Given a system of functions F = (F 1 , . . . , 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 orthogonal expansions of the components F k , k = 1, . . . , d, with respect to a sequence of orthonormal polynomials associated with a measure µ whose support is contained in E. Such sequences of vector rational functions resemble row sequences of type II Hermite-Padé approximants. Under appropriate assumptions on µ, 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 simultaneous approximants is estimated. It is shown that the common denominator of the approximants detect the location of the poles of the system of functions.
We prove a conjecture of T. Erdélyi and E.B. Saff, concerning the form of the dominant term (as N → ∞) of the N -point Riesz d-polarization constant for an infinite compact subset A of a d-dimensional C 1 -manifold embedded in R m (d ≤ m). Moreover, if we assume further that the d-dimensional Hausdorff measure of A is positive, we show that any asymptotically optimal sequence of N -point configurations for the N -point d-polarization problem on A is asymptotically uniformly distributed with respect to H d | A .These results also hold for finite unions of such sets A provided that their pairwise intersections have H d -measure zero.
Given a vector function F = (F 1 , . . . , 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, . . . , 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-Padé 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 give necessary and sufficient conditions for the convergence with geometric rate of the denominators of linear Padé-orthogonal approximants corresponding to a measure supported on a general compact set in the complex plane. Thereby, we obtain an analogue of Gonchar's theorem on row sequences of Padé approximants.
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