}} \int_{-L}^{L} f(x)\, e^{+i n \pi x / L}\, dx \\\\ &= \sqrt{2L}\, c_n, \end{align*} \] where the \(c_n\) are the complex Fourier coefficients taken in the Fourier-series normalization.

Substituting, the abstract Parseval identity becomes \[ \begin{align*} \int_{-L}^{L} |f(x)|^2\, dx &= \sum_{n \in \mathbb{Z}} \bigl| \sqrt{2L}\, c_n \bigr|^2 \\\\ &= 2L \sum_{n \in \mathbb{Z}} |c_n|^2, \end{align*} \] which after dividing by \(2L\) is exactly Parseval's identity in its classical complex-exponential form.

The expansion statement of the abstract Parseval theorem similarly produces the \(L^2\)-convergence of partial sums \[ \begin{align*} S_N f &= \sum_{|n| \leq N} c_n\, e^{-in\pi x/L} \\\\ &\to f \end{align*} \] in \(L^2([-L, L])\) for every \(f \in L^2([-L, L])\). This settles the mean-square convergence claim of the Fourier-series page, whose proof was deferred to "general Hilbert-space theory applied to an orthonormal basis."

The Riemann-Lebesgue Lemma

A recurring theme in classical Fourier analysis is that "high-frequency Fourier coefficients are small." This intuition is made precise by the Riemann-Lebesgue lemma. The Fourier coefficients (for series) or the Fourier transform (for functions on \(\mathbb{R}\)) of an integrable function decay to zero at high frequency. The lemma was invoked, but not proved, in the Fourier series page during the outline of pointwise (Dirichlet-Jordan) convergence, with a forward-pointer to the present page for the proof. With the Hilbert-space framework now in place, the lemma is an almost immediate corollary of Bessel's inequality applied to the trigonometric system.

The \(L^2\) statement: a direct corollary of Bessel's inequality

The abstract Bessel inequality states that for any orthonormal sequence \(\{e_n\}\) in a Hilbert space and any vector \(x\), the series \(\sum_{n} |\langle x, e_n \rangle|^2\) converges and is bounded above by \(\|x\|^2\). Convergence of a non-negative series forces its terms to tend to zero. Specializing to the Hilbert space \(L^2([-L, L])\) with the orthonormal basis \(\{e_n(x) = (2L)^{-1/2} e^{-i n \pi x / L}\}_{n \in \mathbb{Z}}\) of the previous section, the Fourier coefficients of any \(L^2\) function must vanish in the limit:

Theorem: Riemann-Lebesgue Lemma (\(L^2\) version)

If \(f \in L^2([-L, L])\), then its Fourier coefficients \(c_n = \tfrac{1}{2L} \int_{-L}^{L} f(x)\, e^{+i n \pi x / L}\, dx\) satisfy \(c_n \to 0\) as \(|n| \to \infty\).

Proof:

Let \(\langle \cdot, \cdot \rangle\) denote the unnormalized inner product \(\langle f, g \rangle = \int_{-L}^{L} f(x) \overline{g(x)}\, dx\) on \(L^2([-L, L])\). Computing the inner product of \(f\) against \(e_n(x) = (2L)^{-1/2} e^{-in\pi x/L}\), \[ \begin{align*} \langle f, e_n \rangle &= \frac{1}{\sqrt{2L}} \int_{-L}^{L} f(x)\, e^{+i n \pi x / L}\, dx \\\\ &= \sqrt{2L}\, c_n, \end{align*} \] which rearranges to \(|c_n|^2 = (2L)^{-1} |\langle f, e_n \rangle|^2\). By Bessel's inequality applied to the orthonormal sequence \(\{e_n\}_{n \in \mathbb{Z}}\), reindexed by \(\mathbb{N}\) through a bijection as in the proof of completeness of the trigonometric system, \(\sum_{n \in \mathbb{Z}} |\langle f, e_n \rangle|^2 \leq \|f\|_2^2 \lt \infty\). Dividing through by \(2L\), the non-negative series \(\sum |c_n|^2 \leq \|f\|_2^2 / (2L)\) converges, so its general term tends to zero: \(|c_n|^2 \to 0\), hence \(c_n \to 0\).

