Interpolation processes: basic theory and applications
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
[2008]
|
| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | xiv, 444 Seiten Diagramme 235 mm x 155 mm |
| ISBN: | 9783540683469 9783540683490 |
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MARC
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| 100 | 1 | |a Mastroianni, Giuseppe |0 (DE-588)1278926232 |4 aut | |
| 245 | 1 | 0 | |a Interpolation processes |b basic theory and applications |c Giuseppe Mastroianni ; Gradimir V. Milovanović |
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c [2008] | |
| 264 | 4 | |c © 2008 | |
| 300 | |a xiv, 444 Seiten |b Diagramme |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer monographs in mathematics | |
| 650 | 4 | |a Interpolation | |
| 650 | 0 | 7 | |a Approximation |0 (DE-588)4002498-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Interpolation |0 (DE-588)4162121-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Interpolation |0 (DE-588)4162121-9 |D s |
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| 689 | 1 | 0 | |a Approximation |0 (DE-588)4002498-2 |D s |
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| 700 | 1 | |a Milovanović, Gradimir V. |d 1948- |0 (DE-588)136353398 |4 aut | |
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Record in the Search Index
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| adam_text |
Contents
1.1.2
1.1.3
1.1.4
1.1.5
1.1.6
1.1.7
1.2
Preface.
1 Constructive Elements and Approaches in Approximation Theory
1.1 Introduction to Approximation Theory.
1.1.1 Basic Notions.
Algebraic and Trigonometric Polynomials.
Best Approximation by Polynomials.
Chebyshev Polynomials.
Chebyshev Extremal Problems.
Chebyshev Alternation Theorem.
Numerical Methods.
Basic Facts on Trigonometric Approximation.
1.2.1 Trigonometric Kernels.
1.2.2 Fourier Series and Sums.
1.2.3 Moduli of Smoothness, Best Approximation and Besov
Spaces.
Chebyshev Systems and Interpolation.
1.3.1 Chebyshev Systems and Spaces.
Algebraic Lagrange Interpolation.
Trigonometric Interpolation.
Riesz Interpolation Formula.
A General Interpolation Problem.
Interpolation by Algebraic Polynomials.
1.4.1 Representations and Computation of Interpolation
Polynomials.
Interpolation Array and Lagrange Operators.
Interpolation Error for Some Classes of Functions .
Uniform Convergence in the Class of Analytic Functions
Bernstein's Example of Pointwise Divergence.
Lebesgue Function and Some Estimates for the Lebesgue
Constant.
Algorithm for Finding Optimal Nodes .
