Studies in spline functions and approximation theory:
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
New York [u.a.]
Acad. Press
1976
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 500 S. |
| ISBN: | 012398565X |
Staff View
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| 245 | 1 | 0 | |a Studies in spline functions and approximation theory |c Samuel Karlin ... |
| 264 | 1 | |a New York [u.a.] |b Acad. Press |c 1976 | |
| 300 | |a XII, 500 S. | ||
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| 650 | 4 | |a Approximation, Théorie de l' | |
| 650 | 4 | |a Splines, Théorie des | |
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Record in the Search Index
| _version_ | 1804116412446277632 |
|---|---|
| adam_text | CONTENTS
PREFACE XI
ABSTRACTS 1
PART I. BEST APPROXIMATIONS, OPTIMAL QUADRATURE
AND MONOSPLINES
On a Class of Best Non-linear Approximation Problems and
Extended Monosplines 19
Samuel Karlin
§1 Formulation and description of main results. 19
2 Bounds on the number of zeros of extended monosplines. 27
3 The fundamental theorem of algebra for extended monosplines. 36
4 The improvement theorem. 37
5 Existence of minimizing extended splines. 44
6 Proof of the characterization of the extremum, as described 47
in Theorem 1.3 for 1 p °°.
7 Characterization and uniqueness in the case of p = °° 51
in Theorem 1.3.
8 The total positivity nature of the Bergman and Szego reproducing 55
kernels.
9 An equivalent formulation of the best non-linear approximation 59
problem (1.4) in L2.
A Global Improvement Theorem for Polynomial Monosplines 67
Samuel Karlin
§1 Statement of theorem and ramifications. 67
2 Some preliminaries. 72
3 Proof of Theorem 3. 74
Applications of Representation Theorems to Problems of
Chebyshev Approximation with Constraints 83
Allan Pinkus
§1 Introduction. 83
2 Proof of Theorem 2 and extensions of Theorem 1. 87
v
CONTENTS
3 Applications of Theorem 2. 89
4 Representation theorems with interpolation, and
interpolatory approximation. 94
5 Representation theorems with boundary constraints. 98
6 Applications of representation theorems with boundary conditions. 103
Gaussian Quadrature Formulae with Multiple Nodes 113
Samuel Karlin and A llan Pin kus
§1 Formulation and statement of main results. 113
2 Proof of Theorem 1 and ramifications. 121
3 Extensions and remarks. 137
An Extremal Property of Multiple Gaussian Nodes 143
Samuel Karlin and Allan Pinkus
§1 Formulation and statement of results. 143
2 Preliminaries and some determinantal identities. 149
3 Proof of extremal property (Theorem D). 154
PART II. CARDINAL SPLINES AND RELATED MATTERS
Oscillation Matrices and Cardinal Spline Interpolation 163
Charles A. Micchelli
§1 Introduction. 163
2 Quasi-Hermite cardinal interpolation. 168
3 Cardinal X-splines. 184
4 An eigenvalue problem. 189
Cardinal jC-Splines 203
Charles A. Micchelli
§1 Introduction. 203
2 An eigenvalue problem. 205
3 Some special £ -splines. 220
4 Convergence of jC-splines. 224
5 Some applications of the Euler and Bernoulli jC-spline. 228
6 Polynomial spline interpolation on a geometric mesh. 241 *
On Micchelli s Theory of Cardinal jC-Splines 251
/./. Schoenberg
§1 Introduction. 251
2 The class S (jC,t?) of cardinal JC-splines. 253
3 The B-splines. 255
vi
V*
CONTENTS
4 The behavior of the roots of the equation (3.4). 257
5 A proof of Theorem 2. 261
6 Micchelli s cardinal interpolation problem by elements of S(£,n). 266
7 Proof of sufficiency in Theorem 4. 269
8 A few examples. 273
On The Remainders and the Convergence of Cardinal Spline
Interpolation for Almost Periodic Functions 277
/./. Schoenberg
§1 Introduction. 277
2 The kernel of the remainder and some of its properties. 278
3 Further properties of the kernel K2m j (x,t). 283
4 The cardinal spline interpolation formula with remainder. 286
5 A few special choices of f(x). 288
6 Applications. 291
7 Definitions and known results. 295
8 Katznelson s definition and lemma. 297
9 Two applications of Katznelson s lemma. 298
10 A conjecture. 302
PART III. INTERPOLATION WITH SPLINES
Interpolation by Splines with Mixed Boundary Conditions 305
Samuel Karlin and Allan Pinkus
§1 Introduction. 305
2 Proof of Theorem 1. 309
3 Examples of boundary conditions satisfying Postulate J. 314
4 Total positivity properties of Green s function for mixed 318
boundary conditions.
