The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis, dealing with the behavior of the Fourier coefficients of integrable functions. It asserts that if \( f \) is an integrable function on the real line (or on a finite interval), then the Fourier coefficients of \( f \) tend to zero as the frequency goes to infinity.
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If , then its Fourier transform is continuous and tends to zero as . Approximate in by a compactly supported smooth function; integration by parts makes the approximant's transform decay, while transfers the conclusion.