##
**Regular variation, extensions and Tauberian theorems.**
*(English)*
Zbl 0624.26003

CWI Tracts, 40. Centrum voor Wiskunde en Informatica. Amsterdam: Stichting Mathematisch Centrum. V, 132 p.; Dfl. 20.30 (1987).

The aim of this book is to present the theory of regular variation and some extensions, especially the classes \(\Pi\) and \(\Gamma\) (see below) and to apply it to the theory of Abelian and Tauberian theorems for Laplace transforms and some more general integral transforms.

The book contains 4 chapters, where the first one is devoted to the following classes of real functions on \((0,\infty)\): R, i.e. regularly varying functions; \(\Pi\), i.e. functions obeying \(\lim_{\lambda \to \infty}(f(\lambda x)-f(\lambda))/a(\lambda)=\log x,x>0\); \(\Gamma\), i.e. non-decreasing functions with the property \(\lim_{\lambda \to \infty}f(\lambda +x b(\lambda))/f(\lambda)=e^ x,x\in {\mathbb{R}}\), with suitable auxiliary functions a(.) resp. b(.). Uniform convergence and representation theorems, nice candidates and inverse functions in these function classes are mainly discussed. Finally some information on Beurling slow variation and regularly varying sequences are given. Chapter II contains Abelian and Tauberian theorems for the Laplace transform \(\hat f(t)=t\int^{\infty}_{0}e^{-xt}f(x) dx\) in case of regular variation (Karamata, Drasin) and the counter part for the case \(\Pi\). Some extensions to more general integral transforms are given as well. Furthermore Abelian and Tauberian results in between \(\log f\) and \(\log \hat f\) are given, proving as an intermediate step the corresponding relation for the complementary function of \(\log f\) [compare A. A. Balkema, J. L. Geluk and L. de Haan, Q. J. Math., Oxf. II. Ser. 30, 385-416 (1979; Zbl 0464.44002)]. In Chapter III 0-versions of the classes R (bounded variation) and \(\Pi\) (asymptotically balanced functions) are discussed. In the final Chapter IV the corresponding Abelian and Tauberian theorems with respect to these function classes are investigated [see L. de Haan and U. Stadtmüller, J. Math. Anal. Appl. 108, 344-365 (1985; Zbl 0581.44003)].

The book leads through the main circles of Abelian and Tauberian theorems for Laplace transforms with some extensions to more general integral transforms and introduces the basic tools therefor. In contrary to the encyclopedia on regular variation, i.e. the book of N. H. Bingham, C. M. Goldie and J. L. Teugels: “Regular variation” (1987; Zbl 0617.26001) the authors have presented a self-contained text on regular variation as a natural setting for Tauberian theorems for Laplace transforms and have succeeded to present that theory in a well legible and smooth form.

The book contains 4 chapters, where the first one is devoted to the following classes of real functions on \((0,\infty)\): R, i.e. regularly varying functions; \(\Pi\), i.e. functions obeying \(\lim_{\lambda \to \infty}(f(\lambda x)-f(\lambda))/a(\lambda)=\log x,x>0\); \(\Gamma\), i.e. non-decreasing functions with the property \(\lim_{\lambda \to \infty}f(\lambda +x b(\lambda))/f(\lambda)=e^ x,x\in {\mathbb{R}}\), with suitable auxiliary functions a(.) resp. b(.). Uniform convergence and representation theorems, nice candidates and inverse functions in these function classes are mainly discussed. Finally some information on Beurling slow variation and regularly varying sequences are given. Chapter II contains Abelian and Tauberian theorems for the Laplace transform \(\hat f(t)=t\int^{\infty}_{0}e^{-xt}f(x) dx\) in case of regular variation (Karamata, Drasin) and the counter part for the case \(\Pi\). Some extensions to more general integral transforms are given as well. Furthermore Abelian and Tauberian results in between \(\log f\) and \(\log \hat f\) are given, proving as an intermediate step the corresponding relation for the complementary function of \(\log f\) [compare A. A. Balkema, J. L. Geluk and L. de Haan, Q. J. Math., Oxf. II. Ser. 30, 385-416 (1979; Zbl 0464.44002)]. In Chapter III 0-versions of the classes R (bounded variation) and \(\Pi\) (asymptotically balanced functions) are discussed. In the final Chapter IV the corresponding Abelian and Tauberian theorems with respect to these function classes are investigated [see L. de Haan and U. Stadtmüller, J. Math. Anal. Appl. 108, 344-365 (1985; Zbl 0581.44003)].

The book leads through the main circles of Abelian and Tauberian theorems for Laplace transforms with some extensions to more general integral transforms and introduces the basic tools therefor. In contrary to the encyclopedia on regular variation, i.e. the book of N. H. Bingham, C. M. Goldie and J. L. Teugels: “Regular variation” (1987; Zbl 0617.26001) the authors have presented a self-contained text on regular variation as a natural setting for Tauberian theorems for Laplace transforms and have succeeded to present that theory in a well legible and smooth form.

Reviewer: U.Stadtmüller

### MSC:

26A12 | Rate of growth of functions, orders of infinity, slowly varying functions |

40E05 | Tauberian theorems |

44A10 | Laplace transform |