Distributions :: generalized functions with applications in Sobolev spaces /
This book grew out of a course taught in the Department of Mathematics, Indian Institute of Technology, Delhi, which was tailored to the needs of the applied community of mathematicians, engineers, physicists etc., who were interested in studying the problems of mathematical physics in general and t...
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin ; Boston :
De Gruyter,
©2012.
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| Schriftenreihe: | De Gruyter textbook.
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| Schlagworte: | |
| Online-Zugang: | Volltext |
| Zusammenfassung: | This book grew out of a course taught in the Department of Mathematics, Indian Institute of Technology, Delhi, which was tailored to the needs of the applied community of mathematicians, engineers, physicists etc., who were interested in studying the problems of mathematical physics in general and their approximate solutions on computer in particular. Almost all topics which will be essential for the study of Sobolev spaces and their applications in the elliptic boundary value problems and their finite element approximations are presented. Also many additional topics of interests for specific applied disciplines and engineering, for example, elementary solutions, derivatives of discontinuous functions of several variables, delta-convergent sequences of functions, Fourier series of distributions, convolution system of equations etc. have been included along with many interesting examples. |
| Beschreibung: | 1 online resource (xxxviii, 833 pages) : illustrations |
| Bibliographie: | Includes bibliographical references and index. |
| ISBN: | 9783110269291 3110269295 1283857669 9781283857666 |
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| 100 | 1 | |a Bhattacharyya, Pulin K. | |
| 245 | 1 | 0 | |a Distributions : |b generalized functions with applications in Sobolev spaces / |c Pulin Kumar Bhattacharyya. |
| 260 | |a Berlin ; |a Boston : |b De Gruyter, |c ©2012. | ||
| 300 | |a 1 online resource (xxxviii, 833 pages) : |b illustrations | ||
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| 490 | 1 | |a De Gruyter textbook | |
| 504 | |a Includes bibliographical references and index. | ||
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| 546 | |a In English. | ||
| 505 | 0 | |6 880-01 |a Preface ; How to use this book in courses ; Acknowledgment ; Notation ; 1 Schwartz distributions ; 1.1 Introduction: Dirac's delta function e(x) and its properties ; 1.2 Test space D (]) of Schwartz ; 1.2.1 Support of a continuous function ; 1.2.2 Space D (]) ; 1.2.3 Space Dm(]); 1.2.4 Space DK (]) ; 1.2.5 Properties of D (]) ; 1.3 Space D'(]) of (Schwartz) distributions; 1.3.1 Algebraic dual space D*(]). | |
| 505 | 8 | |a 1.3.2 Distributions and the space D'(]) of distributions on ]1.3.3 Characterization, order and extension of a distribution ; 1.3.4 Examples of distributions ; 1.3.5 Distribution defined on test space D(]) of complex-valued functions ; 1.4 Some more examples of interesting distributions ; 1.5 Multiplication of distributions by C -functions ; 1.6 Problem of division of distributions. | |
| 505 | 8 | |a 1.7 Even, odd and positive distributions 1.8 Convergence of sequences of distributions in D'(]); 1.9 Convergence of series of distributions in D'(]) ; 1.10 Images of distributions due to change of variables, homogeneous, invariant, spherically symmetric, constant distributions ; 1.10.1 Periodic distributions. | |
| 505 | 8 | |a 1.11 Physical distributions versus mathematical distributions 1.11.1 Physical interpretation of mathematical distributions ; 1.11.2 Load intensity ; 1.11.3 Electrical charge distribution ; 1.11.4 Simple layer and double layer distributions. | |
| 505 | 8 | |a 1.11.5 Relation with probability distribution [7] 2 Differentiation of distributions and application of distributional derivatives ; 2.1 Introduction: an integral definition of derivatives of C1-functions ; 2.2 Derivatives of distributions. | |
| 650 | 0 | |a Theory of distributions (Functional analysis) |v Textbooks. | |
| 650 | 0 | |a Sobolev spaces |v Textbooks. | |
| 650 | 4 | |a Sobolev spaces |x Textbooks. | |