1.3
1.4
.3.2
.3.3
.3.4
.3.5
1.4.2
1.4.3
1.4.4
1.4.5
1.4.6
1.4.7
Orthogonal Polynomials and Weighted Polynomial Approximation
2.1 Orthogonal Systems and Polynomials.
2.1.1 Inner Product Space and Orthogonal Systems.
2.1.2 Fourier Expansion and Best Approximation .
2.1.3 Examples of Orthogonal Systems.
1
1
1
4
7
9
14
17
20
24
24
30
32
38
38
39
40
44
46
48
48
51
54
56
61
63
68
75
75
75
77
79
Contents
2.1.4 Basic Facts on Orthogonal Polynomials and Extremal
Problems. 89
2.1.5 Zeros of Orthogonal Polynomials. 93
2.2 Orthogonal Polynomials on the Real Line. 95
2.2.1 Basic Properties. 95
2.2.2 Asymptotic Properties of Orthogonal Polynomials. 103
2.2.3 Associated Polynomials and Christoffel Numbers. Ill
2.2.4 Functions of the Second Kind and Stieltjes Polynomials . . 117
2.3 Classical Orthogonal Polynomials. 121
2.3.1 Definition of the Classical Orthogonal Polynomials . 121
2.3.2 General Properties of the Classical Orthogonal
Polynomials. 124
2.3.3 Generating Function. 128
2.3.4 Jacobi Polynomials . 131
2.3.5 Generalized Laguerre Polynomials. 140
2.3.6 Hermite Polynomials . 145
2.4 Nonclassical Orthogonal Polynomials. 146
2.4.1 Semi-classical Orthogonal Polynomials. 146
2.4.2 Generalized Gegenbauer Polynomials. 147
2.4.3 Generalized Jacobi Polynomials. 148
2.4.4 Sonin-Markov Orthogonal Polynomials. 152
2.4.5 Freud Orthogonal Polynomials. 154
2.4.6 Orthogonal Polynomials with Respect to Abel, Lindelof,
and Logistic Weights . 159
2.4.7 Strong Non-classical Orthogonal Polynomials. 159
2.4.8 Numerical Construction of Orthogonal Polynomials . 160
2.5 Weighted Polynomial Approximation. 166
2.5.1 Weighted Functional Spaces, Moduli of Smoothness and
tf'-functionals. 166
2.5.2 Weighted Best Polynomial Approximation on [—1, 1] . . . 170
2.5.3 Weighted Approximation on the Semi-axis. 174
2.5.4 Weighted Approximation on the Real Line. 178
2.5.5 Weighted Polynomial Approximation of Functions Having
Isolated Interior Singularities. 182
Trigonometric Approximation. 193
3.1 Approximating Properties of Operators. 193
3.1.1 Approximation by Fourier Sums . 193
3.1.2 Approximation by Fejer and de la Vallee Poussin Means . . 195
3.2 Discrete Operators. 197
3.2.1 A Quadrature Formula. 197
3.2.2 Discrete Versions of Fourier and de la Vallee Poussin
Sums. 202
3.2.3 Marcinkiewicz Inequalities. 205
Contents
3.2.4 Uniform Approximation. 210
3.2.5 Lagrange Interpolation Error in Lp. 212
3.2.6 Some Estimates of the Interpolation Errors in L! -Sobolev
Spaces. 221
3.2.7 The Weighted Case . 224
Algebraic Interpolation in Uniform Norm. 235
4.1 Introduction and Preliminaries. 235
4.1.1 Interpolation at Zeros of Orthogonal Polynomials . 235
4.1.2 Some Auxiliary Results. 239
4.2 Optimal Systems of Nodes . 248
4.2.1 Optimal Systems of Knots on [-1, 1]. 248
4.2.2 Additional Nodes Method with Jacobi Zeros. 252
4.2.3 Other "Optimal" Interpolation Processes. 264
4.2.4 Some Simultaneous Interpolation Processes . 268
4.3 Weighted Interpolation. 271
4.3.1 Weighted Interpolation at Jacobi Zeros. 271
4.3.2 Lagrange Interpolation in Sobolev Spaces . 276
4.3.3 Interpolation at Laguerre Zeros. 278
4.3.4 Interpolation at Hermite Zeros. 287
4.3.5 Interpolation of Functions with Internal Isolated
Singularities. 292
Applications. 319
5.1 Quadrature Formulae . 319
5.1.1 Introduction. 319
5.1.2 Some Remarks on Newton-Cotes Rules with Jacobi
Weights . 322
5.1.3 Gauss-Christoffel Quadrature Rules. 324
5.1.4 Gauss-Radau and Gauss-Lobatto Quadrature Rules. 328
5.1.5 Error Estimates of Gaussian Rules for Some Classes of
Functions. 332
5.1.6 Product Integration Rules. 345
5.1.7 Integration of Periodic Functions on the Real Line with
Rational Weight. 350
5.2 Integral Equations. 362
5.2.1 Some Basic Facts. 362
5.2.2 Fredholm Integral Equations of the Second Kind. 369
5.2.3 Nystrom Method. 382
5.3 Moment-Preserving Approximation. 385
5.3.1 The Standard L2-Approximation. 385
5.3.2 The Constrained L2-Polynomial Approximation. 388
5.3.3 Moment-Preserving Spline Approximation. 389
5.4 Summation of Slowly Convergent Series. 397
5.4.1 Laplace Transform Method. 398
xiv Contents
5.4.2 Contour Integration Over a Rectangle.401
5.4.3 Remarks on Some Slowly Convergent Power Series . 411
References.415
Index.437
Interpolation
Processes
The classical books on interpolation address name
r
or
s
negative
resalís,
i.e., results on
divergent interpolation
processes, usually
constructed over some equidistant systems or node^., The author:.;
present, with complete proofs, recent results on convergent
interpolation processes, for trigonometric arid algebraic poly¬
nomials of one real
variable,
not yet published in other textbooks
and monographs on approximation theory and
гштега/.а!
mathematics, In this special,but
fondamental
and
importáru
field of real analysis the authors present the state of an. Some
500
references are cited, including many new results of the authors.
