Preface and Dedication
This volume contains the refereed contributions of the international conference
"Continued Fractions: From Analytic Number Theory to Constructive Approxi-
mation", held at the University of Missouri-Columbia on May 20-23, 1998. The
meeting also celebrated Jerry Lange's seventieth birthday and marked his retire-
ment from the University of Missouri.
It
is a great pleasure to dedicate this volume
to Jerry in recognition of his distinguished service and long-lasting impact on the
Mathematics Department at MU.
In spite of their long tradition, continued fractions (whose general definition
appears to go back to the book "Liber Abaci" of Leonardo of Pisa, also called Fi-
bonacci, written around 1202) remain an active area of research in a large number
of fields ranging from pure mathematics to mathematical physics and approxima-
tion theory. The principal purpose of this conference was to focus on continued
fractions as a common interdisciplinary theme bridging gaps between these fields.
As a consequence, the lectures at this conference and the corresponding contribu-
tions to this volume reflect the wide range of applicability of continued fractions in
mathematics and the applied sciences.
More specifically, recurrence relations for orthogonal polynomials and their
relations to continued fractions, appealing to a well-known theorem of Markoff,
are studied in Askey's contribution. Orthogonal Laurent polynomials and ques-
tions of (in)determinacy of the strong Stieltjes moment problem are considered by
Njastad. Strong Stieltjes distributions, orthogonal Laurent polynomials and related
continued fractions appear in Bracciali's article. Generating functions of orthog-
onal Laurent polynomials, associated kernel polynomials, and moment-preserving
approximations are treated by Bojanov and Sri Ranga. Convergence properties of
Laurent polynomials that interpolate functions on the boundary of a circular annu-
lus are investigated by Li. Compact perturbations of reflection coefficients and the
resulting weak asymptotics of orthogonal polynomials on the support of the mea-
sure of orthogonality and asymptotics of the corresponding functions of the second
kind are studied by Peherstorfer and Steinbauer.
Convergence of Stieltjes continued fractions and their asymptotic speed of con-
vergence as a result of analyzing the asymptotic behavior of the corresponding
expansion coefficients, applicable to Whittaker functions, are considered by Jones
and Shen. Various convergence theorems including a new and constructive proof of
an extension of Van Vleck's convergence theorem for continued fractions, including
a sharp truncation error formula, are derived in Lange's contribution. A very ex-
tensive treatment of convergence criteria, including substantial progress in proving
a conjecture by Lorentzen and Ruscheweyh, as well as a simplified approach to
Cordova's extension of the classical parabola theorem, are provided by Lorentzen.
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