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Hilbert's double series theorem extended to the case of non-homogeneous kernels

Sharp version of celebrated Hilbert's double series theorem is given in the case of non--homogeneous kernel. The main mathematical tools are: the integral representation of Mathieu's $(\mathbf a, \boldsymbol \lambda)$-series, the H\"older inequality and an extension of the double series theorem by Yang.

Hilbert's double series theorem; Dirichlet-series; Mathieu's $(\mathbf a; \boldsymbol \lambda)$-series; H\

DOI: 10.1016/j.jmaa.2007.12.051