Continued fractions and certain functional equations (Q1091098)

The paper deals with an application of continued fractions to a functional equation in a Hilbert space equipped with Dirichlet norm. The current effort is an extension of the work of Stesin (1983) who shows that the successive denominators of the S-fraction expansion corresponding to the Taylor series approximate the Fredholm denominator of the kernel. The author extends the above theory to the kernel of the integral equation associated with the Robin-Poincaré problem. This kernel is assumed to be a member of the symmetrizable class of kernels. It is shown that the Neumann series solution of the Fredholm-Poincaré integral equation with a kernel fully symmetrizable by a positive operator gives a Taylor series with Schwarz's constants as coefficients. The paper also contains results relating coefficients associated with Lanczos' minimized iteration scheme and those of J-fraction and S- fraction expansions. Notions of bounds, convergence and truncation errors are discussed using equivalence transformations. The meromorphic character of the solution is established using the properties of continued fractions. This provides an alternate to the Fredholm theory which is generally used for this purpose.