Knowlpedia

Lemma

Mean of a product of periodic Fourier sums

Only pairs of opposite frequencies contribute to the average of a product without conjugation.

Statement

For finite Fourier sums f=∑mamemf=\sum_m a_m e_m and g=∑nbneng=\sum_n b_n e_n,

∫[0,1]df(x)g(x) dx=∑mamb−m.\int_{[0,1]^d}f(x)g(x)\,dx=\sum_m a_m b_{-m}.

This follows by expanding the product and applying . The same identity holds for smooth periodic functions by absolute convergence.

A real oscillatory pair

For a nonzero frequency and complex vectors a,ba,b independent of the averaging variable,

⟨(aem+aˉe−m)⊗(bem+bˉe−m)⟩=a⊗bˉ+aˉ⊗b.\left\langle (ae_m+\bar a e_{-m})\otimes(be_m+\bar b e_{-m})\right\rangle =a\otimes\bar b+\bar a\otimes b.

Using Re⁡(aem)\operatorname{Re}(ae_m) instead introduces a factor 1/41/4 in this displayed product. If amplitudes themselves depend on the averaging variable, their frequencies must also be included; one cannot treat them as constants.