Tauberian and Mercerian Theorems for Systems of Kernels

In [BI3], we introduced the notion of ratio Mercerian theorem, to improve the Mercerian theorem for Fourier and Hankel transforms first proved in [BI1]. In the first half of this paper, we extend (and correct) the ratio Mercerian theorem, and apply it to the proofs of Tauberian theorems for kernels of Korenblum type. The latter can be applied to analytic number theory; this was in fact the motivation for the present paper. The application to analytic number theory will be given in a separate paper [BI5]. In the second half of this paper, we prove a Mercerian counterpart to one of the Tauberian theorems – one in the boundary case. This is done via a further extension of our previous extension of the Drasin–Shea–Jordan theorem (see e.g. [BGT, Chap. 5]). Throughout the paper, the idea of proving assertions for a system of kernels, rather than a single kernel, plays a key role. For measurable functions f, g : (0,∞) → R, the Mellin convolution f ∗ g is the function defined by