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Suppose $f$ is an indefinitely differentiable real valued function on$[0,1]$ which satisfies $\int_0^1 x^k f(x)\, dx=0$ for $k=\{0,1,2,3,.. .\}$, prove $f=0$ .
My attempt :
To prove this assertion , it suffice to prove $$\int_0^1 f^2 (x) \,dx=0$$ Then by approximation theorem , we can find a polynomial such that for every $\epsilon \gt 0$ $$\int_0^1 f^2 (x) \,dx= \int_0^1 (f(x)-\sum_{n=0}^N a_n x^n)f(x) ,dx+\int_0^1 \sum_{n=0}^N f(x) a_n x^n\,dx \le \epsilon M$$ where $M=\sup_{x \in [0,1]}|f(x)|$ .
It seems that we do not need the condition that $f$ is indefinitely differentiable , if $f$ is continuous then the conclution may hold .
My question :
a) Suppose $f$ is a Lebesgue integrable real valued function on$[0,1]$ which satisfies $\int_0^1 x^k f(x)\ dx=0$ for $k=\{0,1,2,3,.. .\}$, can we prove $f=0$ except on a set of measure $0$ ?

b) If a) is not true , suppose $f$ is a Riemann integrable real valued function on$[0,1]$ which satisfies $\int_0^1 x^k f(x)\ dx=0$ for $k=\{0,1,2,3,.. .\}$, can we prove $f=0$ except on a set of measure $0$ ?

EDIT:
To prove $f=0 \text{ a.e.}$ , it suffice to prove all the fourier coefficient of $f$ equal to $0$ , then $$\int_0^1 f(x) \cos(2 \pi nx) \,dx \le \int_0^1 |f(x)||\cos(2 \pi nx)-\sum_{n=0}^{N}a_n x^n| \le \epsilon||f||_{L^1}$$ and the proof is complete.

J.Guo
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  • For a), yes it's true. – Myunghyun Song Jan 14 '19 at 16:58
  • @ MathOverview That one was a really nice question , but I think the question I post is quite different from that one . Since we do not assume that $f$ is continuous and we want to prove $f=0 \text{ a.e.}$ – J.Guo Jan 14 '19 at 17:44
  • Hint: Stone Weirstrass. – operatorerror Jan 14 '19 at 17:48
  • @qbert Thanks for your hint . I've edit an proof which using fourier analysis and approximation theorem , do you mean that there is an alternative way to prove $||f||_{L^1}=0$ without using fourier analysis? – J.Guo Jan 14 '19 at 17:56
  • @J.Guo yes although you could argue a proof using the density of trig polynomials also is using Stone Weirstrass – operatorerror Jan 14 '19 at 18:24
  • @qbert If $f \in L^2$ , then $||f||{L^2}=\int_0^1(f(x)-\sum{n=0}^N a_n x^n)f(x) , dx$ , apply Schwarz inequality we can get the desire conclusion . But if we only assume $f \in L^1$ , can we find polynomials $f_n$ such that $f_n$ converges to $f$ uniformly ? Since we can not apply Weirstrass's theorem to uncontinuous functions , I have no idea how to deal with this . – J.Guo Jan 14 '19 at 19:03

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