Basic tools in this field (orthogonal polynomials, moduli of
smoothness, K-functionals, etc) as well as some selected applica¬
tions in numerical integration integral equations, moment-
ЩЩШ
polynomial
approximationson
(-1,1),
(Ο,+οο)
and
(-«>, +00)
The book addresses
teseárehers
and students in mathematics, ~"
physics3 and other computational'and applied sciences. |
| adam_txt |
Contents
1.1.2
1.1.3
1.1.4
1.1.5
1.1.6
1.1.7
1.2
Preface.
1 Constructive Elements and Approaches in Approximation Theory
1.1 Introduction to Approximation Theory.
1.1.1 Basic Notions.
Algebraic and Trigonometric Polynomials.
Best Approximation by Polynomials.
Chebyshev Polynomials.
Chebyshev Extremal Problems.
Chebyshev Alternation Theorem.
Numerical Methods.
Basic Facts on Trigonometric Approximation.
1.2.1 Trigonometric Kernels.
1.2.2 Fourier Series and Sums.
1.2.3 Moduli of Smoothness, Best Approximation and Besov
Spaces.
Chebyshev Systems and Interpolation.
1.3.1 Chebyshev Systems and Spaces.
Algebraic Lagrange Interpolation.
Trigonometric Interpolation.
Riesz Interpolation Formula.
A General Interpolation Problem.
Interpolation by Algebraic Polynomials.
1.4.1 Representations and Computation of Interpolation
Polynomials.
Interpolation Array and Lagrange Operators.
Interpolation Error for Some Classes of Functions .
Uniform Convergence in the Class of Analytic Functions
Bernstein's Example of Pointwise Divergence.
Lebesgue Function and Some Estimates for the Lebesgue
Constant.
Algorithm for Finding Optimal Nodes .
1.3
1.4
.3.2
.3.3
.3.4
.3.5
1.4.2
1.4.3
1.4.4
1.4.5
1.4.6
1.4.7
Orthogonal Polynomials and Weighted Polynomial Approximation
2.1 Orthogonal Systems and Polynomials.
2.1.1 Inner Product Space and Orthogonal Systems.
2.1.2 Fourier Expansion and Best Approximation .
2.1.3 Examples of Orthogonal Systems.
1
1
1
4
7
9
14
17
20
24
24
30
32
38
38
39
40
44
46
48
48
51
54
56
61
63
68
75
75
75
77
79
Contents
2.1.4 Basic Facts on Orthogonal Polynomials and Extremal
Problems. 89
2.1.5 Zeros of Orthogonal Polynomials. 93
2.2 Orthogonal Polynomials on the Real Line. 95
2.2.1 Basic Properties. 95
2.2.2 Asymptotic Properties of Orthogonal Polynomials. 103
2.2.3 Associated Polynomials and Christoffel Numbers. Ill
2.2.4 Functions of the Second Kind and Stieltjes Polynomials . . 117
2.3 Classical Orthogonal Polynomials. 121
2.3.1 Definition of the Classical Orthogonal Polynomials . 121
2.3.2 General Properties of the Classical Orthogonal
Polynomials. 124
2.3.3 Generating Function. 128
2.3.4 Jacobi Polynomials . 131
2.3.5 Generalized Laguerre Polynomials. 140
2.3.6 Hermite Polynomials . 145
2.4 Nonclassical Orthogonal Polynomials. 146
2.4.1 Semi-classical Orthogonal Polynomials. 146
2.4.2 Generalized Gegenbauer Polynomials. 147
2.4.3 Generalized Jacobi Polynomials. 148
2.4.4 Sonin-Markov Orthogonal Polynomials. 152
2.4.5 Freud Orthogonal Polynomials. 154
2.4.6 Orthogonal Polynomials with Respect to Abel, Lindelof,
and Logistic Weights . 159
2.4.7 Strong Non-classical Orthogonal Polynomials. 159
2.4.8 Numerical Construction of Orthogonal Polynomials . 160
2.5 Weighted Polynomial Approximation. 166
2.5.1 Weighted Functional Spaces, Moduli of Smoothness and
tf'-functionals. 166
