Interpolation and approximation with splines and fractals:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford [u.a.]
Oxford Univ. Press
2010
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XV, 319 S. graph. Darst. |
| ISBN: | 9780195336542 |
Internformat
MARC
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| 245 | 1 | 0 | |a Interpolation and approximation with splines and fractals |c Peter Massopust |
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2010 | |
| 300 | |a XV, 319 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Approximation theory | |
| 650 | 4 | |a Interpolation | |
| 650 | 4 | |a Spline theory | |
| 650 | 4 | |a Fractals | |
| 650 | 0 | 7 | |a Spline-Interpolation |0 (DE-588)4182396-5 |2 gnd |9 rswk-swf |
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Datensatz im Suchindex
| _version_ | 1804142932589019136 |
|---|---|
| adam_text | Contents
Preface
xi
1
The General Interpolation and Approximation Problem
3
1.1
Notation and Terminology
4
1.2
Semi-Normed and Normed Spaces
5
1.3
The Abstract Interpolation Problem
12
1.3.1
Interpolation with Polynomials and Real Parameters
13
1.3.2
Divided Differences
16
1.3.3
Error Estimates for Polynomial Interpolation
19
1.3.4
Runge s Example
20
1.4
The Abstract Approximation Problem
21
1.4.1
Metric Spaces
21
1.4.2
Approximation Methods
23
1.4.3
Existence and Uniqueness of a Best Approximation
25
1.4.4
Pre-Hilbert Spaces
27
1.4.5
Approximation in Pre-Hilbert Spaces
29
1.4.6
Order of Approximation
31
1.4.7
The
Weierstrass
Approximation Theorem
33
1.4.8
Approximation by Step Functions
36
1.4.9
Moduli of Continuity and Smoothness
37
Exercises
39
viii Contents
2
Splines
44
2.1
Definitions
45
2.2
A Basis for Sk(Xn)
46
2.3
B-Splines
47
2.3.1
Properties of B-Splines
49
2.3.2
The Basis Property of B-Splines
53
2.3.3
Derivatives and Integrals of B-Splines
56
2.4
Cardinal B-Splines
57
2.5
The Fourier Transform of Cardinal B-Splines
60
2.6
Cardinal Spline Interpolation
64
2.7
Repeated Knots
67
2.7.1
Properties of the Spaces
Щ
68
2.7.2
Basis for
П|
68
2.8
Hermite Splines
74
2.9
Interpolation with Splines
75
2.9.1
Preliminaries
75
2.9.2
Error Estimates for Spline Interpolation
78
2.10
Approximation with Splines
90
2.11
1}
-Approximation with Splines
93
2.12
Exponential Splines
95
2.12.1
Polynomial Splines and Distributional
Derivatives
95
2.12.2
Linear Differential Operators and Exponential
Splines
99
2.12.3
Exponential B-Splines
105
2.13
¿-Splines
112
2.14
Wavelets and Splines
113
Exercises
119
3
Interpolation in W,
s
> 1
124
3.1
Multivariate Polynomial Interpolation
124
3.2
Spline Interpolation in
W
127
3.3
Tensor Products of B-Splines
130
3.3.1
The Construction of Tensor-Product B-Splines
131
3.3.2
Shortcomings of the Tensor Product Approach
133
3.4
Kergin Interpolation
134
Exercises
139
4
Fractals
141
4.1
Definitions
142
4.1.1
Topological Dimension in R
143
4.1.2
The Hausdorff and Box Dimension in R
144
4.2
Iterated Function Systems
149
4.3
Approximation and the Collage Theorem
154
Contents
ix
4.4
Dimensions
of Fractal Sets
156
4.4.1
The Theoretical Computation of Dimension
157
4.4.2
The Numerical Computation of Dimension
158
4.5
The Code Space of an IFS
160
4.6
Fractal Transformations
164
4.7
Fractal Measures
167
4.7.1
Measures
167
4.7.2
IFSs with Probabilities and Fractal Measures
171
Exercises
177
5
Fractal Functions
182
5.1
Some Examples of Nowhere Differentiable
Functions
183
5.2
Fractal Interpolation Functions
184
5.3
Polynomial Fractal Functions
190
5.4
Bases of Fractal Functions
194
5.5
The Box Dimension of
Affine
Fractal Functions
200
5.6
Code Space and Fractal Functions
207
5.7
Continuous Functions as Special Cases of Fractal
Functions
208
5.8
Fractal Functions of Class Ck
212
5.9
Construction of Fractal Functions of Class Ck from
Splines
214
5.10
Fractal B-Splines
220
5.11
Approximation with Fractal Functions
221
5.12
Indefinite Integrals of Fractal Functions
223
5.13
Fourier Transform of Fractal Functions
226
5.14
Wavelets and Fractal Functions
228
5.15
Fractal Functions and Function Spaces
232
Exercises
236
6
Fractal Surfaces
239
6.1
Tensor Product Fractal Surfaces
240
6.2
Affine
Fractal Surfaces in R2
242
6.3
The Box Dimension of
Affine
Fractal Surfaces
250
6.4
Holder Continuity of
Affine
Fractal Functions
254
6.5
Bilinear Fractal Surfaces in M.2
255
6.6
Fractal Surfaces Arising from Quadratic Forms
260
6.7
Smooth Fractal Surfaces via Indefinite Integrals
262
Exercises
265
7
Superfractals
266
7.1 1
