Let {α 1 , α 2 , . . . } be a sequence of real numbers outside the interval [−1, 1] and µ a positive bounded Borel measure on this interval. We introduce rational functions ϕ n (x) with poles {α 1 , . . . , α n } orthogonal on [−1, 1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of ϕ n+1 (x)/ϕ n (x) as n tends to infinity under certain assumptions on the measure and the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three term recurrence relation satisfied by the orthonormal functions.

No abstract

No abstract

In this note we make a critical comparison of some matlab programs for the digital computation of the fractional Fourier transform that are freely available and we describe our own implementation that filters the best out of the existing ones. Two types of transforms are considered: First the fast approximate fractional Fourier transform algorithm for which two algorithms are available. The method is described in H.M. Ozaktas, M.A. Kutay, and G. Bozdagi. Digital computation of the fractional Fourier transform.

Techniques of Pade approximation and continued fractions have been used often in model reduction problems. An extensive bibliography on this topic is given. The ideas are explained for the simple situation of a scalar function where no singular blocks appear in the Pade table. Extensions to the matrix case and the multivariable case are not explained in detail.