Chebyshev splines and Kolmogorov inequalities:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel ; Boston ; Berlin
Birkhäuser
1998
|
| Series: | Operator theory
105 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 205 S. graph. Darst. |
| ISBN: | 3764359846 0817659846 |
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MARC
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| 100 | 1 | |a Bagdasarov, Sergej K. |d 1969- |e Verfasser |0 (DE-588)120626551 |4 aut | |
| 245 | 1 | 0 | |a Chebyshev splines and Kolmogorov inequalities |c Sergey Bagdasarov |
| 264 | 1 | |a Basel ; Boston ; Berlin |b Birkhäuser |c 1998 | |
| 300 | |a XIII, 205 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
| _version_ | 1804126714874298368 |
|---|---|
| adam_text | Table of Contents
Preface xi
Chapter 0
Introduction
0.1 History of the Kolmogorov Landau problem 1
0.1.1 General setting 1
0.1.2 Cases of the complete solution of the
Kolmogorov problem 1
0.2 Kolmogorov Landau problem in the Sobolev class W^1 (I) 3
0.2.1 Inequalities for derivatives of polynomials 3
0.2.2 Sharp inequalities in the Sobolev class
Wr(I), I = R V R+ V [0,1] 4
0.3 Functional classes WrHu and W^H 6
0.3.1 Definitions 6
0.4 Sharp Kolmogorov Landau inequalities
in WrH {I), I = R+VEV[0,l] 9
Chapter 1
Auxiliary Results
1.1 General facts 13
1.2 General properties of functional classes 16
Chapter 2
Maximization of Functional in H [a, b] and Perfect w Splines
2.1 Introduction to the theory of functional classes H [a, b] 19
2.1.1 Simple kernels ^f and their rearrangements 5R(*; •) 19
2.1.2 Korneichuk lemma 21
2.2 Maximization of integral functionals in
Hu[a, b], oo a b +oo 22
2.2.1 Notations and definitions 23
2.2.2 V^ partitions of the interval [a, b] 24
2.2.3 Theorem X and perfect w splines 25
2.2.4 Structural properties of extremal functions x^,v 27
2.2.5 Limiting properties of extremal functions x^.v 34
2.2.6 Criterion of triviality of the extremal Vn° partition 35
2.2.7 Properties of extremal rearrangements 5RW (ty; •) 36
vi Table of Contents
Chapter 3
Fredholm Kernels
3.1 Kernels of type I 43
3.2 Kernels of type II 47
3.3 Kernels of type III 48
Chapter 4
Review of Classical Chebyshev Polynomial Splines
4.1 Construction of Chebyshev perfect splines 49
4.2 Zero count argument 51
4.3 Application of the Fredholm kernels 53
4.4 Properties of absolutely continuous functions 58
4.5 Sharp Kolmogorov inequalities in W^,(K) and Cavaretta s proof 62
4.5.1 Property of periodic functions 62
4.5.2 Reduction of the Kolmogorov problem
to the periodic case 63
Chapter 5
Additive Kolmogorov Landau Inequalities
5.1 Numerical differentiation formulae 65
5.2 Sufficient conditions of extremality 69
Chapter 6
Proof of the Main Result
6.0 Formulation of the main result 71
6.1 Construction of the Borsuk mapping *r : S2n~r+1 R2 ^1 71
6.2 Continuity of the Borsuk mapping n 75
6.3 Properties of solutions of the equation x{s) = 0 77
6.4 Limiting procedure as e — 0 83
Chapter 7
Properties of Chebyshev w Splines
7.1 Review of the structure of Chebyshev w splines on [0,1] 89
7.2 Rescaled Chebyshev wQ splines of the fixed norm B 90
7.3 Chebyshev u;Q splines of the fixed norm B 92
7.4 Properties of Chebyshev u; splines of the fixed norm 93
7.5 General properties of extremal functions Rn 95
