Theory of approximation:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Russian |
| Published: |
New York
Dover
2003
|
| Edition: | Republ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 307 S. |
| ISBN: | 9780486495439 |
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MARC
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| 245 | 1 | 0 | |a Theory of approximation |c N. I. Achieser |
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Record in the Search Index
| _version_ | 1804149231094595584 |
|---|---|
| adam_text | CONTENTS
Chapter I
APPROXIMATION PROBLEMS
IN LINEAR NORMALIZED SPACES
Page
1.
Formulation of the Principal Problem in the Theory of Approximation
1
2.
The Concept of Metric Space
.................. 1
3.
The Concept of Linear Normalized Space
........... 2
4.
Examples of Linear Normalized Spaces
............. 3
5.
The Inequalities of Holder and Minkowski
........... 4
6.
Additional Examples of Linear Normalized Spaces
........ 7
7.
Hubert Space
........................ 8
8.
The Fundamental Theorem of Approximation Theory in Linear
Normalized Spaces
..................... 10
9.
Strictly Normalized Spaces
................... 11
10.
An Example of Approximation in the Space Lv
.......... 12
11.
Geometric Interpretation
................... 13
12.
Separable and Complete Spaces
................. 14
13.
Approximation Theorems in Hubert Space
............ 15
14.
An Example of Approximation in Hubert Space
......... 19
15.
More About the Approximation Problem in Hilbert Space
..... 21
16.
Orthonormalized Vector Systems in Hilbert Space
........ 22
17.
Orthogonalization of Vector Systems
.............. 23
18.
Infinite Orthonormalized Systems
................ 25
19.
An Example of a Non-Separable System
............. 29
20.
Weierstrass
First Theorem
................... 29
21.
Weierstrass
Second Theorem
.................. 32
22.
The Separability of the Space
С
................ 33
23.
The Separability of the Space Lv
................ 34
24.
Generalization of
Weierstrass
Theorem to the Space Lv
...... 37
25.
The Completeness of the Space Lp
............... 38
26.
Examples
oí
Complete Orthonormalized Systems in L2
,..... 40
27.
Müntz s
Theorem
....................... 43
28.
The Concept of the Linear Functional
.............. 46
29.
F. Riesz s Theorem
...................... 47
30.
A Criterion for the Closure of a Set of Vectors in Linear Normalized
Spaces
........................... 49
Vin
Contents
Chapter II
P.
L. TCHEBYSHEFFS
DOMAIN OF IDEAS
Page
31.
Statement of the Problem
................... 51
32.
A Generalization of the Theorem of
de la Vallée-Poussin
..... 52
33.
The Existence Theorem
.................... 53
34.
Tchebysheff s Theorem
.................... 55
35.
A Special Case of Tchebysheff s Theorem
............ 57
36.
The Tchebysheff Polynomials of Least Deviation from Zero
.... 57
37.
A Further Example of P. Tchebysheff s Theorem
......... 58
38.
An Example for the Application of the General Theorem of
de la
Vallée-Poussin
....................... 60
39.
An Example for the Application of P. L. Tchebysheff
s
General
Theorem
.......................... 62
40.
The Passage to Periodic Functions
............... 64
41.
An Example of Approximating with the Aid of Periodic Functions
. . 66
42.
The
Weierstrass
Function
................... 66
43.
Haar s Problem
........................ 67
44.
Proof of the Necessity of Haar s Condition
............ 68
45.
Proof of the Sufficiency of Haar s Condition
........... 69
46.
An Example Related to Haar s Problem
............. 72
47.
P. L. Tchebysheff s Systems of Functions
............ 73
48.
Generalization of P. L. Tchebysheff s Theorem
. . . ....... 74
49.
On a Question Pertaining to the Approximation of a Continuous
Function in the Space
L
................... 76
50.
A. A.
Markoffs
Theorem
................... 82
51.
Special Cases of the Theorem of A. A. Markoff
.......... 85
Chapter III
ELEMENTS OF HARMONIC ANALYSIS
52.
The Simplest Properties of Fourier Series
............ 89
53.
Fourier Series for Functions of Bounded Variation
........ 93
54.
The
Parseval
Equation for Fourier Series
............ 97
55.
Examples of Fourier Series
................... 98
56.
Trigonometric Integrals
.................... 101
57.
The Riemann-Lebesgue Theorem
................ 103
58.
Plancherel s Theory
...................... 104
59.
Watson s Theorem
...................... 106
60.
Plancherel s Theorem
..................... 108
61.
Fejér s
Theorem
....................... 110
62.
Integral-Operators of the
Fejér
Type
.............. 113
63.
The Theorem of Young and Hardy
............... 116
64.
Examples of Kernels of the
Fejér
Type
............. 118
65.
The Fourier Transformation of
Integrable
Functions
....... 120
Contents
IX
Page
66. The Faltung
of two Functions
................. 122
67^
V. A. Stekloff
s
Functions
................... 123
68.
Multimonotonic Functions
................... 125
69.
Conjugate Functions
..................... 126
Chapter IV
CERTAIN EXTREMAL PROPEUTIES OF INTEGRAL TRANS¬
CENDENTAL FUNCTIONS OF THE EXPONENTIAL TYPE
70.
