Strong asymptotics for extremal polynomials associated with weights on IR:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin, [u.a.]
Springer
[1988]
|
| Series: | Lecture notes in mathematics.
1305 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VII, 153 Seiten |
| ISBN: | 0387189580 3540189580 |
Staff View
MARC
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| 100 | 1 | |a Lubinsky, Doron S. |d 1955- |0 (DE-588)172232589 |4 aut | |
| 245 | 1 | 0 | |a Strong asymptotics for extremal polynomials associated with weights on IR |c D. S. Lubinsky ; E. B. Staff |
| 264 | 1 | |a Berlin, [u.a.] |b Springer |c [1988] | |
| 264 | 4 | |c © 1988 | |
| 300 | |a VII, 153 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Lecture notes in mathematics. |v 1305 | |
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| 650 | 4 | |a Extremal problems (Mathematics) | |
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Record in the Search Index
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| adam_text |
TABLE OF CONTENTS
1. Introduction 1
(i) A brief review 1
(ii) Aims and results of this monograph 7
(iii) Organization of the monograph 10
(iv) Acknowledgements 11
2. Notation and Index of Notation 12
3. Statement of Main Results 17
Theorem 3.1 : The case 1 p ¦» 17
Theorem 3.2 : The cases p = 1,° 18
Corollary 3.3 :The weights exp(-|x|a) 18
Theorem 3.4 :Orthonormal polynomials for
N A. . .
II |x-z | Je~ylx;(l + o(l)) 20
J = l J
Theorem 3.5 :Orthogonal polynomials for a generalized
weight 21
4. Weighted Polynomials and Zeros of Extremal Polynomials 24
Theorem 4.1 Existence of approximation by Pn(x)W(anx) 24
Theorem 4.2 :Approximation with rates 25
Theorem 4.3 :How fast can Pn(x)W(anx) grow? 26
Theorem 4.4 :Largest zeros of extremal polynomials 27
5. Integral Equations 28
Lemma 5.1 :Equation with logarithmic kernel 28
Lemma 5.2 :More on these equations 33
Lemma 5.3 :An application to W(x):= e~Q^x^ 37
6. Polynomial Approximation of Potentials 40
Lemma 6.1 :Discretization of a general potential 40
Lemma 6.2 : Some technical details for Q 43
Lemma 6.3 :Discretization of Un R 47
7. Infinite-Finite Range Inequalities and Their Sharpness 49
Theorem 7.1 :The L^ case 49
Theorem 7.2 :The L case 50
P
Lemma 7.3 :A crude Nikolskii inequality 53
8. The Largest Zeros of Extremal Polynomials 57
Theorem 8.1 :The jth largest zero 58
Corollary 8.2 :The jth largest equiosci1lation point 58
Lemma 8.3 ^Existence of a certain entire function 60
Lemma 8.4 :A Gaussian quadrature estimate 60
Lemma 8.5 :Some inequalities for a 61
Lemma 8.6 :Upper bounds for weighted polynomials 62
9. Further Properties of U o(x) 67
n, K
Theorem 9.1 :Positivity of U o(x) 67
n, K
Lemma 9.2 :Positivity in the unit ball 68
Lemma 9.3 :Positivity in an angle at +1 69
10. Nth Root Asymptotics for Extremal Polynomials 73
Theorem 10.1 :Nth root asymptotics, general R 73
Corollary 10.2 :The case R = an 74
Corollary 10.3 :A technical estimate 74
11. Approximation by Certain Weighted Polynomials, I 80
Theorem 11.1 :Existence of approximations 80
Theorem 11.2 :Rates for smooth functions 81
Lemma 11.4 :A contour integral estimate 83
Lemma 11.5 :A crude Markov-Bernstein inequality 87
12. Approximation by Certain Weighted Polynomials, II 91
Theorem 12.1 Existence of approximations 91
Theorem 12.2 :A precise interval of uniform approximation 93
Theorem 12.3 :Rates for smooth functions 93
Lemma 12.4 :A modulus of continuity estimate 94
Lemma 12.5 ^Weighted polynomials with not too small
logarithm 95
Lemma 12.6 :Another modulus of continuity estimate 104
13. Bernstein's Formula and Bernstein Extremal Polynomials 111
Theorem 13.1 :Bernstein's formula and polynomials 111
Proposition 13.2 :An asymptotic upper bound for E (W) 116
Proposition 13.3 :An asymptotic lower bound for E (W) 117
Lemma 13.4 :Variability of anG[W(ax)] 118
14. Proof of the Asymptottcs for En (W) 120
Theorem 14.1 :An asymptotic upper bound 120
Proof of (3.4) and (3.5) in Theorem 3.1 123
Proof of asymptotics for Er (W), p=l,» in Theorem 3.2 124
15. Proof of the Asymptotics for the L Extremal Polynomials 128
Theorem 15.1 :A comparison theorem, 2 £ p «° 128
Proof of (3.7) and (3.8) of Theorem 3.1 130
Lemma 15.2 :A comparison lemma, p = *• 132
Proof of (3.7) and (3.8) when p = ° 133
16. The Case p = 2 :OrthonormaI Polynomials 136
Lemma 16.1 :A lower bound for E 2(W) 136
Lemma 16.2 :Asymptotics of recurrence relation
coefficients 137
Lemma 16.3 :An infinite-finite range inequality 138
Proof of Theorem 3.5 140
Proof of Theorem 3.4 144
References 146
Subject tndex 151 |
| any_adam_object | 1 |
| author | Lubinsky, Doron S. 1955- Saff, Edward B. 1944- |
