A first kind Fredholm integral equation with nondegenerate kernel is given, which particular solution is the Bessel function of the first kind. This equation is solved by means of Mellin transform pair.

A new version of discrete Hilbert type inequality is given where the kernel function is non-homogeneous. The main mathematical tools are the representation of the Dirichlet series by means of the Laplace integral, and the H?lder inequality with non-conjugated parameters. Numerous special cases are treated and conditional best constants are discussed.

In this article we are interested in the beginnings and the development of the sampling theory in signal analysis of stochastic signals, locating these in the early fifties. Besides the most important papers by Parzen (1956), Balakrishnan (1957), Belyaev (1959), and Lloyd (1959) we expose and report on few other interesting articles not widely known, giving an overview of the topic.