Logarithmic Potentials with External Fields

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Springer Science & Business Media, Oct 9, 1997 - Mathematics - 505 pages
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In 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
LIV
349
LV
352
LVI
359
LVIII
364
LIX
373
LX
379
LXI
381
LXII
382
LXIII
388
LXIV
394
LXV
403
LXVI
409
LXVII
421
LXVIII
426
LXIX
442
LXX
449
LXXII
454
LXXIII
458
LXXIV
465
LXXV
466
LXXVI
471
LXXVII
478
LXXVIII
480
LXXIX
483
LXXX
485
LXXXI
495
LXXXII
501
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