| author_GND | (DE-588)172232589 (DE-588)110736990 |
| author_facet | Lubinsky, Doron S. 1955- Saff, Edward B. 1944- |
| author_role | aut aut |
| author_sort | Lubinsky, Doron S. 1955- |
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| building | Verbundindex |
| bvnumber | BV000806443 |
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| classification_rvk | SI 850 |
| classification_tum | MAT 418f MAT 428f |
| ctrlnum | (OCoLC)310871276 (DE-599)BVBBV000806443 |
| dewey-full | 511/.4 510 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics 510 - Mathematics |
| dewey-raw | 511/.4 510 |
| dewey-search | 511/.4 510 |
| dewey-sort | 3511 14 |
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| discipline | Mathematik |
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| indexdate | 2024-09-12T12:00:34Z |
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| isbn | 0387189580 3540189580 |
| language | English |
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| physical | VII, 153 Seiten |
| publishDate | 1988 |
| publishDateSearch | 1988 |
| publishDateSort | 1988 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics. |
| series2 | Lecture notes in mathematics. |
| spelling | Lubinsky, Doron S. 1955- (DE-588)172232589 aut Strong asymptotics for extremal polynomials associated with weights on IR D. S. Lubinsky ; E. B. Staff Berlin, [u.a.] Springer [1988] © 1988 VII, 153 Seiten txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics. 1305 Développements asymptotiques Polynômes orthogonaux Problèmes extrémaux (Mathématiques) Asymptotic expansions Extremal problems (Mathematics) Orthogonal polynomials Gewicht Mathematik (DE-588)4157270-1 gnd rswk-swf Extremalpolynom (DE-588)4153425-6 gnd rswk-swf Orthogonale Polynome (DE-588)4172863-4 gnd rswk-swf Gewichtete Polynomapproximation (DE-588)4197098-6 gnd rswk-swf Asymptotik (DE-588)4126634-1 gnd rswk-swf Extremalpolynom (DE-588)4153425-6 s Gewicht Mathematik (DE-588)4157270-1 s Asymptotik (DE-588)4126634-1 s DE-604 Gewichtete Polynomapproximation (DE-588)4197098-6 s Orthogonale Polynome (DE-588)4172863-4 s Saff, Edward B. 1944- (DE-588)110736990 aut Erscheint auch als Online-Ausgabe 978-3-540-38857-9 Lecture notes in mathematics. 1305 (DE-604)BV000676446 1305 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000505456&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lubinsky, Doron S. 1955- Saff, Edward B. 1944- Strong asymptotics for extremal polynomials associated with weights on IR Lecture notes in mathematics. Développements asymptotiques Polynômes orthogonaux Problèmes extrémaux (Mathématiques) Asymptotic expansions Extremal problems (Mathematics) Orthogonal polynomials Gewicht Mathematik (DE-588)4157270-1 gnd Extremalpolynom (DE-588)4153425-6 gnd Orthogonale Polynome (DE-588)4172863-4 gnd Gewichtete Polynomapproximation (DE-588)4197098-6 gnd Asymptotik (DE-588)4126634-1 gnd |
| subject_GND | (DE-588)4157270-1 (DE-588)4153425-6 (DE-588)4172863-4 (DE-588)4197098-6 (DE-588)4126634-1 |
| title | Strong asymptotics for extremal polynomials associated with weights on IR |
| title_auth | Strong asymptotics for extremal polynomials associated with weights on IR |
| title_exact_search | Strong asymptotics for extremal polynomials associated with weights on IR |
| title_full | Strong asymptotics for extremal polynomials associated with weights on IR D. S. Lubinsky ; E. B. Staff |
| title_fullStr | Strong asymptotics for extremal polynomials associated with weights on IR D. S. Lubinsky ; E. B. Staff |
| title_full_unstemmed | Strong asymptotics for extremal polynomials associated with weights on IR D. S. Lubinsky ; E. B. Staff |
| title_short | Strong asymptotics for extremal polynomials associated with weights on IR |
| title_sort | strong asymptotics for extremal polynomials associated with weights on ir |
| topic | Développements asymptotiques Polynômes orthogonaux Problèmes extrémaux (Mathématiques) Asymptotic expansions Extremal problems (Mathematics) Orthogonal polynomials Gewicht Mathematik (DE-588)4157270-1 gnd Extremalpolynom (DE-588)4153425-6 gnd Orthogonale Polynome (DE-588)4172863-4 gnd Gewichtete Polynomapproximation (DE-588)4197098-6 gnd Asymptotik (DE-588)4126634-1 gnd |
| topic_facet | Développements asymptotiques Polynômes orthogonaux Problèmes extrémaux (Mathématiques) Asymptotic expansions Extremal problems (Mathematics) Orthogonal polynomials Gewicht Mathematik Extremalpolynom Orthogonale Polynome Gewichtete Polynomapproximation Asymptotik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000505456&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
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