### Abstract

In this paper we use techniques from the theory of ODEs and also from inverse scattering theory to obtain a variety of results on the regularity and support properties of the equilibrium measure for logarithmic potentials on the finite interval [-1, 1], in the presence of an external fieldV. In particular, we show that ifVisC^{2}, then the equilibrium measure is absolutely continuous with respect to Lebesgue measure, with a density which is Hölder-12 on (-1, 1), and with at worst a square root singularity at ±1. Moreover, ifVis real analytic then the support of the equilibrium measure consists of a finite number of intervals. In the cases whereV=tx^{m},m=1, 2, 3, or 4, the equilibrium measure is computed explicitly for allt∈R. For these cases the support of the equilibrium measure consists of 1, 2, or 3 intervals, depending ontandm. We also present detailed results for the general monomial caseV=tx^{m}, for allm∈N. The regularity results for the equilibrium measure are obtained by careful analysis of the Fekete points associated to the weighte^{nV(x)}dx. The results on the support of the equilibrium measure are obtained using two different approaches: (i) an explicit formula of the kind derived by physicists for mean-field theory calculations; (ii) detailed perturbation theoretic results of the kind that are needed to analyze the zero dispersion limit of the Korteweg-de Vries equation in Lax-Levermore theory. The implications of the above results for a variety of related problems in approximation theory and the theory of orthogonal polynomials are also discussed.

Original language | English (US) |
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Pages (from-to) | 388-475 |

Number of pages | 88 |

Journal | Journal of Approximation Theory |

Volume | 95 |

Issue number | 3 |

DOIs | |

State | Published - Dec 1 1998 |

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Mathematics(all)
- Applied Mathematics

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## Cite this

*Journal of Approximation Theory*,

*95*(3), 388-475. https://doi.org/10.1006/jath.1997.3229