## Logarithmic Potentials with External FieldsIn recent years approximation theory and the theory of orthogonal polynomials have witnessed a dramatic increase in the number of solutions of difficult and previously untouchable problems. This is due to the interaction of approximation theoretical techniques with classical potential theory (more precisely, the theory of logarithmic potentials, which is directly related to polynomials and to problems in the plane or on the real line). Most of the applications are based on an exten sion of classical logarithmic potential theory to the case when there is a weight (external field) present. The list of recent developments is quite impressive and includes: creation of the theory of non-classical orthogonal polynomials with re spect to exponential weights; the theory of orthogonal polynomials with respect to general measures with compact support; the theory of incomplete polynomials and their widespread generalizations, and the theory of multipoint Pade approximation. The new approach has produced long sought solutions for many problems; most notably, the Freud problems on the asymptotics of orthogonal polynomials with a respect to weights of the form exp(-Ixl ); the "l/9-th" conjecture on rational approximation of exp(x); and the problem of the exact asymptotic constant in the rational approximation of Ixl. One aim of the present book is to provide a self-contained introduction to the aforementioned "weighted" potential theory as well as to its numerous applications. As a side-product we shall also fully develop the classical theory of logarithmic potentials. |

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### Contents

VII | 23 |

VIII | 24 |

IX | 35 |

X | 43 |

XI | 49 |

XII | 58 |

XIII | 63 |

XIV | 75 |

XLVI | 275 |

XLVII | 277 |

XLVIII | 278 |

XLIX | 301 |

L | 313 |

LI | 326 |

LII | 334 |

LIII | 343 |

XV | 81 |

XVI | 83 |

XVII | 97 |

XVIII | 100 |

XIX | 108 |

XX | 123 |

XXI | 137 |

XXII | 141 |

XXIII | 142 |

XXIV | 153 |

XXV | 162 |

XXVI | 169 |

XXVII | 177 |

XXVIII | 180 |

XXIX | 187 |

XXX | 191 |

XXXI | 192 |

XXXII | 209 |

XXXIII | 221 |

XXXIV | 227 |

XXXV | 238 |

XXXVI | 245 |

XXXVII | 246 |

XXXIX | 249 |

XL | 251 |

XLI | 254 |

XLII | 257 |

XLIV | 267 |

XLV | 273 |

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### Common terms and phrases

admissible weight apply approximation assume assumption asymptotic balayage Borel measure bounded Chebyshev compact set compact subsets compact support conformal mapping consider constant continuous function convergence Corollary defined denotes density derivative Dirichlet problem domain Dr(zo energy problem equality equilibrium measure estimate example exists external field extremal measure fact Fekete points finite logarithmic energy fj.w follows from Theorem formula Furthermore Green function harmonic functions hence holds implies inequality infinity integral interval Lebesgue measure Lemma liminf limsup logarithmic potentials lower semi-continuous maximum minimum principle monic polynomials neighborhood nonnegative norm obtain orthogonal polynomials polynomials Pn positive capacity principle of domination Proof of Theorem prove quasi-every z e quasi-everywhere rational functions respect result Saff satisfies Section sequence signed measure solution support Sw Suppose supremum Theorem 2.1 topology uniformly upper vanishes verify weighted polynomials zero capacity