Extension to \(L^1\) by density

The classical Riemann-Lebesgue lemma is stated for \(L^1\) functions, which form the largest class on which the Fourier transform integral converges absolutely. Crucially, \(L^1\) is neither contained in nor contains \(L^2\) on \(\mathbb{R}\). For instance, \(f(x) = (1 + |x|)^{-1}\) is in \(L^2\) but not \(L^1\), while \(f(x) = |x|^{-1/2}\mathbf{1}_{[0, 1]}(x)\) is in \(L^1\) but not \(L^2\). The bridge between the two is the dense subspace \(C_c(\mathbb{R})\) of continuous compactly supported functions, which lies inside both spaces. The strategy is to prove decay on \(C_c\) using the \(L^2\) result above and then extend to \(L^1\) by approximation.

Theorem: Riemann-Lebesgue Lemma (\(L^1\) version)

If \(f \in L^1(\mathbb{R})\), then its Fourier transform \(\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\, e^{i x \xi}\, dx\) satisfies \[ \lim_{|\xi| \to \infty} \hat{f}(\xi) = 0. \] Moreover \(\hat{f}\) is bounded and continuous on \(\mathbb{R}\).

Proof:

Step 1: continuity of \(\hat{f}\).
For any \(\xi, \xi' \in \mathbb{R}\), \[ |\hat{f}(\xi) - \hat{f}(\xi')| \leq \int |f(x)|\, |e^{i x \xi} - e^{i x \xi'}|\, dx. \] The integrand is bounded by \(2 |f(x)| \in L^1\) and tends pointwise to zero as \(\xi' \to \xi\) (since \(e^{i x \xi'} \to e^{i x \xi}\) pointwise). By the dominated convergence theorem, \(\hat{f}(\xi') \to \hat{f}(\xi)\), so \(\hat{f}\) is continuous on \(\mathbb{R}\). Moreover the trivial bound \(|\hat{f}(\xi)| \leq \|f\|_1\) shows that \(\hat{f}\) is bounded.

Step 2: decay on indicators of bounded intervals.
For the indicator function of a bounded interval, \(f = \mathbf{1}_{[a, b]}\), a direct computation gives \[ \begin{align*} \hat{f}(\xi) &= \int_a^b e^{i x \xi}\, dx \\\\ &= \frac{e^{i b \xi} - e^{i a \xi}}{i \xi}, \quad \xi \neq 0, \end{align*} \] which is bounded in modulus by \(2 / |\xi|\) and therefore tends to zero as \(|\xi| \to \infty\). By linearity, the same holds for every finite linear combination of indicators of bounded intervals, that is, for every step function of bounded support.

Step 3a: step functions of bounded support are dense in \(L^1(\mathbb{R})\).
We argue in three stages, performing truncation first so that all subsequent approximations take place inside a fixed bounded interval.

(1) Truncation to bounded support.
For \(f \in L^1(\mathbb{R})\), set \(f_M := f \cdot \mathbf{1}_{[-M, M]}\). The difference \(f - f_M\) vanishes on \([-M, M]\) and is dominated by \(|f| \in L^1\), so the dominated convergence theorem gives \(\|f - f_M\|_1 \to 0\) as \(M \to \infty\). It therefore suffices to approximate any \(L^1\) function of bounded support.

(2) Simple-function approximation inside \([-M, M]\).
Write the boundedly supported function as a complex combination of four non-negative measurable parts. Each of the four is the pointwise limit of an increasing sequence of non-negative simple functions, by the standard simple-function construction, and the monotone convergence theorem upgrades the pointwise convergence to convergence in \(L^1\). Since each part vanishes outside \([-M, M]\), each approximating simple function may be chosen to vanish there as well, leaving a simple function \(s = \sum_j \alpha_j \mathbf{1}_{E_j}\) with each \(E_j \subseteq [-M, M]\).

(3) Step-function approximation of indicators inside \([-M, M]\).
For each \(E_j \subseteq [-M, M]\), outer regularity of Lebesgue measure produces an open set \(U_j \subseteq \mathbb{R}\) with \(E_j \subseteq U_j\) and \(\mu(U_j \setminus E_j)\) as small as desired. Intersecting with \((-M, M)\), the set \(U_j \cap (-M, M)\) is a countable disjoint union of bounded open intervals. Their total length is at most \(2M\), so the tail can be discarded with arbitrarily small \(L^1\) cost. What remains is a finite disjoint union of bounded open intervals whose indicator approximates \(\mathbf{1}_{E_j}\) in \(L^1\). Chaining the three stages, every \(f \in L^1(\mathbb{R})\) is the \(L^1\)-limit of step functions of bounded support.

