Analytic theory of polynomials:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford [u.a.]
Clarendon Press
2005
|
| Ausgabe: | 1. publ., reprinted |
| Schriftenreihe: | London Mathematical Society monographs
New series ; 26 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIV, 742 S. |
| ISBN: | 0198534930 |
Internformat
MARC
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| 100 | 1 | |a Rahman, Qazi I. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Analytic theory of polynomials |c Q. I. Rahman and G. Schmeisser |
| 250 | |a 1. publ., reprinted | ||
| 264 | 1 | |a Oxford [u.a.] |b Clarendon Press |c 2005 | |
| 300 | |a XIV, 742 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a London Mathematical Society monographs : New series |v 26 | |
| 650 | 4 | |a Analytic functions | |
| 650 | 4 | |a Polynomials | |
| 650 | 0 | 7 | |a Funktionentheorie |0 (DE-588)4018935-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Polynom |0 (DE-588)4046711-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Polynom |0 (DE-588)4046711-9 |D s |
| 689 | 0 | 1 | |a Funktionentheorie |0 (DE-588)4018935-1 |D s |
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| 700 | 1 | |a Schmeißer, Gerhard |e Sonstige |4 oth | |
| 830 | 0 | |a London Mathematical Society monographs |v New series ; 26 |w (DE-604)BV045355493 |9 26 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-019000427 | ||
Datensatz im Suchindex
| _version_ | 1804141210016677888 |
|---|---|
| adam_text | Contents
1
Introduction
1
1.1 The fundamental
theorem of
algebra
2
1.2
Symmetric polynomials
6
1.3
The continuity theorem
9
1.4
Orthogonal polynomials: general properties
13
1.5
The classical orthogonal polynomials
19
1.6
Harmonic and subharmonic functions
30
1.7
Tools from matrix analysis
50
1.8
Notes
61
I Critical Points in Terms of Zeros
Fundamental results on critical points
71
2.1
Convex hulls and the Gauss-Lucas theorem
71
2.2
Extensions of the Gauss-Lucas theorem
75
2.3
Average distances from a line or a point
78
2.4
Real polynomials and Jensen s theorem
85
2.5
Extensions of Jensen s theorem
88
2.6
Notes
91
More sophisticated methods
96
3.1
Circular domains and polar derivative
96
3.2
Laguerre s theorem, its variants, and applications
98
3.3
Apolarity
102
3.4
Grace s theorem and equivalent forms
107
3.5
Notes
114
More specific results on critical points
117
4.1
Products and quotients of polynomials
117
4.2
Derivatives of reciprocals of polynomials
121
4.3
Complex analogues of Rolle s theorem
125
4.4
Bounds for some of the critical points
129
4.5
Converse results
132
4.6
Notes
137
Applications to compositions of polynomials
141
5.1
Linear combination of rational functions
142
5.2
Complex analogues of the intermediate-value theorem
143
5.3
Linear combination of derivatives: Walsh s approach
148
5.4
Linear
combination of derivatives: recursive approach
151
5.5
Multiplicative composition:
Schur-Szegő
approach
158
5.6
Multiplicative composition: Laguerre s approach
164
5.7
Multipliers preserving the reality of zeros
172
5.8
Notes
177
Polynomials with real zeros
184
6.1
The span of a polynomial
184
6.2
Largest zero and largest critical point
189
6.3
Interlacing and the Hermite-Biehler theorem
196
6.4
Consecutive zeros and critical points
201
6.5
Refinement of Rolle s theorem
203
6.6
Notes
209
Conjectures and solutions
212
7.1
A conjecture of Popoviciu
212
7.2
A conjecture of
Smale
214
7.3
The conjecture of Sendov
224
7.4
Notes
237
II Zeros in Terms of Coefficients
8
Inclusion of all zeros
243
8.1
The Cauchy bound and its estimates
243
8.2
Various refinements
249
8.3
Multipliers and the
Eneström-Kakeya
theorem
252
8.4
More general expansions
255
8.5
Orthogonal expansions with real coefficients
259
8.6
Alternative approach by matrix methods
263
8.7
Notes
270
9
Inclusion of some of the zeros
275
9.1
Inclusions in terms of a norm
275
9.2
Pellet s theorem and its consequences
284
9.3
Bounds in terms of some of the coefficients
290
9.4
Orthogonal expansions with real coefficients
294
9.5
The Landau-Montel problem
304
9.6
Notes
309
10
Number of zeros in an interval
315
10.1
The Budan-Fourier theorem and Descartes rule
315
10.2
Exact count under a side condition
320
10.3
Extensions to pairs of conjugate zeros
