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Register a new userDetermining Singularities Using Row Sequences of Pade-orthogonal Approximants
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Guillermo L\\'opez Lagomasino
Publication date
2013
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English
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Starting from the orthogonal polynomial expansion of a function $F$ corresponding to a finite positive Borel measure with infinite compact support, we study the asymptotic behavior of certain associated rational functions (Pad{e}-orthogonal approximants). We obtain both direct and inverse results relating the convergence of the poles of the approximants and the singularities of $F.$ Thereby, we obtain analogues of the theorems of E. Fabry, R. de Montessus de Ballore, V.I. Buslaev, and S.P. Suetin.
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We give necessary and sufficient conditions for the convergence with geometric rate of the denominators of linear Pade-orthogonal approximants corresponding to a measure supported on a general compact set in the complex plane. Thereby, we obtain an analogue of Gonchars theorem on row sequences of Pade approximants.
Given a vector function ${bf F}=(F_1,ldots,F_d),$ analytic on a neighborhood of some compact subset $E$ of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in terms of the expansions of the components $F_k, k=1,ldots,d,$ with respect to the sequence of Faber polynomials associated with $E$. Such sequences of vector rational functions are analogous to row sequences of type II Hermite-Pade approximation. We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of the sequence of vector rational functions so constructed. The exact rate of convergence of these denominators is provided and the rate of convergence of the approximants is estimated. It is shown that the common denominators of the approximants detect the poles of the system of functions closest to $E$ and their order.
We consider row sequences of vector valued Pad{e}-Faber approximants (simultaneous Pad{e}-Faber approximants) and prove a Montessus de Ballore type theorem.
Let $f$ be a power series with positive radius of convergence. In the present paper, we study the phenomenon of overconvergence of sequences of classical Pade approximants pi{n,m_n} associated with f, where m(n)<=m(n+1)<=m(n) and m(n) = o(n/log n), resp. m(n) = 0(n) as n is going to infiity. We extend classical results by J. Hadamard and A. A. Ostrowski related to overconvergent Taylor polynomials, as well as results by G. Lopez Lagomasino and A. Fernandes Infante concerning overconvergent subsequences of a fixed row of the Pade table.
Given a system of functions f = (f1, . . . , fd) analytic on a neighborhood of some compact subset E of the complex plane, we give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of multipoint Hermite-Pade approximants. The exact rate of convergence of these denominators and of the approximants themselves is given in terms of the analytic properties of the system of functions. These results allow to detect the location of the poles of the system of functions which are in some sense closest to E.
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