| 650 | 4 | |a Theory of distributions (Functional analysis) |x Textbooks. | |
| 650 | 7 | |a MATHEMATICS |x Functional Analysis. |2 bisacsh | |
| 650 | 7 | |a Sobolev spaces |2 fast | |
| 650 | 7 | |a Theory of distributions (Functional analysis) |2 fast | |
| 650 | 7 | |a Distribution. |2 gnd | |
| 650 | 7 | |a Sobolev-Raum |2 gnd |0 http://d-nb.info/gnd/4055345-0 | |
| 655 | 7 | |a Textbooks |2 fast | |
| 758 | |i has work: |a Distributions (Text) |1 https://id.oclc.org/worldcat/entity/E39PCGpPwR48PVBJMt9xXF7Xgq |4 https://id.oclc.org/worldcat/ontology/hasWork | ||
| 776 | 0 | 8 | |i Print version: |a Bhattacharyya, Pulin Kumar. |t Distributions : Generalized Functions with Applications in Sobolev Spaces. |d Berlin : De Gruyter, ©2012 |z 9783110269277 |
| 830 | 0 | |a De Gruyter textbook. |0 http://id.loc.gov/authorities/names/n94049545 | |
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| 880 | 0 | 0 | |6 505-01/(S |t Frontmatter -- |t Preface -- |t Contents -- |t How to use this book in courses -- |t Acknowledgment -- |t Notation -- |t Chapter 1. Schwartz distributions -- |t Chapter 2. Differentiation of distributions and application of distributional derivatives -- |t Chapter 3. Derivatives of piecewise smooth functions, Green's formula, elementary solutions, applications to Sobolev spaces -- |t Chapter 4. Additional properties of Dʹ(Ω) -- |t Chapter 5. Local properties, restrictions, unification principle, space ℰʹ(ℝn) of distributions with compact support -- |t Chapter 6. Convolution of distributions -- |t Chapter 7. Fourier transforms of functions of L1(ℝn) and S(ℝn) -- |t Chapter 8. Fourier transforms of distributions and Sobolev spaces of arbitrary order HS(ℝn) -- |t 8.1 Motivation for a possible definition of the Fourier transform of a distribution -- |t 8.2 Space Sʹ (Rn) of tempered distributions -- |t 8.3 Fourier transform of tempered distributions -- |t 8.4 Fourier transform of distributions with compact support -- |t 8.5 Fourier transform of convolution of distributions -- |t 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives of tempered distributions -- |t 8.7 Fourier transform methods for differential equations and elementary solutions in Sʹ(ℝn) -- |t 8.8 Laplace transform of distributions on ℝ -- |t 8.9 Applications -- |t 8.10 Sobolev spaces on Ω ≠ Rn revisited -- |t 8.11 Compactness results in Sobolev spaces -- |t 8.12 Sobolev's imbedding results -- |t 8.13 Sobolev spaces Hs.(Γ), Ws;p(Γ) on a manifold boundary Γ -- |t 8.14 Trace results in Sobolev spaces on Ω⊊ℝn -- |t Chapter 9. Vector-valued distributions -- |t Appendix A. Functional analysis (basic results) -- |t Appendix B. Lp-spaces -- |t Appendix C. Open cover and partition of unity -- |t Appendix D. Boundary geometry -- |t Bibliography -- |t Index. |
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| author | Bhattacharyya, Pulin K. |
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| contents | Preface ; How to use this book in courses ; Acknowledgment ; Notation ; 1 Schwartz distributions ; 1.1 Introduction: Dirac's delta function e(x) and its properties ; 1.2 Test space D (]) of Schwartz ; 1.2.1 Support of a continuous function ; 1.2.2 Space D (]) ; 1.2.3 Space Dm(]); 1.2.4 Space DK (]) ; 1.2.5 Properties of D (]) ; 1.3 Space D'(]) of (Schwartz) distributions; 1.3.1 Algebraic dual space D*(]). 1.3.2 Distributions and the space D'(]) of distributions on ]1.3.3 Characterization, order and extension of a distribution ; 1.3.4 Examples of distributions ; 1.3.5 Distribution defined on test space D(]) of complex-valued functions ; 1.4 Some more examples of interesting distributions ; 1.5 Multiplication of distributions by C -functions ; 1.6 Problem of division of distributions. 1.7 Even, odd and positive distributions 1.8 Convergence of sequences of distributions in D'(]); 1.9 Convergence of series of distributions in D'(]) ; 1.10 Images of distributions due to change of variables, homogeneous, invariant, spherically symmetric, constant distributions ; 1.10.1 Periodic distributions. 1.11 Physical distributions versus mathematical distributions 1.11.1 Physical interpretation of mathematical distributions ; 1.11.2 Load intensity ; 1.11.3 Electrical charge distribution ; 1.11.4 Simple layer and double layer distributions. 1.11.5 Relation with probability distribution [7] 2 Differentiation of distributions and application of distributional derivatives ; 2.1 Introduction: an integral definition of derivatives of C1-functions ; 2.2 Derivatives of distributions. |
