Convergence of Associated Continued Fractions Revised

Abstract

This is an expository article which contains alternative proofs of many theorems concerning convergence of a continued fraction to a holomorphic function. The continued fractions which are studied are continued fractions of the form $$K_\infty {\lambda ) = \frac {1|}{|\lambda } - b_1 } - \frac{{a_1^2 |}}{{|{\lambda } - b_2 }} - \frac{{a_2^2 |}}{{|{\lambda } - b_3 }} - \cdots ,$$ where {an}, {bn} are real sequences with an>0 (associated continued fractions). The proofs rely on the properties of the resolvent (λ−T)−1, where T is the symmetric tridiagonal operator corresponding to {an} and {bn}, and avoid most of technical aspects of earlier work. A variety of well-known results is proved in a unified way using operator methods. Many proofs can be regarded as functional analytic proofs of important classical theorems.