Rational approximation of real functions:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
1987
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| Edition: | 1. publ. |
| Series: | Encyclopedia of mathematics and its applications
28 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 371 S. |
| ISBN: | 0521331072 |
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| 245 | 1 | 0 | |a Rational approximation of real functions |c P. P. Petrushev ; V. A. Popov |
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Record in the Search Index
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| adam_text | Titel: Rational approximation of real functions
Autor: Petrušev, Penčo P
Jahr: 1987
CONTENTS Preface ix Acknowledgements xi 1 Qualitative theory of linear approximation 1 1.1 Approximation in normed linear spaces 1 1.2 Characterization of the algebraic polynomial of best uniform approximation 6 1.3 Numerical methods 9 1.4 Notes 15 2 Qualitative theory of the best rational approximation 17 2.1 Existence 17 2.2 Uniqueness and characterization of the best uniform approximation 20 2.3 Nonuniqueness in L p (a, b), 1 ^ p oo 26 2.4 Properties of the rational metric projection in C[a, 6] 27 2.5 Numerical methods for best uniform approximation 33 2.6 Notes 37 3 Some classical results in the linear theory 40 3.1 Moduli of continuity and smoothness in C and L p 40 3.2 Direct theorems: Jackson’s theorem 46 3.3 Converse theorems 53 3.4 Direct and converse theorems for algebraic polynomial approximation in a finite interval 57 3.5 Direct and converse theorems and the K-functional of J. Peetre 65 3.6 Notes 71 4 Approximation of some important functions 73 4.1 Newman’s theorem 73 4.2 The exact order of the best uniform approximation to |x| 78 4.3 Zolotarjov’s results 86 4.4 Uniform approximation of e v on [—1,1]: Meinardus conjecture 94 4.5 Uniform approximation of e - * on [0, co) 100 4.6 Notes 105
Contents vi 5 Uniform approximation of some function classes 107 5.1 Preliminaries 107 5.2 Functions with rth derivative of bounded variation 114 5.3 Some classes of absolutely continuous functions and functions of bounded variation 124 5.3.1. One technical result 124 5.3.2. Sobolev classes Wj,,p l 128 5.3.3. Absolutely continuous functions with derivative in Orlicz space LlogL 131 5.3.4. Functions with bounded variation and given modulus of continuity 134 5.4 DeVore’s method 137 5.4.1. Hardy-Littlewood maximal function 137 5.4.2. More on the class W p ,p 1 141 5.4.3. Functions with derivative of bounded variation 144 5.5 Convex functions 146 5.6 Functions with singularities 153 5.7 Notes 158 6 Converse theorems for rational approximation 161 6.1 Gonchar’s and Dolzenko’s results 161 6.2 Estimates for L r norms for the derivatives of rational functions and their Hilbert transforms 164 6.3 Estimation for higher derivatives of rational functions and its applications 169 6.4 Notes 179 7 Spline approximation and Besov spaces 185 7.1 L p (0 p 1) spaces 185 7.2 Besov spaces 201 7.3 Spline approximation 203 7.3.1. Introduction 203 7.3.2. Direct and converse theorems in L p (0 p co) 207 7.3.3. Direct and converse theorems in uniform metric and in BMO 219 7.4 Notes 223 8 Relations between rational and spline approximations 224 8.1 The rational functions are not worse than spline functions as a tool for approximation in L p (1 ^ p co) metric 224 8.2 Estimate of spline approximation by means of rational in L p , 1 p oo 240 8.3 Relations between rational and spline approximations of functions and rational and spline approximations of their derivatives 243 8.4 Notes 260 9 Approximation with respect to Hausdorff distance 263 9.1 Hausdorff distance and its properties 264 9.2 Hausdorff approximation of the jump 266 9.3 Bounded functions 271 9.4 Notes 275
Contents vii 10 The o-effect 278 10.1 Existence and characterization for uniform approximation of individual functions of the class V r and for approximation of functions of bounded variation 278 10.1.1. One new functional characteristic 219 10.1.2. The o-effect in some spline approximations 285 10.1.3. Uniform approximation of individual functions of the class V r and Li approximation of functions of bounded variation 294 10.2 Uniform approximation of absolutely continuous functions 296 10.3 Uniform approximation of convex functions 300 10.4 L p approximation of functions of bounded variation 302 10.5 Notes 304 11 Lower bounds 308 11.1 Some simple lower bounds 308 11.1.1. Negative results for uniform approximation of continuous functions with given modulus of smoothness 308 11.1.2. One negative result for uniform approximation of absolutely continuous functions 311 11.1.3. Lower bound for uniform approximation of the functions of the class V r 312 11.1.4. Uniform approximation of convex functions 314 11.1.5. L p approximation of functions of bounded variation 316 11.2 Uniform approximation of functions of bounded variation and given modulus of continuity 317 11.3 Notes 328 12 Pad£ approximations 329 12.1 Definition and properties of the Pade approximants 329 12.2 Direct theorem for the rows of the Pade-table 332 12.3 Converse theorem for the rows of the Pade-table 336 12.4 The diagonal of the Pade-table 342 12.5 Notes 346 Appendix Some numerical results 348 References 357 Author index 368 Notation and subject index 370
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| any_adam_object | 1 |
| author | Petrušev, Penčo P. 1949- Popov, Vasil A. 1942-1990 |
| author_GND | (DE-588)129250058 (DE-588)119350270 |
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| dewey-full | 515.8'3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.8'3 |
| dewey-search | 515.8'3 |
| dewey-sort | 3515.8 13 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| isbn | 0521331072 |
| language | English |
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| physical | X, 371 S. |
| publishDate | 1987 |
| publishDateSearch | 1987 |
| publishDateSort | 1987 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Encyclopedia of mathematics and its applications |
| series2 | Encyclopedia of mathematics and its applications |
| spelling | Petrušev, Penčo P. 1949- Verfasser (DE-588)129250058 aut Rational approximation of real functions P. P. Petrushev ; V. A. Popov 1. publ. Cambridge [u.a.] Cambridge Univ. Press 1987 X, 371 S. txt rdacontent n rdamedia nc rdacarrier Encyclopedia of mathematics and its applications 28 Reelle Funktion (DE-588)4048918-8 gnd rswk-swf Rationale Approximation (DE-588)4135759-0 gnd rswk-swf Reelle Funktion (DE-588)4048918-8 s Rationale Approximation (DE-588)4135759-0 s DE-604 Popov, Vasil A. 1942-1990 Verfasser (DE-588)119350270 aut Encyclopedia of mathematics and its applications 28 (DE-604)BV000903719 28 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000504976&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Petrušev, Penčo P. 1949- Popov, Vasil A. 1942-1990 Rational approximation of real functions Encyclopedia of mathematics and its applications Reelle Funktion (DE-588)4048918-8 gnd Rationale Approximation (DE-588)4135759-0 gnd |
| subject_GND | (DE-588)4048918-8 (DE-588)4135759-0 |
| title | Rational approximation of real functions |
| title_auth | Rational approximation of real functions |
| title_exact_search | Rational approximation of real functions |
| title_full | Rational approximation of real functions P. P. Petrushev ; V. A. Popov |
| title_fullStr | Rational approximation of real functions P. P. Petrushev ; V. A. Popov |
| title_full_unstemmed | Rational approximation of real functions P. P. Petrushev ; V. A. Popov |
| title_short | Rational approximation of real functions |
| title_sort | rational approximation of real functions |
| topic | Reelle Funktion (DE-588)4048918-8 gnd Rationale Approximation (DE-588)4135759-0 gnd |
| topic_facet | Reelle Funktion Rationale Approximation |
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