| ctrlnum | (OCoLC)808342072 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.782 |
| dewey-search | 515/.782 |
| dewey-sort | 3515 3782 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Electronic eBook |
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Fourier transforms of distributions and Sobolev spaces of arbitrary order HS(ℝn) --</subfield><subfield code="t">8.1 Motivation for a possible definition of the Fourier transform of a distribution --</subfield><subfield code="t">8.2 Space Sʹ (Rn) of tempered distributions --</subfield><subfield code="t">8.3 Fourier transform of tempered distributions --</subfield><subfield code="t">8.4 Fourier transform of distributions with compact support --</subfield><subfield code="t">8.5 Fourier transform of convolution of distributions --</subfield><subfield code="t">8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives of tempered distributions --</subfield><subfield code="t">8.7 Fourier transform methods for differential equations and elementary solutions in Sʹ(ℝn) --</subfield><subfield code="t">8.8 Laplace transform of distributions on ℝ --</subfield><subfield code="t">8.9 Applications --</subfield><subfield code="t">8.10 Sobolev spaces on Ω ≠ Rn revisited --</subfield><subfield code="t">8.11 Compactness results in Sobolev spaces --</subfield><subfield code="t">8.12 Sobolev's imbedding results --</subfield><subfield code="t">8.13 Sobolev spaces Hs.(Γ), Ws;p(Γ) on a manifold boundary Γ --</subfield><subfield code="t">8.14 Trace results in Sobolev spaces on Ω⊊ℝn --</subfield><subfield code="t">Chapter 9. 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| id | ZDB-4-EBA-ocn808342072 |
| illustrated | Illustrated |
| indexdate | 2024-11-27T13:24:54Z |
| institution | BVB |
| isbn | 9783110269291 3110269295 1283857669 9781283857666 |
| language | English |
| oclc_num | 808342072 |
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| owner | MAIN DE-863 DE-BY-FWS |
| owner_facet | MAIN DE-863 DE-BY-FWS |
| physical | 1 online resource (xxxviii, 833 pages) : illustrations |
| psigel | ZDB-4-EBA |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | De Gruyter, |
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| spelling | Bhattacharyya, Pulin K. Distributions : generalized functions with applications in Sobolev spaces / Pulin Kumar Bhattacharyya. Berlin ; Boston : De Gruyter, ©2012. 1 online resource (xxxviii, 833 pages) : illustrations text txt rdacontent computer c rdamedia online resource cr rdacarrier text file De Gruyter textbook Includes bibliographical references and index. This book grew out of a course taught in the Department of Mathematics, Indian Institute of Technology, Delhi, which was tailored to the needs of the applied community of mathematicians, engineers, physicists etc., who were interested in studying the problems of mathematical physics in general and their approximate solutions on computer in particular. Almost all topics which will be essential for the study of Sobolev spaces and their applications in the elliptic boundary value problems and their finite element approximations are presented. Also many additional topics of interests for specific applied disciplines and engineering, for example, elementary solutions, derivatives of discontinuous functions of several variables, delta-convergent sequences of functions, Fourier series of distributions, convolution system of equations etc. have been included along with many interesting examples. In English. 880-01 Preface ; How to use this book in courses ; Acknowledgment ; Notation ; 1 Schwartz distributions ; 1.1 Introduction: Dirac's delta function e(x) and its properties ; 1.2 Test space D (]) of Schwartz ; 1.2.1 Support of a continuous function ; 1.2.2 Space D (]) ; 1.2.3 Space Dm(]); 1.2.4 Space DK (]) ; 1.2.5 Properties of D (]) ; 1.3 Space D'(]) of (Schwartz) distributions; 1.3.1 Algebraic dual space D*(]). 