7.6 Restricted action of the generating kernel Tn(t) 98
Table of Contents vii
Chapter 8
Chebyshev w Splines on the Half line R+
8.1 Limiting procedure 101
8.2 Structure of Chebyshev w splines 108
Chapter 9
Maximization of Integral Functionals
in if [01,02], —00 ai a i +00
9.0 Formulation of the extremal problem Ill
9.1 Definitions Ill
9.2 Structure of perfect w splines 112
9.3 Kernels ^ and their rearrangements ^( P; •) 114
Chapter 10
Sharp Kolmogorov Inequalities in WrHa(R)
10.1 Chebyshev w splines of the problem (P.0) 123
10.1.1 Numerical differentiation formulae 123
10.1.2 Chebyshev w splines on the symmetric interval 128
10.2 Kolmogorov inequalities in WrHa(R) 128
Chapter 11
Landau and Hadamard Inequalities in WrHu (R+) and WrH^(R)
11.1 Landau inequalities in W1HUJ(R+) 131
11.2 Hadamard inequalities in W1HUJ{R) 134
11.3 Specific feature of Holder classes W1Ha{R) and W1Ha{R+) 135
11.4 Extrapolation problem in W1Huj( lj, t] 136
Chapter 12
Sharp Kolmogorov Landau inequalities
in W2H (R) AND W2H (R+)
12.1 Kolmogorov Stechkin inequalities in W2HW(R) 139
12.1.1 Estimates of the first derivative 139
12.1.2 Estimates of the second derivative 141
12.2 Sharp estimates of derivatives in W2HU (R+), W2Hu[0,1] 144
12.2.1 Extensions from V^if^O, A] to W2HUJ(R+) 144
12.2.2 Stechkin Matorin inequalities in W^if IO, 1], W2HUJ(R+) .... 144
12.2.3 Inequalities for ||/ ||l^(k+) in W2H {R+) 148
viii Table of Contents
Chapter 13
Chebyshev w Splines in the Problem
of TV Width of the Functional Class WrHu [0,1]
13.1 TV widths of Sobolev classes W^l[Q,1] 151
13.1.1 Estimates of dN (W^+^O, 1],C[0,1]) from below 151
13.1.2 Estimates of dN (W^+1[0,1], C[0,1]) from above 153
13.2 Chebyshev w splines in the problem of n widths of WrHu[0,1] 155
Chapter 14
Function in WrHu [ l, 1] Deviating Most
from Polynomials of Degree r
14.1 Preliminary observations 160
14.2 Generating kernels 160
14.3 Preliminary remarks 161
14.3.1 Chebyshev w polynomials 161
14.4 Concluding remarks 163
14.4.1 Norm of the Chebyshev function in WTH [ 1,1] 163
14.4.2 Solution of one extremal problem 163
Chapter 15
TV Widths of the Class WrHu[ l, 1]
15.1 Formulation of the main results 165
15.2 Estimate of TV widths from below 168
15.3 Estimate of TV widths from above. Optimal subspaces 174
Chapter 16
Lower Bounds for the TV Widths of the Class WrHw[n]
16.1 Definition of the class WrHu[n 179
16.2 Linear spaces 1Zn+2 and Mn+2 and their properties 180
16.3 Lower bounds for dn+1 (WrHu[n], C[ l, 1]) 185
Appendix A
Kolmogorov Problem for Functions / e WrHUJ(R+) : ||/||Lp(K+) +oc
A.I Differentiation formulae for /(m (0), 0 m r 187
A.2 Differentiation formula for f^ (0) 188
A.3 Sufficient conditions of extremality in the problem (K — L) 189
A.3.1 Corollaries of differentiation formulas 189
A.3.2 Extremality conditions in the form of an operator equation ... 189
Table of Contents ix
A.4 Sharp inequalities in problems (K) and (K — L) 190
A.4.1 Kolmogorov Landau inequalities in WHU (R+) 190
A.4.2 Solution of the problem (K) 191
A.4.3 Problem (K) in the Holder classes 191
Appendix B
Kolmogorov Problems in W1HU (R+) and WlH (R)
B.I Preliminary remarks 193
B.2 Maximization of the norm ||/||loc(]r+) 193
B.2.1 Differentiation formulae and inequalities 194
B.2.2 Rearrangements ^(Z; t) and shifts Xj( ) :=X( +£j) 194
