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Stahl H., Totik V. — General Orthogonal Polynomials
Stahl H., Totik V. — General Orthogonal Polynomials

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Название: General Orthogonal Polynomials

Авторы: Stahl H., Totik V.


In this treatise, the authors present the general theory of orthogonal polynomials on the complex plane and several of its applications. The assumptions on the measure of orthogonality are general, the only restriction is that it has compact support on the complex plane. In the development of the theory the main emphasis is on asymptotic behavior and the distribution of zeros. In the first two chapters exact upper and lower bounds are given for the orthonormal polynomials and for the location of their zeros. The next three chapters deal with regular n-th root asymptotic behavior, which plays a key role both in the theory and in its applications. Orthogonal polynomials with this behavior correspond to classical orthogonal polynomials in the general case, and many extremal properties of measures in mathematical analysis and approximation theory with this type of regularity turn out to be equivalent. Several easy-to-use criteria are presented for regular behavior. The last chapter contains applications of the theory, including exact rates for convergence of rational interpolants, best rational approximants and non-diagonal Pade approximants to Markov functions (Cauchy transforms of measures). The results are based on potential theoretic methods, so both the methods and the results can be extended to extremal polynomials in norms other than L]2 norms. A sketch of the theory of logarithmic potentials is given in an appendix.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1992

Количество страниц: 264

Добавлена в каталог: 28.07.2008

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Предметный указатель
$\mu$-quasi everywhere      11
$\mu$-regular point      140
$\mu\in Reg$      61
Admissible interpolation scheme      153
Admissible, definition of      233
Asymptotic distribution      153
Balayage      230
Bernstein — Walsh lemma      112 205 234
Best $L^2(\mu)$ polynomial approximation      211
Best rational approximant      161
Cantor set      104
Capacity      1 221
Carrier      2
Carrier-related measures      6
Chebyshev constant      234
Chebyshev number(s)      209 234
Condenser      163 171 231
Condenser capacity      161 231
Condenser potential      232
Continued fraction      150
Convergents      150
Convex hull      2
Criterion $\lambda$      110
Criterion $\lambda^*$      110
Criterion, Erdos — Turan      101
Criterion, Ullman's      102
Criterion, Widom's      105
Dirichlet's problem      229
Dirichlet's problem, generalized      229
Domain of convergence      191
Domain of divergence      191
Equilibrium distribution      7 163 204
Equilibrium measure(s)      224 232 233
External field      233
Extremal measure      233
Fine topology      222
Finite logarithmic energy      221
First maximum principle      223
Fourier coefficients      212
Green function      1 7 15 227 230
Green potential      15 78 154 231
Harnack's inequality      22 224
Interpolation defect      152
Interpolation points      150
Interpolation scheme      153
Leading coefficient      2 5 14
localization      144
Localization at a point      140
Locally uniformly, notion of      3 223
Logarithmic energy      102 220
Logarithmic potential      221
Lower envelope theorem      223
Markov function      150 161
Maximum principles      223
Minimal carrier      8
Minimal-carrier capacity      2
Minimal-carrier equilibrium distribution      10
Minimal-carrier Green function      2
Moments      150
Monic orthogonal polynomial      14
Multiset      151
Norm asymptotics      42
Normalized counting measure      30
Orthonormal polynomials      2
Outer boundary      2 205
Outer domain      2
Pade approximant(s)      150 189
Pade table      190
Polynomial convex hull      2
Potential      7
Principle of descent      222
Principle of domination      223
Qu.e.      1
Quasi everywhere      1 221
Rational interpolant      150
Ray sequences      190
Regular (nth-root) asymptotic behavior      61
Regular asymptotic zero distribution      62
Regular exterior asymptotic behavior      59
Regular interior asymptotic behavior      60
Regular measure      61
Regular point      229
Regular with respect to the Dirichlet problem      229
Regularity of a measure      61
Remainder formula      151
Remez's inequality      115
Riesz' representation theorem      222
Second maximum principle      223
Superharmonic function      221
Uniformly on compact subsets of D, notion of      3
Uniqueness      224
Varying measure(s)      78 209
Weak* convergence      7
Weighted Chebyshev constants      209
Weighted Chebyshev problem      233
Weighted energy      233
Weighted polynomials      204
Zero distribution      7 30
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