2.5.2 Weighted Best Polynomial Approximation on [—1, 1] . . . 170
2.5.3 Weighted Approximation on the Semi-axis. 174
2.5.4 Weighted Approximation on the Real Line. 178
2.5.5 Weighted Polynomial Approximation of Functions Having
Isolated Interior Singularities. 182
Trigonometric Approximation. 193
3.1 Approximating Properties of Operators. 193
3.1.1 Approximation by Fourier Sums . 193
3.1.2 Approximation by Fejer and de la Vallee Poussin Means . . 195
3.2 Discrete Operators. 197
3.2.1 A Quadrature Formula. 197
3.2.2 Discrete Versions of Fourier and de la Vallee Poussin
Sums. 202
3.2.3 Marcinkiewicz Inequalities. 205
Contents
3.2.4 Uniform Approximation. 210
3.2.5 Lagrange Interpolation Error in Lp. 212
3.2.6 Some Estimates of the Interpolation Errors in L! -Sobolev
Spaces. 221
3.2.7 The Weighted Case . 224
Algebraic Interpolation in Uniform Norm. 235
4.1 Introduction and Preliminaries. 235
4.1.1 Interpolation at Zeros of Orthogonal Polynomials . 235
4.1.2 Some Auxiliary Results. 239
4.2 Optimal Systems of Nodes . 248
4.2.1 Optimal Systems of Knots on [-1, 1]. 248
4.2.2 Additional Nodes Method with Jacobi Zeros. 252
4.2.3 Other "Optimal" Interpolation Processes. 264
4.2.4 Some Simultaneous Interpolation Processes . 268
4.3 Weighted Interpolation. 271
4.3.1 Weighted Interpolation at Jacobi Zeros. 271
4.3.2 Lagrange Interpolation in Sobolev Spaces . 276
4.3.3 Interpolation at Laguerre Zeros. 278
4.3.4 Interpolation at Hermite Zeros. 287
4.3.5 Interpolation of Functions with Internal Isolated
Singularities. 292
Applications. 319
5.1 Quadrature Formulae . 319
5.1.1 Introduction. 319
5.1.2 Some Remarks on Newton-Cotes Rules with Jacobi
Weights . 322
5.1.3 Gauss-Christoffel Quadrature Rules. 324
5.1.4 Gauss-Radau and Gauss-Lobatto Quadrature Rules. 328
5.1.5 Error Estimates of Gaussian Rules for Some Classes of
Functions. 332
5.1.6 Product Integration Rules. 345
5.1.7 Integration of Periodic Functions on the Real Line with
Rational Weight. 350
5.2 Integral Equations. 362
5.2.1 Some Basic Facts. 362
5.2.2 Fredholm Integral Equations of the Second Kind. 369
5.2.3 Nystrom Method. 382
5.3 Moment-Preserving Approximation. 385
5.3.1 The Standard L2-Approximation. 385
5.3.2 The Constrained L2-Polynomial Approximation. 388
5.3.3 Moment-Preserving Spline Approximation. 389
5.4 Summation of Slowly Convergent Series. 397
5.4.1 Laplace Transform Method. 398
xiv Contents
5.4.2 Contour Integration Over a Rectangle.401
5.4.3 Remarks on Some Slowly Convergent Power Series . 411
References.415
Index.437
Interpolation
Processes
The classical books on interpolation address name
r
or
s
negative
resalís,
i.e., results on
divergent interpolation
processes, usually
constructed over some equidistant systems or node^., The author:.;
present, with complete proofs, recent results on convergent
interpolation processes, for trigonometric arid algebraic poly¬
nomials of one real
variable,
not yet published in other textbooks
and monographs on approximation theory and
гштега/.а!
mathematics, In this special,but
fondamental
and
importáru
field of real analysis the authors present the state of an. Some
500
references are cited, including many new results of the authors.
