L2 Convergence Of Fourier Partial Sums

What L2 Convergence Says

Fourier partial sums are finite trigonometric polynomials, even when the function being approximated is not itself a polynomial. Their convergence in $L^2$ is a statement about the average squared error: the squared difference has integral tending to zero. The key ingredients are that partial sums are orthogonal projections and that the trigonometric system is complete. Completeness ensures that trigonometric polynomials can approximate every $L^2$ function; the projection property ensures that the partial sum is at least as good as any other polynomial of the same degree.

Let $\mathbb{T}$ denote the circle parametrized by $[-\pi,\pi)$, with normalized Lebesgue measure $d\theta/(2\pi)$. We work in the complex Hilbert space $L^2(\mathbb{T})=L^2(\mathbb{T},\mathcal{L},m)$, where functions are identified if they agree $m$-a.e. and $\|f\|_2^2=\int_{-\pi}^{\pi}|f(\theta)|^2\,d\theta/(2\pi)$. Inner products use the convention $\langle f,g\rangle=\int f\overline{g}\,dm$, linear in the first variable. For $k\in\mathbb{Z}$, set $e_k(\theta)=e^{ik\theta}$. Direct integration gives $\langle e_j,e_k\rangle=1$ if $j=k$ and $0$ otherwise, so $(e_k)_{k\in\mathbb{Z}}$ is orthonormal.

Definition: For $f\in L^2(\mathbb{T})$, its $k$-th Fourier coefficient is $\widehat f(k)=\langle f,e_k\rangle=\int_{\mathbb{T}} f(\theta)e^{-ik\theta}\,dm(\theta)$. For an integer $N\ge0$, its $N$-th Fourier partial sum is $S_Nf=\sum_{k=-N}^{N}\widehat f(k)e_k$. The sum is finite, and each coefficient is well-defined because $f\in L^2(\mathbb{T})$ and $m(\mathbb{T})=1$, so Cauchy–Schwarz implies $f\in L^1(\mathbb{T})$.

Write $T_N=\operatorname{span}\{e_k:-N\le k\le N\}$. Completeness of the trigonometric system means that the union of these finite-dimensional spaces is dense in $L^2(\mathbb{T})$: for every $f\in L^2(\mathbb{T})$ and every $\varepsilon>0$, there is a trigonometric polynomial $p\in T_M$ for some $M\ge0$ such that $\|f-p\|_2<\varepsilon$. We use this standard completeness result from the earlier material on the trigonometric system.

The Projection Error at a Fixed Degree

Proposition: For every $f\in L^2(\mathbb{T})$, every $N\ge0$, and every $p\in T_N$, $$ \|f-p\|_2^2=\|f-S_Nf\|_2^2+\|S_Nf-p\|_2^2. $$ Consequently $S_Nf$ is the unique element of $T_N$ that minimizes the distance to $f$, and $\|f-S_Nf\|_2\le\|f-p\|_2$.

Proof. Write $p=\sum_{k=-N}^{N}c_ke_k$. Orthonormality gives $\langle f-S_Nf,e_j\rangle=\widehat f(j)-\widehat f(j)=0$ for every $-N\le j\le N$. By linearity in the first variable, $f-S_Nf$ is therefore orthogonal to every element of $T_N$, including $S_Nf-p$. Since $f-p=(f-S_Nf)+(S_Nf-p)$, expanding its squared norm using the inner product gives $$ \|f-p\|_2^2 =\|f-S_Nf\|_2^2+\|S_Nf-p\|_2^2, $$ because the two cross terms vanish by orthogonality. The second term on the right is nonnegative, which proves the minimizing inequality. If another $p\in T_N$ attains the same minimum, the identity forces $\|S_Nf-p\|_2^2=0$, so $p=S_Nf$ as an element of $L^2(\mathbb{T})$. $\square$

Taking $p=0$ in the identity also gives a useful finite-stage error formula: $\|f-S_Nf\|_2^2=\|f\|_2^2-\|S_Nf\|_2^2$. Orthonormality computes the latter norm as a finite sum. Thus

\|f-S_Nf\|_2^2 = \|f\|_2^2-\sum_{k=-N}^{N}|\widehat f(k)|^2.

