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is there an example of a non-even real $2\pi$-periodic continuous function $f(x)$ such that $$\int^{\pi}_{-\pi}f(x)\cdot sin(nx)dx = 0$$ for every $n\in \mathbb{Z}$.

I can easily construct a non-continuous example, and I managed to put up a proof that such a function cannot exist if $f(x)$ is continuously differentiable using convergence of the Fourier series.

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  • $\begingroup$ Modulo null sets, every square-integrable $2\pi$-periodic function orthogonal to all $x\mapsto \sin (nx)$ is even. $\endgroup$ – Daniel Fischer Jul 3 '18 at 11:03
  • $\begingroup$ what do you mean by 'modulo null sets'? $\endgroup$ – Rei Henigman Jul 3 '18 at 11:54
  • $\begingroup$ You can arbitrarily change a function on a null set (set of measure $0$) without affecting integrals. So you can for example take the characteristic function of $1 + \pi\mathbb{Q}$. This is not an even function, but from the perspective of the Lebesgue integral, it's equivalent to the zero function, which is even. Thus, a class in $L^2([-\pi,\pi])$ is orthogonal to all $\sin (nx)$ if and only if it has an even representative. $\endgroup$ – Daniel Fischer Jul 3 '18 at 12:01
  • $\begingroup$ I didn't study Lebesgue integrals yet, so I'll have to take your word for it. Do you by any chance have an idea of how to approach proving this without using Lebesgue integrals? $\endgroup$ – Rei Henigman Jul 3 '18 at 12:14
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    $\begingroup$ If you're using the Riemann integral, it's similar. Up to changes that don't affect any integral, such a function must be even. If you want such a function to be continuous, it must be even, as follows for example from Fejér's theorem. $\endgroup$ – Daniel Fischer Jul 3 '18 at 12:24
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You can arbitrarily change a function on a null set (set of measure 0) without affecting integrals. So you can for example take the characteristic function of 1+πQ. This is not an even function, but from the perspective of the Lebesgue integral, it's equivalent to the zero function, which is even. Thus, a class in L2([−π,π]) is orthogonal to all sin(nx) if and only if it has an even representative.

If you're using the Riemann integral, it's similar. Up to changes that don't affect any integral, such a function must be even. If you want such a function to be continuous, it must be even, as follows for example from Fejér's theorem.

Thanks Daniel Fischer for the answer!

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