Basic tools in this field (orthogonal polynomials, moduli of
smoothness, K-functionals, etc) as well as some selected applica¬
tions in numerical integration integral equations, moment-
ЩЩШ
polynomial
approximationson
(-1,1),
(Ο,+οο)
and
(-«>, +00)
The book addresses
teseárehers
and students in mathematics, ~"
physics3 and other computational'and applied sciences. |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Mastroianni, Giuseppe Milovanović, Gradimir V. 1948- |
| author_GND | (DE-588)1278926232 (DE-588)136353398 |
| author_facet | Mastroianni, Giuseppe Milovanović, Gradimir V. 1948- |
| author_role | aut aut |
| author_sort | Mastroianni, Giuseppe |
| author_variant | g m gm g v m gv gvm |
| building | Verbundindex |
| bvnumber | BV035069738 |
| callnumber-first | Q - Science |
| callnumber-label | QA281 |
| callnumber-raw | QA281 |
| callnumber-search | QA281 |
| callnumber-sort | QA 3281 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 470 SK 905 |
| ctrlnum | (OCoLC)233788977 (DE-599)DNB989061876 |
| dewey-full | 511.42 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.42 |
| dewey-search | 511.42 |
| dewey-sort | 3511.42 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| id | DE-604.BV035069738 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T22:03:44Z |
| indexdate | 2025-01-30T17:02:32Z |
| institution | BVB |
| isbn | 9783540683469 9783540683490 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016738135 |
| oclc_num | 233788977 |
| open_access_boolean | |
| owner | DE-703 DE-824 DE-83 DE-11 DE-188 DE-20 DE-739 |
| owner_facet | DE-703 DE-824 DE-83 DE-11 DE-188 DE-20 DE-739 |
| physical | xiv, 444 Seiten Diagramme 235 mm x 155 mm |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Mastroianni, Giuseppe (DE-588)1278926232 aut Interpolation processes basic theory and applications Giuseppe Mastroianni ; Gradimir V. Milovanović Berlin ; Heidelberg Springer [2008] © 2008 xiv, 444 Seiten Diagramme 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Interpolation Approximation (DE-588)4002498-2 gnd rswk-swf Interpolation (DE-588)4162121-9 gnd rswk-swf Interpolation (DE-588)4162121-9 s DE-604 Approximation (DE-588)4002498-2 s 1\p DE-604 Milovanović, Gradimir V. 1948- (DE-588)136353398 aut Erscheint auch als Online-Ausgabe 978-3-540-68349-0 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016738135&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 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=016738135&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Mastroianni, Giuseppe Milovanović, Gradimir V. 1948- Interpolation processes basic theory and applications Interpolation Approximation (DE-588)4002498-2 gnd Interpolation (DE-588)4162121-9 gnd |
| subject_GND | (DE-588)4002498-2 (DE-588)4162121-9 |
| title | Interpolation processes basic theory and applications |
| title_auth | Interpolation processes basic theory and applications |
| title_exact_search | Interpolation processes basic theory and applications |
| title_exact_search_txtP | Interpolation processes basic theory and applications |
| title_full | Interpolation processes basic theory and applications Giuseppe Mastroianni ; Gradimir V. Milovanović |
| title_fullStr | Interpolation processes basic theory and applications Giuseppe Mastroianni ; Gradimir V. Milovanović |
| title_full_unstemmed | Interpolation processes basic theory and applications Giuseppe Mastroianni ; Gradimir V. Milovanović |
| title_short | Interpolation processes |
| title_sort | interpolation processes basic theory and applications |
| title_sub | basic theory and applications |
| topic | Interpolation Approximation (DE-588)4002498-2 gnd Interpolation (DE-588)4162121-9 gnd |
| topic_facet | Interpolation Approximation |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016738135&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016738135&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT mastroiannigiuseppe interpolationprocessesbasictheoryandapplications AT milovanovicgradimirv interpolationprocessesbasictheoryandapplications |