Let $f : [−π, π] → \mathbb{R}$ be continuous. Pick out the case(s) which imply that $f ≡ 0$.

Question

Let $f : [−π, π] → \mathbb{R}$ be continuous. Pick out the case(s) which imply that $f ≡ 0$.

(a) $\int_{-\pi}^{\pi}{x^nf(x)\,dx} = 0$, for all $n ≥ 0$.

(b)$\int_{-\pi}^{\pi}{\cos (nx)\, f(x)\,dx} = 0$, for all $n ≥ 0$.

(c)$\int_{-\pi}^{\pi}{\sin (nx)\, f(x)\,dx} = 0$, for all $n ≥ 1$ .

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If we take $\sin nx$ in (b) and $\cos nx$ in (a), then the integral becomes zero. So, (b) and (c) are not true.
But I can not verify (a). Can I get some help?
Also, in the answer it is given that all are true.
Where did I make a mistake?

Answer 1

For $(a)$, use Weierstrass' approximation theorem. You know $$\int_{-\pi}^\pi fp=0$$ for any polynomial $p$. Pick $\varepsilon >0$; and approximate $f$ uniformly by a polynomial $p$, that is make $M(p)= \sup\{|f(t)-p(t)|:t\in[-\pi,\pi]\}<\varepsilon$. Now, observe that $$\int_{-\pi}^\pi f^2=\int_{-\pi}^\pi f^2-\int_{-\pi}^\pi fp=\int_{-\pi}^\pi f(f-p)$$

Answer 2

I just complete the answer of others. This is the approximation theorem related to the polynomial and the trigonometric polynomial. And $(b)$ and $(c)$ is related to the approximation of trigonometric polynomial.

$f(x)=\frac{a_0}{2}+\sum (a_ncons(nx)+b_nsin(nx))$ and $a_n,b_n$ is above in the choice $(b)(c)$

So the $(b)(c)$ just provide the half information of $f$, so it is wrong.