Fourier Transform 22 | Riemann–Lebesgue Lemma for Fourier Series

Riemann–Lebesgue lemma pins down a key asymptotic behavior of Fourier series: for any function f in L1, its Fourier coefficients form a sequence that always tends to 0 as the frequency k goes to ±∞. That matters because it guarantees that high-frequency oscillations “wash out” in the Fourier coefficient sense, even when f is not square-integrable—so the result reaches beyond the comfortable L2 setting.

The starting point is the Fourier coefficient formula for an L1 function: the coefficients are defined by an integral and can be packaged as a sequence {f̂(k)} of complex numbers. The lemma asserts that this sequence converges and that its limit is exactly zero. In practical terms, the absolute values |f̂(k)| eventually become arbitrarily small, though the lemma does not quantify how fast the decay occurs.

The proof strategy hinges on a gap: for L2 functions, the conclusion is already known. Parseval’s identity implies that the Fourier coefficients are square-summable, meaning {f̂(k)} lies in ℓ2. Since any square-summable sequence must go to zero at infinity, the lemma follows automatically for f ∈ L2. The real work begins when f is in L1 but not in L2, where f may be unbounded and the square-integrability argument breaks.

To bridge that gap, the argument constructs truncated approximations of f that become square-integrable. For each natural number n, define a 2π-periodic function f_n by “cutting off” the magnitude of f: when |f(x)| ≤ n, keep f(x); when |f(x)| > n, replace it by n (in magnitude). This truncation produces bounded functions, and bounded periodic functions on a finite interval are in L2. As a result, Parseval’s identity applies to each f_n, giving control over the tail of their Fourier coefficients.

Next comes the convergence back to the original f. The proof uses Lebesgue’s dominated convergence theorem: f_n(x) → f(x) almost everywhere as n → ∞, and |f_n(x)| is dominated by |f(x)|, which is integrable because f ∈ L1. Dominated convergence then yields convergence in L1, i.e., ∫|f_n − f| → 0.

That L1 convergence transfers to the Fourier coefficients. For each fixed k, the difference f̂_n(k) − f̂(k) is an integral of (f_n − f) against the complex exponential e^{-ikx}, whose modulus is 1. Taking absolute values and using the triangle inequality shows that |f̂_n(k) − f̂(k)| can be made small uniformly in k once n is large enough.

Finally, the proof combines two tail controls: (1) for large n, the Fourier coefficients of f_n approximate those of f, and (2) for each such f_n, Parseval’s identity forces f̂_n(k) to be small when |k| is large. A careful ε/2 bookkeeping then shows that |f̂(k)| itself becomes smaller than any prescribed ε for all sufficiently large |k|. The lemma is established for complex Fourier coefficients, and the same conclusion follows for real sine/cosine coefficients by standard relationships between the two Fourier representations.