×

General kernel convolutions with slowly varying functions. (English) Zbl 1119.26004

Let \(B_\rho(z)=\int_0^{+\infty}t^\rho| K(t,z)|\, dt\), where \(K(t,z)\) is a complex-valued kernel. Suppose that \(B_\rho(z)\) exists for \(-a\leq \rho\leq1\), \(a>0\), in some complex region \(D\). The author proves that, under the conditions presented in the paper, the following asymptotic relation holds \[ \int_0^{+\infty}L(t)K(t,z)\,dt=L(B_1(z))\int_0^{+\infty}K(t,z)\,dt \quad(1+o(1)), \] for all \(z\in D\), where \(L\) is a slowly varying function. The paper ends with examples showing that some classical theorems, such as Karamata’s theorem for the Laplace transform, can be extended to some regions of the complex plane.

MSC:

26A12 Rate of growth of functions, orders of infinity, slowly varying functions
PDFBibTeX XMLCite
Full Text: DOI EuDML