B.2.3 Special properties of extremal functions
in Holder classes W1Ha(R+) 195
B.3 Maximization of the norm ||/ ||loc(R+) 196
B.4 Maximization of the norm H/Hl^r) 197
B.5 Maximization of the norm H/ HloJIR) 199
Bibliography 201
Index 207
|
| any_adam_object | 1 |
| author | Bagdasarov, Sergej K. 1969- |
| author_GND | (DE-588)120626551 |
| author_facet | Bagdasarov, Sergej K. 1969- |
| author_role | aut |
| author_sort | Bagdasarov, Sergej K. 1969- |
| author_variant | s k b sk skb |
| building | Verbundindex |
| bvnumber | BV012110252 |
| classification_rvk | SK 620 |
| ctrlnum | (OCoLC)438858288 (DE-599)BVBBV012110252 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV012110252 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:21:51Z |
| institution | BVB |
| isbn | 3764359846 0817659846 |
| language | English |
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| physical | XIII, 205 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
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| publisher | Birkhäuser |
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| series | Operator theory |
| series2 | Operator theory |
| spelling | Bagdasarov, Sergej K. 1969- Verfasser (DE-588)120626551 aut Chebyshev splines and Kolmogorov inequalities Sergey Bagdasarov Basel ; Boston ; Berlin Birkhäuser 1998 XIII, 205 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Operator theory 105 Approximation (DE-588)4002498-2 gnd rswk-swf Kolmogorov-Ungleichung (DE-588)4528419-2 gnd rswk-swf Extremwert (DE-588)4137272-4 gnd rswk-swf Čebyšev-Spline (DE-588)4528418-0 gnd rswk-swf Extremwert (DE-588)4137272-4 s Approximation (DE-588)4002498-2 s Kolmogorov-Ungleichung (DE-588)4528419-2 s Čebyšev-Spline (DE-588)4528418-0 s DE-604 Operator theory 105 (DE-604)BV000000970 105 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008200019&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bagdasarov, Sergej K. 1969- Chebyshev splines and Kolmogorov inequalities Operator theory Approximation (DE-588)4002498-2 gnd Kolmogorov-Ungleichung (DE-588)4528419-2 gnd Extremwert (DE-588)4137272-4 gnd Čebyšev-Spline (DE-588)4528418-0 gnd |
| subject_GND | (DE-588)4002498-2 (DE-588)4528419-2 (DE-588)4137272-4 (DE-588)4528418-0 |
| title | Chebyshev splines and Kolmogorov inequalities |
| title_auth | Chebyshev splines and Kolmogorov inequalities |
| title_exact_search | Chebyshev splines and Kolmogorov inequalities |
| title_full | Chebyshev splines and Kolmogorov inequalities Sergey Bagdasarov |
| title_fullStr | Chebyshev splines and Kolmogorov inequalities Sergey Bagdasarov |
| title_full_unstemmed | Chebyshev splines and Kolmogorov inequalities Sergey Bagdasarov |
| title_short | Chebyshev splines and Kolmogorov inequalities |
| title_sort | chebyshev splines and kolmogorov inequalities |
| topic | Approximation (DE-588)4002498-2 gnd Kolmogorov-Ungleichung (DE-588)4528419-2 gnd Extremwert (DE-588)4137272-4 gnd Čebyšev-Spline (DE-588)4528418-0 gnd |
| topic_facet | Approximation Kolmogorov-Ungleichung Extremwert Čebyšev-Spline |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008200019&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000970 |
| work_keys_str_mv | AT bagdasarovsergejk chebyshevsplinesandkolmogorovinequalities |