I am quite interested about a problem which seems intuitively direct but hard for me to verify. Let $f\in L^2[-\pi,\pi]$ and has Fourier series $f\sim \sum_{n=1,2,...}A_n\cos(ns)$. (Then $\sum A_n^2<\infty$ and thus by completeness of $L^2$), there is always a function $g\in L^2[-\pi,\pi]$ with Fourier series $\sum_{n=1,2,...}A_n\sin(ns)$. My question is about the smoothness property: i.e., whether $f$ is continuous if and only if $g$ is continuous? whether $f$ is $C^k$($k$-times continuously differentiable) if and only if $g$ is $C^k$? The same question applies for Lipschitz continuity, Absolute continuity?
I know this may be related to Hilbert transformation. But I am still not sure about whether the above statements are correct or not after reading some materials about Hilbert transformation.