Step 3b: \(\varepsilon\)-extension.
Given \(f \in L^1(\mathbb{R})\) and \(\varepsilon \gt 0\), choose a step function \(g\) of bounded support with \(\|f - g\|_1 \lt \varepsilon / 2\). Then \[ \begin{align*} |\hat{f}(\xi)| &\leq |\hat{f}(\xi) - \hat{g}(\xi)| + |\hat{g}(\xi)| \\\\ &\leq \|f - g\|_1 + |\hat{g}(\xi)| \\\\ &\leq \tfrac{\varepsilon}{2} + |\hat{g}(\xi)|, \end{align*} \] where the bound on the difference uses \(|\hat{f}(\xi) - \hat{g}(\xi)| \leq \int |f - g|\, dx = \|f - g\|_1\). By Step 2, \(|\hat{g}(\xi)| \lt \varepsilon / 2\) for \(|\xi|\) sufficiently large, hence \(|\hat{f}(\xi)| \lt \varepsilon\) for all sufficiently large \(|\xi|\). Since \(\varepsilon\) was arbitrary, \(\hat{f}(\xi) \to 0\).

What was Discharged

In the Fourier-series page, the proof outline of the Dirichlet-Jordan pointwise convergence theorem used the Riemann-Lebesgue lemma to control a tail integral. That outline explicitly deferred the proof to "the forthcoming page on Fourier analysis in Hilbert spaces." Both the \(L^2\) form (used directly in the tail estimate) and its \(L^1\) extension (the standard form of the lemma) are now established. The proof reveals what kind of theorem the lemma actually is. Rather than a theorem about oscillatory integrals, it is a theorem about orthonormal bases. Once one views Fourier coefficients as inner products against an orthonormal sequence, their decay is a one-line consequence of the convergence of the Bessel series.

Plancherel: The Fourier Transform as a Unitary

The Fourier-transform page proved Plancherel's identity \(\|f\|_2^2 = \tfrac{1}{2\pi}\|\hat{f}\|_2^2\) on \(L^1 \cap L^2(\mathbb{R})\), using Schwartz functions and Fubini. Two structural questions were left open and, in fact, explicitly flagged for the present page:

The unified answer to both questions, and the structural punchline of this page, is that the normalized Fourier transform \(\mathcal{U} := \mathcal{F}/\sqrt{2\pi}\) extends to a unitary operator on \(L^2(\mathbb{R})\), that is, to a bijective isometry whose inverse coincides with its adjoint. Once this is established, the Fourier transform on \(L^2\) is no longer an integral but an abstract Hilbert-space isomorphism.

Step 1: extending the Fourier transform to \(L^2(\mathbb{R})\)

The natural domain of the classical Fourier transform integral is \(L^1(\mathbb{R})\), while the natural target of an "energy-preserving" theory is \(L^2(\mathbb{R})\). The bridge is the intersection \(L^1 \cap L^2(\mathbb{R})\), on which both the integral definition and Plancherel's identity make sense.

Crucially, \(L^1 \cap L^2\) is a dense subspace of \(L^2(\mathbb{R})\). It contains \(C_c(\mathbb{R})\), which is itself dense in \(L^2(\mathbb{R})\) by the same chain of approximations used in the proof of trigonometric ONB completeness above. That chain runs through simple functions in \(L^2\) via dominated convergence, indicators of measurable sets via outer regularity, and indicators of intervals via trapezoidal cutoffs. We use this density to extend the Fourier transform from its initial domain \(L^1 \cap L^2\) to the entirety of \(L^2(\mathbb{R})\).

On \(L^1 \cap L^2\), set \(\mathcal{U} f := \tfrac{1}{\sqrt{2\pi}}\hat{f}\). The classical Plancherel identity was proved on the Schwartz class and extended to \(L^1 \cap L^2\) by Schwartz density together with norm continuity. It gives \(\|f\|_2^2 = \tfrac{1}{2\pi}\|\hat{f}\|_2^2\) for every \(f \in L^1 \cap L^2\), which rewrites as \(\|\mathcal{U} f\|_2 = \|f\|_2\). Thus \(\mathcal{U}\) is an isometry on \(L^1 \cap L^2\) with values in \(L^2\). Density of \(L^1 \cap L^2\) in \(L^2(\mathbb{R})\), together with completeness of \(L^2\), allows us to extend \(\mathcal{U}\) by continuity to a unique bounded linear operator on all of \(L^2\):

Theorem: The Fourier-Plancherel Transform on \(L^2(\mathbb{R})\)

There exists a unique bounded linear operator \(\mathcal{U} : L^2(\mathbb{R}) \to L^2(\mathbb{R})\) such that for every \(f \in L^1 \cap L^2(\mathbb{R})\), \[ (\mathcal{U} f)(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)\, e^{i x \xi}\, dx \quad \text{(a.e. in } \xi\text{)}, \] and this operator satisfies \(\|\mathcal{U} f\|_2 = \|f\|_2\) for every \(f \in L^2(\mathbb{R})\).

