Weighted integrability of double cosine series with nonnegative coefficients
Weighted integrability of double cosine series with nonnegative coefficients
Let $f_c(x,y)\equiv \sum _{j=1}^\infty \sum _{k=1}^\infty a_{jk}(1-\mathop {\rm cos}\nolimits jx)(1-\mathop {\rm cos}\nolimits ky)$ with $a_{jk}\ge 0$ for all $j,k\ge 1$. We estimate the integral $ \int _0^\pi \int _0^\pi x^{\alpha -1} y^{\beta -1} \phi (