$f:[a,b]\to \mathbb R$ be Riemann Integrable , then $g_n(x):= \int _a^x f(t)\cos(nt)dt$ converges uniformly , on $[a,b]$ , to the zero function ?

Question

Let $f:[a,b]\to \mathbb R$ be Riemann Integrable ; define a sequence of functions $\{g_n\}$ with domain $[a,b]$ as $g_n(x):= \int _a^x f(t)\cos(nt)dt , \forall x \in [a,b] , \forall n \in \mathbb N$ . Then is it true that $\{g_n\}$ converges uniformly , on $[a,b]$ , to the zero function ? If not , then what extra condition(s) do we need to impose on $f$ to make it happen ?

Answer 1

Hint: If f is Riemann integrable we can uniformly approximate it by step functions. Then we can use this uniform convergence to swap the limits with the integrals. Hope this allows you to make some headway.

Answer 2

We know by the Riemann-Lebesgue lemma that $g_n$ converges pointwise to $0$ on $[a,b]$. Since $f$ is Riemann integrable, it is bounded. Say $|f(x)|\le M$ for all $x\in[a,b]$. Then $$ |g_n(x)|\le M(b-a),\quad a\le x\le b, $$ and $$ |g_n(x)-g_n(y)|=\Bigl|\int_x^yf(t)\cos(n\,t)\,dt\Bigr|\le M\,|x-y|,\quad a\le x,y\le b. $$ The sequence $\{g_n\}$ is uniformly bounded and equicontinuous. ¿Can you follow from here?

Answer 3

Here is another idea:

• Prove that $\{g_n\}$ converges pointwise to the zero function on $[a,b]$.
• Prove that $\{g_n\}$ is a Cauchy sequence of the Banach space of Riemann-integrable functions on $[a,b]$ endowed with the sup norm.

Hint: For all real number $x$ and for all integers $n$ and $m$, $$\cos(nx)-\cos(mx)=2\sin\left(\dfrac{n+m}2x\right)\sin\left(\dfrac{n-m}2x\right)$$