$f \in L_1([0,1],m)$ such that $\int_0^1 f \sin (n^2x) \ dm= 1$

Question

I have the space of $\mathbb{K}$-valued integrable functions with respect to a Lebesgue measure $m$ and I need to find a function $f$ such that $\int_0^1 |f| \ dm=1$ and $\int_0^1 f \sin(n^2x) \ dm=1 $, $n \geq 2, n \in \mathbb{N}$. I was thinking to take a continuous function so it's Riemann integrable on $[0,1]$ and then I can "forget" the Lebesgue measure, but I don't know if it's a good idea.

Answer 1

You won't find such a function.

If $f \in L^1[0,1]$ the Riemann-Lebesgue Lemma tells you that $$\lim_{n \to \infty} \int_0^1 f(x) \sin(nx) \, dx = 0.$$