Integral Functions of the Exponential Type
........... 130
71.
The
Borei
Transformation
................... 132
72.
The Theorem of Wiener and Paley
............... 134
73.
Integral Functions of the Exponential Type which are Bounded
along the Real Axis
..................... 137
74.
S.
N.
Bernstein s Inequality
.................. 140
75.
B. M. Levitan s Polynomials
.................. 146
76.
The Theorem of
Fejér
and Riesz. A Generalization of This Theorem
152
77.
A Criterion for the Representation of Continuous Functions as
Fourier-Stieltjes Integrals
.................. 154
Chapter V
QUESTIONS REGARDING THE BEST HARMONIC
APPROXIMATION OF FUNCTIONS
78.
Preliminary Remarks
..................... 160
79.
The Modulus of Continuity
................... 161
80.
The Generalization to the Space Lp (p ^
1)........... 162
81.
An Example of Harmonic Approximation
............ 165
82.
Some Estimates for Fourier Coefficients
............. 169
83.
More about V. A. Stekloff s Functions
.............. 173
84.
Two Lemmas
......................... 175
85.
The Direct Problem of Harmonic Approximation
......... 176
86.
A Criterion due to B. Sz.-Nagy
.................. 183
87.
The Best Approximation of Differentiable Functions
....... 187
88.
Direct Observations Concerning Periodic Functions
........ 195
89.
Jackson s Second Theorem
................... 199
90.
The Generalized
Fejér
Method
................. 201
9J. Berstein s Theorem
...................... 206
92.
Priwaloff s Theorem
...................... 210
93.
Generalizations of Bernstein s Theorems to the Space Lp (p ^
1) . . 211
94.
The Best Harmonic Approximation
oí
Analytic Functions
.... 214
95.
A Different Formulation of the Result of the Preceding Section.
. . 218
96.
The Converse of Bernstein s Theorem
.............. 221
Contenta
Chapter VI
WIENER S THEOREM ON APPROXIMATION
Page
97.
Wiener s Problem
....................... 224
98.
The Necessity of Wiener s Condition
.............. 224
99.
Some Definitions and Notation
................. 225
100.
Several Lemmas
....................... 227
101.
The Wiener-Levy Theorem
................... 230
102.
Proof of the Sufficiency of Wiener s Condition
.......... 233
103.
Wiener s General
Tauber
Theorem
............... 234
104.
Weakly Decreasing Functions
................. 235
105.
Remarks on the Terminology
.................. 237
106.
Ikehara s Theorem
...................... 238
107.
Carleman s
Tauber
Theorem
.................. 241
VARIOUS ADDENDA AND PROBLEMS
A. Elementary Extremal Problems and Certain Closure Criteria
.... 243
B.
Szegö s
Theorem and Some of Its Applications
.......... 256
C. Further Examples of Closed Sequences of Functions
....... 267
D. The
Carathéodory-Fejér
Problem and Similar Problems
...... 270
E. Solotareff s Problems and Related Problems
........... 280
F. The Best Harmonic Approximation of the Simplest Analytic
Functions
......................... 289
Notes
.............................. 296
Index
.............................. 306
|
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| callnumber-subject | QA - Mathematics |
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| classification_tum | MAT 410f |
| ctrlnum | (OCoLC)796271631 (DE-599)BVBBV040239910 |
| dewey-full | 511/.4 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511/.4 |
| dewey-search | 511/.4 |
| dewey-sort | 3511 14 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Republ. |
| format | Book |
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| isbn | 9780486495439 |
| language | English Russian |
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| spelling | Achiezer, Naum I. 1901-1980 Verfasser (DE-588)122043014 aut Lekcii po teorii approksimacii Theory of approximation N. I. Achieser Republ. New York Dover 2003 X, 307 S. txt rdacontent n rdamedia nc rdacarrier Approximationstheorie (DE-588)4120913-8 gnd rswk-swf Approximation (DE-588)4002498-2 gnd rswk-swf 1\p (DE-588)4143413-4 Aufsatzsammlung gnd-content Approximationstheorie (DE-588)4120913-8 s DE-604 Approximation (DE-588)4002498-2 s 2\p DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025096126&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Achiezer, Naum I. 1901-1980 Theory of approximation Approximationstheorie (DE-588)4120913-8 gnd Approximation (DE-588)4002498-2 gnd |
| subject_GND | (DE-588)4120913-8 (DE-588)4002498-2 (DE-588)4143413-4 |
| title | Theory of approximation |
| title_alt | Lekcii po teorii approksimacii |
| title_auth | Theory of approximation |
| title_exact_search | Theory of approximation |
| title_full | Theory of approximation N. I. Achieser |
| title_fullStr | Theory of approximation N. I. Achieser |
| title_full_unstemmed | Theory of approximation N. I. Achieser |
| title_short | Theory of approximation |
| title_sort | theory of approximation |
| topic | Approximationstheorie (DE-588)4120913-8 gnd Approximation (DE-588)4002498-2 gnd |
| topic_facet | Approximationstheorie Approximation Aufsatzsammlung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025096126&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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