Proof:

Given \(f \in L^2(\mathbb{R})\), choose any sequence \(\{f_n\} \subset L^1 \cap L^2\) with \(f_n \to f\) in \(L^2\) (possible by density). The classical Plancherel identity on \(L^1 \cap L^2\) gives \(\|\mathcal{U} f_n - \mathcal{U} f_m\|_2 = \|f_n - f_m\|_2\) for all \(n, m\), since \(f_n - f_m \in L^1 \cap L^2\) and \(\mathcal{U}\) is linear on this subspace. The Cauchy property of \(\{f_n\}\) in \(L^2\) therefore transfers to \(\{\mathcal{U} f_n\}\). By completeness of \(L^2\), there exists \(\mathcal{U} f \in L^2(\mathbb{R})\) with \(\mathcal{U} f_n \to \mathcal{U} f\) in \(L^2\).

Well-definedness.
If \(\{f_n'\}\) is another sequence in \(L^1 \cap L^2\) with \(f_n' \to f\), then \(\|f_n - f_n'\|_2 \to 0\), so by isometry on \(L^1 \cap L^2\), \(\|\mathcal{U} f_n - \mathcal{U} f_n'\|_2 \to 0\), forcing the two limit candidates to agree.

Linearity, boundedness, isometry.
Linearity passes to the limit termwise. Boundedness follows from the isometric estimate \[ \begin{align*} \|\mathcal{U} f\|_2 &= \lim \|\mathcal{U} f_n\|_2 \\\\ &= \lim \|f_n\|_2 \\\\ &= \|f\|_2, \end{align*} \] proving \(\|\mathcal{U} f\|_2 = \|f\|_2\) for every \(f \in L^2(\mathbb{R})\).

Compatibility with the integral definition.
If \(f \in L^1 \cap L^2\), choosing the constant sequence \(f_n = f\) gives \(\mathcal{U} f = \tfrac{1}{\sqrt{2\pi}}\hat{f}\) immediately from the construction. Uniqueness: any bounded linear operator agreeing with \(\mathcal{U}\) on the dense subspace \(L^1 \cap L^2\) must agree with the limit defined above by continuity, so the extension is the unique one.

The operator \(\mathcal{U}\) constructed above is the Fourier-Plancherel transform on \(L^2(\mathbb{R})\). On \(L^1 \cap L^2\) it coincides with the normalized integral. On the rest of \(L^2\), where the integral may fail to converge absolutely, \(\mathcal{U}\) is defined via the \(L^2\)-limit construction above. We will continue to write \(\hat{f}\) for \(\sqrt{2\pi}\, \mathcal{U} f\) with this understanding, and so keep the classical normalization of the Fourier transform.

Step 2: surjectivity via Fourier inversion

Isometry alone does not yield a bijection on an infinite-dimensional Hilbert space. On the sequence space \(\ell^2\) of square-summable sequences, for instance, the right-shift operator \((x_1, x_2, \ldots) \mapsto (0, x_1, x_2, \ldots)\) is an isometry but not surjective. To upgrade \(\mathcal{U}\) from an isometry to a unitary, we must show that its image is all of \(L^2(\mathbb{R})\). This is exactly where the classical Fourier inversion formula enters.

Define the "inverse" candidate operator \(\mathcal{V}\) on \(L^1 \cap L^2\) by \[ (\mathcal{V} g)(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g(\xi)\, e^{-i x \xi}\, d\xi, \] differing from \(\mathcal{U}\) only by the sign in the exponent. The substitution \(\eta = -\xi\) carried out below gives \(\mathcal{V} g = \mathcal{U} \tilde g\) with \(\tilde g(\eta) = g(-\eta)\), and \(\|\tilde g\|_2 = \|g\|_2\), so \(\mathcal{V}\) is likewise an isometry on \(L^1 \cap L^2\). The density argument of Step 1 then extends \(\mathcal{V}\) to a bounded linear isometry \(\mathcal{V} : L^2(\mathbb{R}) \to L^2(\mathbb{R})\).

