Let $(a_n)$ be a sequence of real numbers such that the series $\sum_n n a_n$ converges absolutely. Suppose also that the function $$f(x)=\sum_n n a_n \sin nx$$ is non negative on the interval $[0,\pi]$.
Then is the function $$g(x,y)=\sum_n a_n \sin nx \sin ny $$ also non negative on $[0,\pi]\times [0,\pi]$?