In this note, a general formula is proved. It expresses the integral on the line of the product of a function $f$ and a periodic function $g$ in terms of the Fourier transform of $f$ and the Fourier coefficients of $g$. This allows the evaluation of some oscillatory integrals.
Change of variable and the rapidity of decrease of Fourier coefficients
Uniform bounds for Fourier transforms of surface measures in R^3 with nonsmooth density
A pointwise estimate for the Fourier transform and the number of maxima of a function
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