This is a finite identity for each $N$. It describes the error remaining after the frequencies from $-N$ through $N$ have been included; it does not assert an infinite-series identity for $\|f\|_2^2$.

Worked Example: A Finite Fourier Sum and Its Exact Error

Let $f=2e_0+3e_1-2i e_{-2}$ in $L^2(\mathbb{T})$. Compute its partial sums and errors for $N=0,1,2$.

Orthonormality shows that the only nonzero coefficients are $\widehat f(0)=2$, $\widehat f(1)=3$, and $\widehat f(-2)=-2i$. For $N=0$, $S_0f=2e_0$, so the residual is $3e_1-2i e_{-2}$. Its squared norm is $3^2+|-2i|^2=9+4=13$, and the error is $\sqrt{13}$. For $N=1$, $S_1f=2e_0+3e_1$; the residual is $-2i e_{-2}$, whose squared norm is $|-2i|^2=4$, giving error $2$. For $N=2$, all three nonzero frequencies are included, so $S_2f=f$ and the error is $0$.

Check: the full squared norm is $2^2+3^2+|-2i|^2=4+9+4=17$. The finite error identity gives $17-2^2=13$ at $N=0$, $17-(2^2+3^2)=4$ at $N=1$, and $17-(2^2+3^2+|-2i|^2)=0$ at $N=2$, agreeing with the residual calculations.

Convergence of the Partial Sums

Theorem: For every complex-valued $f\in L^2(\mathbb{T})$, the Fourier partial sums converge to $f$ in $L^2(\mathbb{T})$: $$ \lim_{N\to\infty}\|f-S_Nf\|_2=0. $$ This is convergence in norm, so it asserts that the integral of the squared error tends to zero.

Proof. Fix $\varepsilon>0$. By completeness of the trigonometric system, choose $M\ge0$ and $p\in T_M$ such that $\|f-p\|_2<\varepsilon$. For every $N\ge M$, the inclusion $T_M\subseteq T_N$ gives $p\in T_N$. The projection error proposition, applied to $p$ at degree $N$, yields $$ \|f-S_Nf\|_2\le\|f-p\|_2<\varepsilon. $$ Thus for each $\varepsilon>0$, all $N\ge M$ satisfy $\|f-S_Nf\|_2<\varepsilon$. This is exactly the definition of convergence of the errors to zero. No exchange of a limit and an integral is used. $\square$

The proof separates the two roles in the result. Completeness supplies a polynomial approximation $p$, while orthogonal projection guarantees that $S_Nf$ does no worse than $p$ once $N$ is large enough to contain its frequencies. In particular, if a polynomial $p\in T_M$ approximates $f$ within a known error $\delta$, then every $N\ge M$ satisfies $\|f-S_Nf\|_2\le\delta$. A rate of convergence follows when such approximation errors can themselves be bounded as the permitted degree grows.

Worked Example: A Geometric Rate from an L2-Convergent Series

For $n\ge1$, consider the functions $2^{-n}e_n$ on $\mathbb{T}$, and define $f$ to be the $L^2$ limit of their partial sums. Find the Fourier partial sums and their errors.

For integers $b\ge a\ge1$, orthonormality gives $$ \left\|\sum_{n=a}^{b}2^{-n}e_n\right\|_2^2 =\sum_{n=a}^{b}4^{-n} \le\sum_{n=a}^{\infty}4^{-n} =\frac{4^{-a}}{1-1/4} =\frac{4^{1-a}}{3}. $$ This bound tends to zero as $a\to\infty$, so the partial sums form a Cauchy sequence in $L^2(\mathbb{T})$. The space $L^2(\mathbb{T})$ is complete, hence the limit $f$ exists. Continuity of the inner product in $L^2$, which follows from Cauchy–Schwarz, gives $\widehat f(k)=2^{-k}$ for $k\ge1$ and $\widehat f(k)=0$ for $k\le0$. Indeed, the coefficient of each finite partial sum at a fixed frequency $k$ is eventually $2^{-k}$ if $k\ge1$, and is always zero if $k\le0$.

