Simić, Slavko General kernel convolutions with slowly varying functions. (English) Zbl 1119.26004 Publ. Inst. Math., Nouv. Sér. 78(92), 73-77 (2005). 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. Reviewer: Slobodanka Janković (Beograd) MSC: 26A12 Rate of growth of functions, orders of infinity, slowly varying functions Keywords:complex valued kernel PDFBibTeX XMLCite \textit{S. Simić}, Publ. Inst. Math., Nouv. Sér. 78(92), 73--77 (2005; Zbl 1119.26004) Full Text: DOI EuDML