For a Schwartz function \(f \in \mathcal{S}(\mathbb{R})\), composing the operators directly, \[ \begin{align*} (\mathcal{V} \mathcal{U} f)(x) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} (\mathcal{U} f)(\xi)\, e^{-i x \xi}\, d\xi \\\\ &= \frac{1}{2\pi} \int_{-\infty}^{\infty} \hat{f}(\xi)\, e^{-i x \xi}\, d\xi \\\\ &= f(x), \end{align*} \] where the second equality uses \(\mathcal{U} f = \hat{f}/\sqrt{2\pi}\) and the third is the classical Fourier inversion formula applied to \(f\).

The reverse identity \(\mathcal{U} \mathcal{V} g = g\) on Schwartz \(g\) follows by relating \(\mathcal{V}\) to \(\mathcal{U}\) and a reflection. Substituting \(\eta = -\xi\) in the definition of \(\mathcal{V}\), \[ \begin{align*} (\mathcal{V} g)(x) &= \frac{1}{\sqrt{2\pi}}\int g(-\eta)\, e^{i x \eta}\, d\eta \\\\ &= (\mathcal{U} \tilde g)(x), \quad \text{where } \tilde g(\eta) := g(-\eta). \end{align*} \] Iterating, \(\mathcal{U}\mathcal{V} g = \mathcal{U}^2 \tilde g\). Now the corollary \(\mathcal{F}\{\mathcal{F}\{f\}\}(x) = 2\pi\, f(-x)\) stated with the Fourier-transform definition translates to \(\mathcal{U}^2 f(x) = f(-x)\), so \(\mathcal{U}^2 \tilde g(x) = \tilde g(-x) = g(x)\). Hence \(\mathcal{U}\mathcal{V} g = g\) on the Schwartz class. Since the Schwartz class is dense in \(L^2(\mathbb{R})\) (it contains \(C_c^\infty\), which is itself dense in \(L^2\)), the same density-and-limit argument extends both identities to all of \(L^2(\mathbb{R})\): \(\mathcal{V} \mathcal{U} = \mathcal{U} \mathcal{V} = I\) on \(L^2(\mathbb{R})\). In particular, \(\mathcal{U}\) is surjective with two-sided inverse \(\mathcal{V}\).

Step 3: the adjoint and the unitarity

The final ingredient is the identification of \(\mathcal{V}\) with the Hilbert-space adjoint \(\mathcal{U}^*\) of \(\mathcal{U}\). For Schwartz functions \(f, g \in \mathcal{S}(\mathbb{R})\), we compute \(\langle \mathcal{U} f, g \rangle\) by applying the polarized form of the classical Plancherel identity, \(\int f\, \overline{h}\, dx = \tfrac{1}{2\pi}\int \hat{f}\, \overline{\hat{h}}\, d\xi\), to the pair \((f, \mathcal{V} g)\). The Fourier transform of \(\mathcal{V} g\) is computable directly. Comparing definitions with the Fourier-transform page shows \(\mathcal{V} g = \sqrt{2\pi}\, \mathcal{F}^{-1}(g)\), hence \(\mathcal{F}(\mathcal{V} g) = \sqrt{2\pi}\, \mathcal{F}\mathcal{F}^{-1}(g) = \sqrt{2\pi}\, g\) by Fourier inversion on the Schwartz class. Therefore \[ \begin{align*} \langle f,\, \mathcal{V} g \rangle &= \int f(x)\, \overline{(\mathcal{V} g)(x)}\, dx \\\\ &= \frac{1}{2\pi} \int \hat{f}(\xi)\, \overline{\sqrt{2\pi}\, g(\xi)}\, d\xi \\\\ &= \frac{1}{\sqrt{2\pi}} \int \hat{f}(\xi)\, \overline{g(\xi)}\, d\xi \\\\ &= \int (\mathcal{U} f)(\xi)\, \overline{g(\xi)}\, d\xi \\\\ &= \langle \mathcal{U} f,\, g \rangle, \end{align*} \] where the second equality is classical Plancherel and the fourth uses \(\mathcal{U} f = \hat{f}/\sqrt{2\pi}\) by definition.

