Proof of Riemann-Lebesgue lemma

Question

I read a book, and this mention to the following lemma of Rieman-Lebesgue type.

Lemma. Let $-\infty<a<b<\infty$ and $f(x,y):[a,b]^2\to\mathbb R$ be an integrable and nonnegative function. Then we the followings.

a) $\lim\limits_{r\to\infty}\int\limits_{[a,b]^2}f(x,y)\sin\left(\pi r x\right) \, dx \, dy=0$

b) $\lim\limits_{r\to\infty}\int\limits_{[a,b]^2}f(x,y)\sin\left(\pi r x\right)\sin(\pi ry) \, dx \, dy=0$.

I try to find a proof, my idea is to use step function, but I fail. Does anyone know some proof for this lemma? Thank you.

Answer 1

You can see Bochner, Chandrasekharan, "Fourier Transforms"

Answer 2

Hint. Use the fact that every integrable function $f:[a,b]^2\to\mathbb R$ can be $L^1-$approximated by linear combinations of the form $$ f_n=\sum_{j=1}^n c_n\chi_{R_n}, $$ where $R_j\subset [a,b]^2$, rectangle.

Answer 3

Write $f = g + h$, where $g$ is a linear combination of characteristic functions of bounded rectangles and $||h|| < \epsilon$ in $L_1$. The statements are obvious with $f$ replaced by $g$ (as in the 1-dimensional case), and the $L_1$ bound for $h$ ensures that the contribution of $h$ to the original integrals is sufficiently small.

The approximation exists for Lebesgue integrals because the definition of an abstract measure-theory integral gives an approximation by characteristic functions of Borel sets, and the definition of Lebesgue measure gives approximations of Borel sets by unions of rectangles.

The approximation exists for Riemann integrals because the definition of the integral gives suitable characteristic functions even more directly.

A previous version of this answer overcomplicated things for (a) using Fubini's theorem. It is important that the statements are for double integrals, not for iterated integrals, since the iterated integrals don't exist in general for Riemann integration and require Fubini's theorem for Lebesgue integration. Using Fubini's theorem in the proof only gave technical complications.