Divided Differences and Other Non-linear Existence Problems
at Extremal Points 327
Samuel Karlin and Allan Pinkus
§1 Preliminaries and statement of main results. 327
2 Polynomials with prescribed k* order divided differences:
Proof of Theorem 1. 330
3 Extensions of Theorem 1 to various classes of spline
functions and Chebyshev systems. 339
4 A general formulation and applications. 345
5 Open problems. 349
vii
CONTENTS
PART IV. GENERALIZED LANDAU AND MARKOV TYPE
INEQUALITIES AND GENERALIZED PERFECT
SPLINES.
Notes on Spline Function VI. Extremum Problems of the
Landau-Type for the Differential Operators D2 ± 1 353
/./. Schoenberg
§0 Introduction. 353
1 The differential operator D2 + 1. 354
2 The differential operator D2 - 1. 359
3 Weak extremum functions in Theorem 3. 364
Oscillatory Perfect Splines and Related Extremal Problems 371
Samuel Karlin j
§ 1 Introduction and statement of main results. 371
2 The construction of certain generalized perfect splines
with special oscillatory properties. 385
3 Some special oscillatory perfect splines. 394
4 Uniqueness criteria, proof of Theorem 4.1 and related matters. 396
5 A special one parameter family of equi-oscillation perfect splines. 403
6 Results on equi-oscillating perfect splines with variable interval
length. 412
7 Some symmetry considerations. 417 ,
8 Some extremal properties of the equi-oscillating perfect splines. 418
9 Perfect L-splines. 423 .
10 Equi-oscillating perfect L-splines on [0,°°) and on (-°°, °°)
with an infinite number of knots. 426
11 Perfect L-splines for n-th order differential operators with
constant coefficients on the half line and sharp Landau
Kolmogorov type inequalities. 432
12 Equi-oscillating perfect L-splines for the full line where L is
a differential operator with constant coefficients. 443
13 Landau Kolmogorov type inequalities for certain convolution 453
operators.
Generalized Markov Bernstein Type Inequalities for Spline
Functions 461
Samuel Karlin
§ 1 Introduction and statement of main results. 461
2 Markov inequalities for perfect splines: Proof of Theorem 1. 469
3 Markov inequalities for one-sided and two-sided Cardinal splines . 476
viii
CONTENTS
Some One-sided Numerical Differentiation Formulae and
Applications 485
Samuel Karlin
§0 Introduction. 485
1 The finite interval case. 486
2 The one-sided infinite interval case. 494
3 Remarks. 499
ix
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| id | DE-604.BV001968734 |
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| indexdate | 2024-07-09T15:38:06Z |
| institution | BVB |
| isbn | 012398565X |
| language | English |
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| physical | XII, 500 S. |
| publishDate | 1976 |
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| spelling | Studies in spline functions and approximation theory Samuel Karlin ... New York [u.a.] Acad. Press 1976 XII, 500 S. txt rdacontent n rdamedia nc rdacarrier Approximation, Théorie de l' Splines, Théorie des Approximation theory Spline theory Spline-Funktion (DE-588)4056332-7 gnd rswk-swf Approximationstheorie (DE-588)4120913-8 gnd rswk-swf Spline-Funktion (DE-588)4056332-7 s Approximationstheorie (DE-588)4120913-8 s DE-604 Karlin, Samuel 1924-2007 Sonstige (DE-588)118918672 oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001283980&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Studies in spline functions and approximation theory Approximation, Théorie de l' Splines, Théorie des Approximation theory Spline theory Spline-Funktion (DE-588)4056332-7 gnd Approximationstheorie (DE-588)4120913-8 gnd |
| subject_GND | (DE-588)4056332-7 (DE-588)4120913-8 |
| title | Studies in spline functions and approximation theory |
| title_auth | Studies in spline functions and approximation theory |
| title_exact_search | Studies in spline functions and approximation theory |
| title_full | Studies in spline functions and approximation theory Samuel Karlin ... |
| title_fullStr | Studies in spline functions and approximation theory Samuel Karlin ... |
| title_full_unstemmed | Studies in spline functions and approximation theory Samuel Karlin ... |
| title_short | Studies in spline functions and approximation theory |
| title_sort | studies in spline functions and approximation theory |
| topic | Approximation, Théorie de l' Splines, Théorie des Approximation theory Spline theory Spline-Funktion (DE-588)4056332-7 gnd Approximationstheorie (DE-588)4120913-8 gnd |
| topic_facet | Approximation, Théorie de l' Splines, Théorie des Approximation theory Spline theory Spline-Funktion Approximationstheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001283980&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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