non-even continuous real function orthogonal to sin(nx) for every n

Question

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.

Answer 1

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!