Fourier Transform 8 | Bessel's Inequality and Parseval's Identity

TL;DR

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

Briefing Cornell Notes

Briefing

Fourier series in the square-integrable setting come with a clean geometric guarantee: the partial Fourier sums act like orthogonal projections, so their “error” is controlled by a Pythagorean-type identity. That geometry leads directly to Bessel’s inequality (an always-true bound on the Fourier coefficients) and sets up the condition for full convergence via Parseval’s identity (an equality that turns the bound into an exact energy balance). The practical payoff is that convergence in L2 is no longer a mystery—it becomes a question about whether the Fourier coefficients capture exactly the total “energy” of the function.

Working in L2 for 2π-periodic functions, the partial sum FN(f) is built from the orthonormal system of complex exponentials ek(x)=e^{ikx}. Each coefficient CK is the inner product of f with ek, using the normalized L2 inner product (1/2π)∫ f(x)·e^{-ikx} dx. Because FN(f) is the orthogonal projection of f onto the span of {e_k : |k|≤n}, the difference f−FN(f) is orthogonal to FN(f). That orthogonality yields the key formula for the L2 error: ||f−FN(f)||^2 equals ||f||^2 minus the sum_{k=-n}^n |Ck|^2. In other words, the squared length of the “normal component” (the part not captured by the projection) is exactly the leftover energy after accounting for the squared magnitudes of the included Fourier coefficients.

From this identity, Bessel’s inequality follows immediately: for every n, the partial energy sum ∑_{k=-n}^n |Ck|^2 cannot exceed ||f||^2. Geometrically, the projection can never be longer than the original vector. Analytically, this means the sequence of partial sums is monotone increasing and bounded above, so it converges; equivalently, the Fourier coefficients must tend to zero in magnitude as |k| grows. Still, the inequality alone doesn’t guarantee that the approximation error ||f−FN(f)|| goes to zero.

To get L2 convergence of the Fourier series to f, the error must vanish in the limit n→∞, which happens exactly when the inequality becomes an equality in the limit. That requirement is precisely Parseval’s identity: the total sum of squared Fourier coefficients equals the squared L2 norm of f, i.e., ∑_{k=-∞}^{∞} |Ck|^2 = ||f||^2. When this equality holds, the Pythagorean leftover term disappears, forcing ||f−FN(f)||^2→0 and giving L2 convergence. The transcript ends by flagging that Parseval’s identity is the missing step needed to conclude convergence, with the proof reserved for later videos.

Cornell Notes

In L2, Fourier partial sums behave like orthogonal projections onto the span of the exponentials e^{ikx}. With the normalized inner product (1/2π)∫ f(x)·e^{-ikx} dx, the Fourier coefficients Ck are exactly the inner products ⟨f, e_k⟩. Orthogonality gives a Pythagorean error formula: ||f−FN(f)||^2 = ||f||^2 − ∑_{k=-n}^n |Ck|^2. This immediately implies Bessel’s inequality, ∑_{k=-n}^n |Ck|^2 ≤ ||f||^2, so the coefficient energy is bounded. L2 convergence requires the bound to become an equality in the limit, which is exactly Parseval’s identity: ∑_{k=-∞}^{∞} |Ck|^2 = ||f||^2.

Why does the L2 error ||f−FN(f)||^2 reduce to a difference involving the Fourier coefficients?

Because FN(f) is an orthogonal projection of f onto the span of {e_k : |k|≤n}. The “captured” part FN(f) and the “leftover” part f−FN(f) are orthogonal in L2, so the Pythagorean theorem applies: ||f||^2 = ||FN(f)||^2 + ||f−FN(f)||^2. Since ||FN(f)||^2 equals the sum of squared coefficients ∑_{k=-n}^n |Ck|^2 (orthonormality of the exponentials), rearranging gives ||f−FN(f)||^2 = ||f||^2 − ∑_{k=-n}^n |Ck|^2.

What exactly is Bessel’s inequality, and why is it always true?

Bessel’s inequality says that for every n, the partial coefficient energy satisfies ∑_{k=-n}^n |Ck|^2 ≤ ||f||^2. It’s always true because the projection FN(f) cannot have greater L2 length than the original function f. In the error formula ||f−FN(f)||^2 = ||f||^2 − ∑_{k=-n}^n |Ck|^2, the left side is a squared norm and therefore nonnegative, forcing the right side to be nonnegative as well.

How does Bessel’s inequality imply that Fourier coefficients must eventually get small?

Since ∑_{k=-n}^n |Ck|^2 is increasing in n and bounded above by ||f||^2, it converges to a finite limit. For a convergent series of nonnegative terms, the individual terms must go to zero. So |Ck|^2 → 0 as |k|→∞, meaning the Fourier coefficients themselves tend to zero in magnitude.

Why doesn’t Bessel’s inequality alone guarantee that FN(f) converges to f in L2?

Bessel’s inequality only provides an upper bound on ∑_{k=-n}^n |Ck|^2. The L2 error is ||f−FN(f)||^2 = ||f||^2 − ∑_{k=-n}^n |Ck|^2. Even if the coefficient sums stay below ||f||^2, the difference might not shrink to zero unless the coefficient sums approach ||f||^2 exactly. That missing “approach to the full energy” is what Parseval’s identity supplies.

What is the precise role of Parseval’s identity in proving L2 convergence?

L2 convergence requires ||f−FN(f)||^2 → 0. Using the error formula, this happens if and only if ∑_{k=-n}^n |Ck|^2 → ||f||^2 as n→∞. Parseval’s identity states exactly that equality for the infinite sum: ∑_{k=-∞}^{∞} |Ck|^2 = ||f||^2. When that holds, the “leftover energy” term disappears, forcing the L2 error to vanish.

Review Questions

State the orthogonality-based formula for ||f−FN(f)||^2 in terms of ||f||^2 and the Fourier coefficients Ck.
Explain why Bessel’s inequality implies that |Ck| → 0 as |k|→∞.
What equality must hold for the Fourier series to converge to f in L2, and how does it relate to the error formula?

Key Points

1
In L2, the partial Fourier sum FN(f) is the orthogonal projection of f onto the span of {e^{ikx} : |k|≤n}.
2
Fourier coefficients Ck are given by the normalized inner product Ck = ⟨f, e_k⟩ with ⟨g,h⟩ = (1/2π)∫ g(x)·h(x) dx.
3
Orthogonality yields the error identity ||f−FN(f)||^2 = ||f||^2 − ∑_{k=-n}^n |Ck|^2.
4
Bessel’s inequality follows immediately: ∑_{k=-n}^n |Ck|^2 ≤ ||f||^2 for every n.
5
Because the partial sums of |Ck|^2 are bounded and increasing, the coefficients satisfy |Ck|→0 as |k|→∞.
6
L2 convergence requires the error to go to zero, which is equivalent to the infinite energy balance given by Parseval’s identity: ∑_{k=-∞}^{∞} |Ck|^2 = ||f||^2.

Highlights

Orthogonal projection turns Fourier approximation into geometry: the L2 error is the leftover squared length after projecting onto finitely many exponentials.

Bessel’s inequality is not a guess—it drops out of nonnegativity of ||f−FN(f)||^2.

Parseval’s identity is the exact condition that upgrades a bound into full L2 convergence.

The squared L2 norm of f becomes an “infinite Pythagorean theorem” once the Fourier coefficients capture all the energy.