-Variable Fractal Sets
267
7.2 1
-Variable Fractal Measures
272
Contents
7.3 V-Variable
Fractal Sets and Fractal Measures
273
7.3.1
У
-Variable Fractal Sets
274
7.3.2
V-Variable Fractal Measures
276
7.4
Graph Theory and Random Fractals
278
7.4.1
Code Trees
278
7.4.2
Random Fractal Sets and Random Fractal Measures
281
Exercises
287
8
Superfractal Functions
289
8.1
Preliminaries
290
8.2
V-Variable Fractal Interpolation
292
Exercises
300
Bibliography
302
Nomenclature
310
Index
315
his unique textbook emphasizes the commonalities between splines and frac¬
tals in interpolation and approximation theory, with particular emphasis on
fractal functions and fractal surfaces. In addition to providing a somewhat clas¬
sical introduction to the main issues involving approximation and interpolation,
the book also describes fractals, fractal functions, and their properties. The theory
of splines is well-established but the relationship to fractal functions is novel and
has not before been discussed in one volume. Throughout the book, connections
and applications between these two apparently different areas are exposed and
presented. In this way, more options are given to the prospective reader who will
encounter complex approximation and interpolation problems in his or her chosen
career. The material in the chapters on superfractals and superfractal functions is
on the forefront of research. Numerous examples, figures, and exercises accom¬
pany the material.
The book is intended for advanced undergraduate and graduate students,
researchers, and professionals in applied and numerical mathematics, physics,
electrical engineering, biomathematics, and other computational sciences who are
involved with the mathematical modeling of complex natural phenomena, includ¬
ing bioinformatics.
ABOUT THE AUTHOR
Peter
Massopust
is a senior research scientist on the Marie Curie Excellence in
Research Team MAMEBIA (Mathematical Methods in Biological Image Analysis)
at the Helmholtz
Zentrum
München-Germán
Research Center for Environmental
Health, and guest professor at the Centre of Mathematics, Research Unit M6,
Tech¬
nische Universität München.
|
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| discipline | Mathematik |
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| id | DE-604.BV036447308 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:39:37Z |
| institution | BVB |
| isbn | 9780195336542 |
| language | English |
| lccn | 2009007637 |
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| spelling | Massopust, Peter Robert 1958- Verfasser (DE-588)140877789 aut Interpolation and approximation with splines and fractals Peter Massopust Oxford [u.a.] Oxford Univ. Press 2010 XV, 319 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Approximation theory Interpolation Spline theory Fractals Spline-Interpolation (DE-588)4182396-5 gnd rswk-swf Spline-Approximation (DE-588)4182394-1 gnd rswk-swf Spline-Approximation (DE-588)4182394-1 s Spline-Interpolation (DE-588)4182396-5 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020319527&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020319527&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Massopust, Peter Robert 1958- Interpolation and approximation with splines and fractals Approximation theory Interpolation Spline theory Fractals Spline-Interpolation (DE-588)4182396-5 gnd Spline-Approximation (DE-588)4182394-1 gnd |
| subject_GND | (DE-588)4182396-5 (DE-588)4182394-1 |
| title | Interpolation and approximation with splines and fractals |
| title_auth | Interpolation and approximation with splines and fractals |
| title_exact_search | Interpolation and approximation with splines and fractals |
| title_full | Interpolation and approximation with splines and fractals Peter Massopust |
| title_fullStr | Interpolation and approximation with splines and fractals Peter Massopust |
| title_full_unstemmed | Interpolation and approximation with splines and fractals Peter Massopust |
| title_short | Interpolation and approximation with splines and fractals |
| title_sort | interpolation and approximation with splines and fractals |
| topic | Approximation theory Interpolation Spline theory Fractals Spline-Interpolation (DE-588)4182396-5 gnd Spline-Approximation (DE-588)4182394-1 gnd |
| topic_facet | Approximation theory Interpolation Spline theory Fractals Spline-Interpolation Spline-Approximation |
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