Extension of the Drasin-Shea-Jordan theorem
Consider a positive function \(f\) and a kernel \(K\) such that the Mellin convolution \(k*f(x):=\int^\infty_0K(x/t)f(t)dt/t\) exists absolutely for \(x>0\). We may consider the statements i) \(f\in RV_\rho\), i.e., is regularly varying of order \(\rho\); ii) \(K*f\in RV_\rho\); iii) \((K*f)(x)/f(x)\to c\neq 0\), as \(x\to\infty\). Under mild conditions i) implies ii) (Abelian conclusion), whereas the Tauberian counterpart ii) implies i) needs additional assumptions on \(f\), so-called Tauberian conditions. Finally, the Drasin-Shea theorem says, that for positive kernels \(K\) iii) implies i) and hence ii) (Mercerian conclusion). By a result of Jordan \(K\) may also have sign changes but has to obey other technical conditions [see e.g., the book of \textit{N. H. Bingham}, \textit{C. M. Goldie} and \textit{J. L. Teugels}, Regular variation (1987; Zbl 0617.26001)]. The aim of the present paper is to give a Mercerian result for kernels which may have infinitely many changes of signs, but satisfy otherwise only some mild conditions and functions \(f\) being of bounded increase or decrease. Particular attention is given to Hankel transforms with index \(\nu>-1/2\). The proofs are simpler than before, in particular the Polyá peak theorem is not used, but absolute convergence of the integrals is assumed instead. This paper continues earlier work of the authors in this context [see \textit{N. H. Bingham} and \textit{A. Inoue}, Q. J. Math., Oxf. II. Ser. 48, No. 191, 279-307 (1997; Zbl 0889.42005) and, Ratio Mercerian theorems with applications to Fourier and Hankel transforms, Proc. Lond. Math. Soc., III. Ser. 79, No. 3, 626-648 (1999; Zbl 1030.44005)].
scientific article; zbMATH DE number 1529887