Weighted integrability of double cosine series with nonnegative coefficients
Volume 156 / 2003
Studia Mathematica 156 (2003), 133-141
MSC: Primary 42A32, 42B05.
DOI: 10.4064/sm156-2-4
Abstract
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 (f_c(x,y))\, dx\, dy $ in terms of the coefficients $a_{jk}$, where $\alpha ,\beta \in {\mathbb R}$ and $\phi :[0,\infty ]\to [0,\infty ]$. Our results can be regarded as the trigonometric analogues of those of Mazhar and Móricz [MM]. They generalize and extend Boas [B, Theorem 6.7].
