Proof
(related to Lemma: Riemann Integral of a Product of Continuously Differentiable Functions with Sine)
- By hypothesis, [a,b] is a closed real interval and f:[a,b]\to\mathbb R is a continuously differentiable function.
- We define for y\in\mathbb R, the Riemann integral F(y):=\int_{a}^{b}f(x)\sin(yx)dx.
- Please note that the derivative of \frac {d}{dx}-\frac {\cos(yx)}{y} is sin(xy).
- With this result, if y\neq 0, then the partial integration gives us F(y)=\underbrace{-f(x)\frac{\cos(yx)}{y}\Rule{1px}{4ex}{2ex}^b_a}_{=:A}+\underbrace{\frac 1y\int_a^b f'(x)\cos(yx)dx}_{=:B}.\label{eq:E20241}\tag{1}
- Because f and the derivative f' are continuous and because [a,b] is closed, both continuous functions are bounded.
- By definition of bounded functions there is a constant M\ge 0 such that for all |f(x)|\le M,\quad |f'(x)|\le M,\quad\text{for all }x\in[a,b].
- Since \cos(yx)\le 1 for all x\in[a,b], the term A in (\ref{eq:E20241}) can be approximated to |A|\le \frac{2M}{|y|} and the term B to |B|\le \frac{(b-a)M}{|y|}, where |y| is the absolute value of y.
- Since both upper bounds tend to zero as |y| tends to infinity, we have \lim_{|y|\to\infty}F(y)=0.
∎
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983