It follows that $S_Nf=\sum_{n=1}^{N}2^{-n}e_n$, with the sum empty when $N=0$. The residual is the $L^2$ limit of the terms with $n\ge N+1$. The finite orthogonal tail sums have squared norms $\sum_{n=N+1}^{b}4^{-n}$; passing to the limit is valid because convergence in $L^2$ implies convergence of norms. Therefore $$ \|f-S_Nf\|_2^2 =\sum_{n=N+1}^{\infty}4^{-n} =\frac{4^{-(N+1)}}{1-1/4} =\frac{4^{-N}}{3}, \qquad \|f-S_Nf\|_2=\frac{2^{-N}}{\sqrt{3}}. $$

Check: at $N=0$ the error is $1/\sqrt{3}$; at $N=1$ it is $1/(2\sqrt{3})$; at $N=2$ it is $1/(4\sqrt{3})$. Each value follows by substituting $N$ into $2^{-N}/\sqrt{3}$, and each successive error is half the preceding one. This explicitly verifies L2 convergence and its rate.

Worked Example: A Constant Partial Sum as Best Approximation

Let $f(\theta)=1+3\cos(2\theta)$, a real-valued element of $L^2(\mathbb{T})$. Find its degree-zero Fourier partial sum and its error, and compare it with all constant approximations.

The mean of $\cos(2\theta)$ on the circle is zero, so $\widehat f(0)=1$ and $S_0f=1$. Since the normalized integral of $\cos^2(2\theta)$ is $1/2$, the squared error is $\|f-S_0f\|_2^2=\int_{\mathbb{T}}9\cos^2(2\theta)\,dm=9/2$. Thus the error is $3/\sqrt{2}$.

For any complex constant $c$, the cross term between $1-c$ and $\cos(2\theta)$ integrates to zero. Consequently, $$ \|f-c\|_2^2 =\int_{\mathbb{T}}|(1-c)+3\cos(2\theta)|^2\,dm =|1-c|^2+\frac{9}{2} \ge\frac{9}{2}. $$ Equality holds exactly when $c=1$, so $S_0f$ is the unique best constant approximation. At degree $1$, no new nonzero coefficient is included and $S_1f=1$; at degree $2$, the frequency $2$ terms are included and $S_2f=f$. Check: the degree-two residual is zero, while the degree-zero squared error $9/2$ agrees with the direct integral and the minimization calculation.

What the Result Does—and Does Not—Guarantee

The convergence theorem has a precise error interpretation: for every $\varepsilon>0$, there is an $N_0$ such that $\int_{\mathbb{T}}|f-S_Nf|^2\,dm<\varepsilon^2$ whenever $N\ge N_0$. It does not assert that $S_Nf(\theta)\to f(\theta)$ at every point, nor does it assert uniform convergence. Those are different modes of convergence and require separate analysis.

Common mistake: using orthogonality without completeness. Orthogonality alone does not show that projection errors tend to zero. For example, consider only the orthonormal family $\{e_{2k}:k\in\mathbb{Z}\}$ and the function $f=e_1$. Every inner product $\langle e_1,e_{2k}\rangle$ is zero, so every finite projection onto these even frequencies is zero and the error remains $\|e_1\|_2=1$. The family is not complete. For the full trigonometric system, completeness is exactly the additional fact that rules out a nonzero residual orthogonal to every frequency. The convergence proof depends on that fact as well as on the projection identity.

The finite error identity also provides a practical diagnostic. Increasing $N$ enlarges the approximation space, so the best-approximation error cannot increase. The theorem says more: density forces that nonincreasing error to approach zero for every $f\in L^2(\mathbb{T})$. Parseval’s formula gives a further infinite-series description of the coefficients; it is a separate result and is not needed for this convergence argument.

Check Your Understanding

Use the projection identity and the convergence proof to explain each answer.

Why does the minimizing property of $S_Nf$ compare it only with elements of $T_N$, rather than with every function in $L^2(\mathbb{T})$?
If $p\in T_M$ and $\|f-p\|_2<\delta$, what bound follows for $\|f-S_Nf\|_2$ when $N\ge M$?
For the geometric-series example, compute the squared error at $N=3$ from the formula for the tail.
Which hypothesis fails in the even-frequency example where every projection of $e_1$ is zero?
Does convergence of Fourier partial sums in $L^2$, by itself, assert pointwise convergence? State what mode of convergence the theorem proves.

L2 Convergence Of Fourier Partial Sums

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