Continued fraction solution of Krein's inverse problem

The spectral data of a vibrating string are encoded in its so-called characteristic function. We consider the problem of recovering the distribution of mass along the string from its characteristic function. It is well-known that Stieltjes' continued fraction provides a solution of this inverse problem in the particular case where the distribution of mass is purely discrete. We show how to adapt Stieltjes' method to solve the inverse problem for a related class of strings. An application to the excursion theory of diffusion processes is presented.Comment: 18 pages, 2 figure

Continued Fraction as a Discrete Nonlinear Transform

The connection between a Taylor series and a continued-fraction involves a nonlinear relation between the Taylor coefficients $\{ a_n \}$ and the continued-fraction coefficients $\{ b_n \}$. In many instances it turns out that this nonlinear relation transforms a complicated sequence $\{a_n \}$ into a very simple one $\{ b_n \}$. We illustrate this simplification in the context of graph combinatorics.Comment: 6 pages, OKHEP-93-0

Continued fraction digit averages an Maclaurin's inequalities

A classical result of Khinchin says that for almost all real numbers $\alpha$, the geometric mean of the first $n$ digits $a_i(\alpha)$ in the continued fraction expansion of $\alpha$ converges to a number $K = 2.6854520\ldots$ (Khinchin's constant) as $n \to \infty$. On the other hand, for almost all $\alpha$, the arithmetic mean of the first $n$ continued fraction digits $a_i(\alpha)$ approaches infinity as $n \to \infty$. There is a sequence of refinements of the AM-GM inequality, Maclaurin's inequalities, relating the $1/k$-th powers of the $k$-th elementary symmetric means of $n$ numbers for $1 \leq k \leq n$. On the left end (when $k=n$) we have the geometric mean, and on the right end ($k=1$) we have the arithmetic mean. We analyze what happens to the means of continued fraction digits of a typical real number in the limit as one moves $f(n)$ steps away from either extreme. We prove sufficient conditions on $f(n)$ to ensure to ensure divergence when one moves $f(n)$ steps away from the arithmetic mean and convergence when one moves $f(n)$ steps away from the geometric mean. For typical $\alpha$ we conjecture the behavior for $f(n)=cn$, $0<c<1$. We also study the limiting behavior of such means for quadratic irrational $\alpha$, providing rigorous results, as well as numerically supported conjectures.Comment: 32 pages, 7 figures. Substantial additions were made to previous version, including Theorem 1.3, Section 6, and Appendix

33-dimensional Continued Fraction Algorithms Cheat Sheets

Multidimensional Continued Fraction Algorithms are generalizations of the Euclid algorithm and find iteratively the gcd of two or more numbers. They are defined as linear applications on some subcone of $\mathbb{R}^d$. We consider multidimensional continued fraction algorithms that acts symmetrically on the positive cone $\mathbb{R}^d_+$ for $d=3$. We include well-known and old ones (Poincar\'e, Brun, Selmer, Fully Subtractive) and new ones (Arnoux-Rauzy-Poincar\'e, Reverse, Cassaigne). For each algorithm, one page (called cheat sheet) gathers a handful of informations most of them generated with the open source software Sage with the optional Sage package \texttt{slabbe-0.2.spkg}. The information includes the $n$-cylinders, density function of an absolutely continuous invariant measure, domain of the natural extension, lyapunov exponents as well as data regarding combinatorics on words, symbolic dynamics and digital geometry, that is, associated substitutions, generated $S$-adic systems, factor complexity, discrepancy, dual substitutions and generation of digital planes. The document ends with a table of comparison of Lyapunov exponents and gives the code allowing to reproduce any of the results or figures appearing in these cheat sheets.Comment: 9 pages, 66 figures, landscape orientatio