On sequences of rational interpolants of the exponential function with unbounded interpolation points
- Autores: T. Claeys, F. Wielonsky
- Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 171, Nº 1, 2013, págs. 1-32
- Idioma: inglés
- DOI: 10.1016/j.jat.2013.03.007
- Enlaces
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Resumen
We consider sequences of rational interpolants rn(z) of degree n to the exponential function ez associated to a triangular scheme of complex points {z(2n) j }2n j=0, n > 0, such that, for all n, |z(2n) j | . cn1.�¿, j = 0, . . . , 2n, with 0 < �¿ . 1 and c > 0. We prove the local uniform convergence of rn(z) to ez in the complex plane, as n tends to infinity, and show that the limit distributions of the conveniently scaled zeros and poles of rn are identical to the corresponding distributions of the classical Pad�Le approximants. This extends previous results obtained in the case of bounded (or growing like log n) interpolation points. To derive our results, we use the Deift.Zhou steepest descent method for Riemann.Hilbert problems. For interpolation points of order n, satisfying |z(2n) j | . cn, c > 0, the above results are false if c is large, e.g.
c . 2�Î. In this connection, we display numerical experiments showing how the distributions of zeros and poles of the interpolants may be modified when considering different configurations of interpolation points with modulus of order n.