The defining property of the adjoint is \(\langle \mathcal{U} f, g \rangle = \langle f, \mathcal{U}^* g \rangle\) for all \(f, g\), so \(\mathcal{V} = \mathcal{U}^*\) on the Schwartz class, and by density on all of \(L^2(\mathbb{R})\). Combined with Step 2, \(\mathcal{U}^* \mathcal{U} = \mathcal{U} \mathcal{U}^* = I\), which is exactly the condition that \(\mathcal{U}\) be a unitary operator on \(L^2(\mathbb{R})\).

Theorem: Plancherel's Theorem (Unitarity on \(L^2(\mathbb{R})\))

The normalized Fourier transform \(\mathcal{U} : L^2(\mathbb{R}) \to L^2(\mathbb{R})\), \(\mathcal{U} = \mathcal{F}/\sqrt{2\pi}\), extending the classical Fourier transform from \(L^1 \cap L^2\) by density, is a unitary operator. It is bijective, preserves the \(L^2\) inner product \[ \langle \mathcal{U} f, \mathcal{U} g \rangle = \langle f, g \rangle \quad \text{for all } f, g \in L^2(\mathbb{R}), \] and satisfies \(\mathcal{U}^{-1} = \mathcal{U}^*\), the Hilbert-space adjoint. On the dense subspace \(L^1 \cap L^2\), the adjoint admits the integral representation \((\mathcal{U}^* g)(x) = (1/\sqrt{2\pi}) \int_{-\infty}^{\infty} g(\xi)\, e^{-i x \xi}\, d\xi\), differing from \(\mathcal{U}\) only by the sign of the exponent.

Proof:

The four conclusions correspond to the three steps above together with a final polarization argument. Step 1 (Fourier-Plancherel transform on \(L^2\)) gave the existence of \(\mathcal{U}\) as a bounded linear operator with \(\|\mathcal{U} f\|_2 = \|f\|_2\). Step 2 (surjectivity via Fourier inversion) gave bijectivity, with \(\mathcal{V}\) as two-sided inverse: \(\mathcal{V}\mathcal{U} = \mathcal{U}\mathcal{V} = I\). Step 3 (adjoint identification) established \(\mathcal{V} = \mathcal{U}^*\), so \(\mathcal{U}^{-1} = \mathcal{V} = \mathcal{U}^*\) and the integral representation on \(L^1 \cap L^2\) is the defining formula for \(\mathcal{V}\).

Finally, preservation of the inner product follows from norm-preservation by polarization. The identity \(\langle f, g \rangle = \tfrac{1}{4}\sum_{k=0}^3 i^k \|f + i^k g\|^2\) (valid in any complex inner-product space under the first-slot-linear convention) determines \(\langle f, g \rangle\) from norms alone, and each norm term is preserved by \(\mathcal{U}\).

What the Unitarity Buys

The classical Plancherel identity \(\|f\|_2^2 = \tfrac{1}{2\pi}\|\hat{f}\|_2^2\) is a statement about a single function. It says that integrating the modulus-squared of either \(f\) or \(\hat{f}\) gives the same answer, up to a constant. The unitary reformulation upgrades this to a statement about the entire structure of \(L^2(\mathbb{R})\). The Fourier transform is an isomorphism of Hilbert spaces from \(L^2(\mathbb{R})\) (as a space of "spatial" functions) to \(L^2(\mathbb{R})\) (as a space of "frequency" functions), and the two spaces are not merely isomorphic abstractly. They are isomorphic via this specific concrete map.

Every Hilbert-space construction one performs on functions has a frequency-domain mirror. Orthogonal decompositions of \(L^2\) correspond to orthogonal decompositions of the frequency space. Self-adjoint operators commuting with translations correspond to multiplication operators in frequency, which is the spectral theorem for translation-invariant operators. Probability densities with characteristic functions in \(L^2\) inherit the same structural framework. This is the viewpoint that makes the Fourier transform the prototype, in spectral theory and in quantum mechanics, of "diagonalizing a self-adjoint family of operators by a unitary change of basis."

With this, the program announced in the introduction is complete. Three claims of classical \(L^2\) Fourier theory were stated, used, or sketched in earlier pages. They are Parseval's identity for Fourier series, the decay of Fourier coefficients, and the unitary nature of the Fourier transform on \(L^2(\mathbb{R})\). Each has now been identified, by direct proof, as a specific instance of Hilbert-space machinery: respectively, the Parseval identity applied to a concrete orthonormal basis, the Bessel inequality applied to that same basis, and the bounded-extension construction applied to an isometric operator on a dense subspace. What was previously a chapter of classical analysis is now a chapter of operator theory.