Fourier Transform 8 | Bessel's Inequality and Parseval's Identity [dark version]

TL;DR

In L2, the partial Fourier sum FN(f) is the orthogonal projection of f onto span{e^{ikx} : −n ≤ k ≤ n}.

Briefing Cornell Notes

Briefing

Fourier coefficients in L2 don’t just come from integrals—they measure how much of a function lies in the span of the first 2n+1 complex exponentials. That geometric viewpoint turns convergence questions into a clean distance computation, leading to Bessel’s inequality and setting up Parseval’s identity as the missing “limit equals length” step.

Starting with a 2π-periodic function f in L2, the partial Fourier sum FN(f) is the orthogonal projection of f onto the finite-dimensional space spanned by {e^{ikx} : k = −n,…,n}. With the L2 inner product normalized by 1/(2π), the Fourier coefficients are exactly the inner products Ck = ⟨e^{ikx}, f⟩. Because projection and the residual (the “normal component”) are orthogonal, the difference between f and its partial sum has a norm that can be expressed by a Pythagorean theorem in an inner product space.

Concretely, the L2 distance satisfies ‖f − FN(f)‖^2 = ‖f‖^2 − Σ_{k=−n}^n |Ck|^2. This identity is the engine behind everything that follows: as n grows, the partial projection captures more of f, so the residual norm can only shrink. From the nonnegativity of ‖f − FN(f)‖^2, it follows immediately that Σ_{k=−n}^n |Ck|^2 ≤ ‖f‖^2 for every n. That inequality is Bessel’s inequality, and it has two key consequences. First, the captured “energy” in the coefficients can never exceed the total energy ‖f‖^2. Second, because the left-hand side increases with n but stays bounded, the coefficient contributions must eventually become small—equivalently, the Fourier coefficients Ck tend toward 0 as |k| grows.

Yet Bessel’s inequality alone doesn’t guarantee that the Fourier series actually converges to f in L2. The goal is stronger: the residual norm ‖f − FN(f)‖ should go to 0 as n → ∞. From the distance formula, that happens exactly when the inequality becomes an equality in the limit, meaning Σ_{k=−∞}^{∞} |Ck|^2 = ‖f‖^2. This limiting equality is Parseval’s identity, described as an infinite-sided generalization of the Pythagorean theorem. At this stage, the framework shows what must be true for L2 convergence—Parseval’s identity is the final missing piece, which will be established in later material. The upshot is that Fourier series convergence in L2 is not a mysterious analytic phenomenon; it is the statement that the orthogonal projections eventually recover the full “length” of f.

Cornell Notes

For a 2π-periodic function f in L2, the partial Fourier sum FN(f) is the orthogonal projection of f onto the span of {e^{ikx} : −n ≤ k ≤ n}. Using the L2 inner product (normalized by 1/(2π)), the Fourier coefficients are Ck = ⟨e^{ikx}, f⟩. Orthogonality yields the key distance formula: ‖f − FN(f)‖^2 = ‖f‖^2 − Σ_{k=−n}^n |Ck|^2. Since the left side is always nonnegative, Σ_{k=−n}^n |Ck|^2 ≤ ‖f‖^2 (Bessel’s inequality), and the coefficient contributions must shrink. Full L2 convergence requires the limit equality Σ_{k=−∞}^{∞} |Ck|^2 = ‖f‖^2, which is Parseval’s identity.

Why is the partial Fourier sum FN(f) best viewed as an orthogonal projection in L2?

In L2, the complex exponentials e^{ikx} form an orthonormal system under the normalized inner product ⟨g,h⟩ = (1/2π)∫_{0}^{2π} g(x)\overline{h(x)} dx. The finite set {e^{ikx} : −n,…,n} spans a subspace, and FN(f) is the unique element of that subspace whose difference from f is orthogonal to the subspace. That is exactly the definition of an orthogonal projection.

How does orthogonality produce the formula ‖f − FN(f)‖^2 = ‖f‖^2 − Σ|Ck|^2?

The residual f − FN(f) is orthogonal to the projection FN(f). By the Pythagorean theorem in inner product spaces, ‖f‖^2 = ‖FN(f)‖^2 + ‖f − FN(f)‖^2. Meanwhile, ‖FN(f)‖^2 equals the sum of squared magnitudes of its orthonormal expansion coefficients, giving ‖FN(f)‖^2 = Σ_{k=−n}^n |Ck|^2. Rearranging yields the distance identity.

What does Bessel’s inequality say, and why does it hold for every n?

Bessel’s inequality is Σ_{k=−n}^n |Ck|^2 ≤ ‖f‖^2 for all n. It follows directly from the distance formula because ‖f − FN(f)‖^2 is nonnegative. Geometrically, the length of the projection onto a subspace cannot exceed the length of the original vector f.

What can be concluded about the Fourier coefficients from Bessel’s inequality?

Because Σ_{k=−n}^n |Ck|^2 is increasing in n and bounded above by ‖f‖^2, the sequence of partial sums converges. That forces the individual “new” terms to vanish in the limit, so the Fourier coefficients must approach 0 as |k| grows.

Why is Parseval’s identity the exact condition needed for L2 convergence of the Fourier series?

L2 convergence means ‖f − FN(f)‖ → 0. Using ‖f − FN(f)‖^2 = ‖f‖^2 − Σ_{k=−n}^n |Ck|^2, this happens if and only if Σ_{k=−n}^n |Ck|^2 → ‖f‖^2. Writing the limit as an infinite sum gives Parseval’s identity: Σ_{k=−∞}^{∞} |Ck|^2 = ‖f‖^2.

Review Questions

State the orthogonality-based distance formula relating ‖f − FN(f)‖^2, ‖f‖^2, and the coefficients Ck.
Explain how Bessel’s inequality follows from nonnegativity of ‖f − FN(f)‖^2.
What equality must hold in the limit for the Fourier series to converge to f in L2?

Key Points

1
In L2, the partial Fourier sum FN(f) is the orthogonal projection of f onto span{e^{ikx} : −n ≤ k ≤ n}.
2
Fourier coefficients satisfy Ck = ⟨e^{ikx}, f⟩ under the normalized inner product (1/2π)∫_{0}^{2π} · dx.
3
Orthogonality yields the exact identity ‖f − FN(f)‖^2 = ‖f‖^2 − Σ_{k=−n}^n |Ck|^2.
4
Bessel’s inequality follows immediately: Σ_{k=−n}^n |Ck|^2 ≤ ‖f‖^2 for every n.
5
The bounded, increasing partial sums imply the Fourier coefficients must tend to 0 as |k| → ∞.
6
L2 convergence of the Fourier series is equivalent to the limiting equality Σ_{k=−∞}^{∞} |Ck|^2 = ‖f‖^2 (Parseval’s identity).

Highlights

The residual energy splits cleanly: ‖f‖^2 = ‖FN(f)‖^2 + ‖f − FN(f)‖^2.

Bessel’s inequality is just the nonnegativity of the projection error.

L2 convergence happens exactly when the coefficient energy fills the entire ‖f‖^2—Parseval’s identity.

Fourier coefficients measure “how much” of f sits in each orthonormal frequency mode.