Lebl-Basic-Analysis-I · Problem 5.5.7

The Riemann Integral

Mathematics · improper-integrals-fourier-transform

Problem

Let f:R→Rf : \mathbb{R} \to \mathbb{R} be a Riemann integrable function on every interval [a,b][a, b], and such that ∫−∞∞∣f(x)∣dx<∞\int_{-\infty}^{\infty} \left|f(x)\right| dx < \infty. Show that the Fourier sine and cosine transforms exist. That is, for every ω≥0\omega \geq 0 the following integrals converge Fs(ω)≔1π∫−∞∞f(t)sin⁡(ωt) dt,Fc(ω)≔1π∫−∞∞f(t)cos⁡(ωt) dt.F^s(\omega) \coloneqq \frac{1}{\pi} \int_{-\infty}^{\infty} f(t) \sin(\omega t) \, dt, \qquad F^c(\omega) \coloneqq \frac{1}{\pi} \int_{-\infty}^{\infty} f(t) \cos(\omega t) \, dt. Furthermore, show that FsF^s and FcF^c are bounded functions.

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