Knowlpedia

Lemma

Decay of smooth periodic Fourier coefficients

Integration by parts gives decay faster than every inverse power of frequency.

Statement

For a smooth Zd\mathbb Z^d-periodic function ff and an integer N≥0N\ge0,

∣f^(m)∣≤(1+4π2∣m∣2)−N∥(1−Δ)Nf∥L1([0,1]d).|\widehat f(m)|\le(1+4\pi^2|m|^2)^{-N} \|(1-\Delta)^N f\|_{L^1([0,1]^d)}.

In particular, the decay faster than every inverse power of 1+∣m∣1+|m|.

Proof and parameters

Periodic integration by parts has canceling boundary terms and gives (1−Δ)Nf^(m)=(1+4π2∣m∣2)Nf^(m)\widehat{(1-\Delta)^N f}(m)=(1+4\pi^2|m|^2)^N\widehat f(m). Bound the coefficient of (1−Δ)Nf(1-\Delta)^N f by its L1L^1 norm. Uniform bounds on the indicated derivatives of a parameter family give uniform coefficient decay; no uniform conclusion follows without those bounds.

References