Hilbert’s Double Series Theorem’s Extensions via the Mathieu Series Approach

Abstract

The author’s research devoted to the Hilbert’s double series theorem and its various further extensions are the focus of a recent survey article. The sharp version of double series inequality result is extended in the case of a not exhaustively investigated non-homogeneous kernel, which mutually covers the homogeneous kernel cases as well. Particularly, novel Hilbert’s double series inequality results are presented, which include the upper bounds built exclusively with non-weighted ℓp–norms. The main mathematical tools are the integral expression of Mathieu (a,λ)-series, the Hölder inequality and a generalization of the double series theorem by Yang.

Keywords

• Hilbert’s double series theorem
• Hölder inequality
• Dirichlet series
• Jacobi theta function
• Riemann zeta function
• Mathieu (<b><i>a</i></b>,<b><i>λ</i></b>)-series