323
10.4
More general expansions
330
10.5
Exact count by Sturm sequences
335
10.6
Exact count via quadratic forms
339
10.7
Notes
350
11
Number of zeros in a domain
357
11.1
General principles
357
11.2
Number of zeros in a sector
359
11.3
Number of zeros in a half-plane
362
11.4
The Routh-Hurwitz problem
366
11.5
Number of zeros in a disc
374
11.6
Distribution of zeros
384
11.7
Notes
392
III Extremal Properties
12
Growth estimates
403
12.1
The Bernstein-Walsh lemma
403
12.2
The convolution method
408
12.3
The method of functionals
416
12.4
Various refinements
428
12.5
Local behaviour
447
12.6
Extensions to functions of exponential type
454
12.7
Notes
456
13
Mean values
460
13.1
Mean values on circles
460
13.2
A class of linear operators
468
13.3
Mean values on the unit interval
486
13.4
Notes
504
14
Derivative estimates on the unit disc
508
14.1
Bernstein s inequality and generalizations
508
14.2
Refinements
515
14.3
Conditions on the coefficients
526
14.4
Conditions on the zeros
532
14.5
Some special operators
538
14.6
Inequalities involving mean values
552
14.7
Notes
557
15
Derivative estimates on the unit interval
566
15.1
Inequalities of S. Bernstein and A. Markov
566
15.2
Extensions to higher-order derivatives
568
15.3
Two other extensions
577
15.4
Dependence of the bounds on the zeros
585
15.5
Some special classes
601
15.6
Lp analogues of Markov s inequality
611
15.7
Notes
622
16
Coefficient estimates
636
16.1
Polynomials on the unit circle
636
16.2
Coefficients of real trigonometric polynomials
653
16.3
Polynomials on the unit interval
672
16.4
Notes
677
References
681
List of notation
729
Index
733
|
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| author | Rahman, Qazi I. |
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| dewey-search | 512.9/42 |
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| discipline | Mathematik |
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| id | DE-604.BV036110228 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T22:12:15Z |
| institution | BVB |
| isbn | 0198534930 |
| language | English |
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| physical | XIV, 742 S. |
| publishDate | 2005 |
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| publisher | Clarendon Press |
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| series | London Mathematical Society monographs |
| series2 | London Mathematical Society monographs : New series |
| spelling | Rahman, Qazi I. Verfasser aut Analytic theory of polynomials Q. I. Rahman and G. Schmeisser 1. publ., reprinted Oxford [u.a.] Clarendon Press 2005 XIV, 742 S. txt rdacontent n rdamedia nc rdacarrier London Mathematical Society monographs : New series 26 Analytic functions Polynomials Funktionentheorie (DE-588)4018935-1 gnd rswk-swf Polynom (DE-588)4046711-9 gnd rswk-swf Polynom (DE-588)4046711-9 s Funktionentheorie (DE-588)4018935-1 s DE-604 Schmeißer, Gerhard Sonstige oth London Mathematical Society monographs New series ; 26 (DE-604)BV045355493 26 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019000427&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rahman, Qazi I. Analytic theory of polynomials London Mathematical Society monographs Analytic functions Polynomials Funktionentheorie (DE-588)4018935-1 gnd Polynom (DE-588)4046711-9 gnd |
| subject_GND | (DE-588)4018935-1 (DE-588)4046711-9 |
| title | Analytic theory of polynomials |
| title_auth | Analytic theory of polynomials |
| title_exact_search | Analytic theory of polynomials |
| title_full | Analytic theory of polynomials Q. I. Rahman and G. Schmeisser |
| title_fullStr | Analytic theory of polynomials Q. I. Rahman and G. Schmeisser |
| title_full_unstemmed | Analytic theory of polynomials Q. I. Rahman and G. Schmeisser |
| title_short | Analytic theory of polynomials |
| title_sort | analytic theory of polynomials |
| topic | Analytic functions Polynomials Funktionentheorie (DE-588)4018935-1 gnd Polynom (DE-588)4046711-9 gnd |
| topic_facet | Analytic functions Polynomials Funktionentheorie Polynom |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=019000427&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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