Let $f\in H(0,2\pi)$, with inner product $<f,g>=\int_0^{2\pi}{f(t)g(t)dt}$
$S_f=a_0 + \sum_{n=1}^{\infty}{a_ncos(nx)}+\sum_{n=1}^{\infty}{b_nsin(nx)}$, is the fourier series for f.
Where $a_0=\frac{1}{\sqrt{2\pi}}<f,cos(nx)>$, $a_n=\frac{1}{\sqrt{\pi}}<f,cos(nx)>$,$b_n=\frac{1}{\sqrt{\pi}}<f,sin(nx)>$
Does the sum $\sum_{n=1}^{\infty}{a_nb_n}$ converge?
Since $\{a_n\}_{n=0}^{\infty}\cup{\{b_n\}_{n=1}^{\infty}}$ is a complete orthonormal system, I was thinking if there meight be a function which has a fourier series coefficients $A_n=a_nb_n$, and then determine the convergence for the mentioned sum.
Or perhaps using a known test, since $\sum_{n=1}^{\infty}{b_n}$ is bounded, but then again, the sequence $a_n$ is not monotonic...
Any help would be appreciated =]