1.3.2 Distributions and the space D'(]) of distributions on ]1.3.3 Characterization, order and extension of a distribution ; 1.3.4 Examples of distributions ; 1.3.5 Distribution defined on test space D(]) of complex-valued functions ; 1.4 Some more examples of interesting distributions ; 1.5 Multiplication of distributions by C -functions ; 1.6 Problem of division of distributions. 1.7 Even, odd and positive distributions 1.8 Convergence of sequences of distributions in D'(]); 1.9 Convergence of series of distributions in D'(]) ; 1.10 Images of distributions due to change of variables, homogeneous, invariant, spherically symmetric, constant distributions ; 1.10.1 Periodic distributions. 1.11 Physical distributions versus mathematical distributions 1.11.1 Physical interpretation of mathematical distributions ; 1.11.2 Load intensity ; 1.11.3 Electrical charge distribution ; 1.11.4 Simple layer and double layer distributions. 1.11.5 Relation with probability distribution [7] 2 Differentiation of distributions and application of distributional derivatives ; 2.1 Introduction: an integral definition of derivatives of C1-functions ; 2.2 Derivatives of distributions. Theory of distributions (Functional analysis) Textbooks. Sobolev spaces Textbooks. MATHEMATICS Functional Analysis. bisacsh Sobolev spaces fast Theory of distributions (Functional analysis) fast Distribution. gnd Sobolev-Raum gnd http://d-nb.info/gnd/4055345-0 Textbooks fast has work: Distributions (Text) https://id.oclc.org/worldcat/entity/E39PCGpPwR48PVBJMt9xXF7Xgq https://id.oclc.org/worldcat/ontology/hasWork Print version: Bhattacharyya, Pulin Kumar. Distributions : Generalized Functions with Applications in Sobolev Spaces. Berlin : De Gruyter, ©2012 9783110269277 De Gruyter textbook. http://id.loc.gov/authorities/names/n94049545 FWS01 ZDB-4-EBA FWS_PDA_EBA https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&AN=471060 Volltext 505-01/(S Frontmatter -- Preface -- Contents -- How to use this book in courses -- Acknowledgment -- Notation -- Chapter 1. Schwartz distributions -- Chapter 2. Differentiation of distributions and application of distributional derivatives -- Chapter 3. Derivatives of piecewise smooth functions, Green's formula, elementary solutions, applications to Sobolev spaces -- Chapter 4. Additional properties of Dʹ(Ω) -- Chapter 5. Local properties, restrictions, unification principle, space ℰʹ(ℝn) of distributions with compact support -- Chapter 6. Convolution of distributions -- Chapter 7. Fourier transforms of functions of L1(ℝn) and S(ℝn) -- Chapter 8. Fourier transforms of distributions and Sobolev spaces of arbitrary order HS(ℝn) -- 8.1 Motivation for a possible definition of the Fourier transform of a distribution -- 8.2 Space Sʹ (Rn) of tempered distributions -- 8.3 Fourier transform of tempered distributions -- 8.4 Fourier transform of distributions with compact support -- 8.5 Fourier transform of convolution of distributions -- 8.6 Derivatives of Fourier transforms and Fourier transforms of derivatives of tempered distributions -- 8.7 Fourier transform methods for differential equations and elementary solutions in Sʹ(ℝn) -- 8.8 Laplace transform of distributions on ℝ -- 8.9 Applications -- 8.10 Sobolev spaces on Ω ≠ Rn revisited -- 8.11 Compactness results in Sobolev spaces -- 8.12 Sobolev's imbedding results -- 8.13 Sobolev spaces Hs.(Γ), Ws;p(Γ) on a manifold boundary Γ -- 8.14 Trace results in Sobolev spaces on Ω⊊ℝn -- Chapter 9. Vector-valued distributions -- Appendix A. Functional analysis (basic results) -- Appendix B. Lp-spaces -- Appendix C. Open cover and partition of unity -- Appendix D. Boundary geometry -- Bibliography -- Index. |
| spellingShingle | Bhattacharyya, Pulin K. Distributions : generalized functions with applications in Sobolev spaces / De Gruyter textbook. Preface ; How to use this book in courses ; Acknowledgment ; Notation ; 1 Schwartz distributions ; 1.1 Introduction: Dirac's delta function e(x) and its properties ; 1.2 Test space D (]) of Schwartz ; 1.2.1 Support of a continuous function ; 1.2.2 Space D (]) ; 1.2.3 Space Dm(]); 1.2.4 Space DK (]) ; 1.2.5 Properties of D (]) ; 1.3 Space D'(]) of (Schwartz) distributions; 1.3.1 Algebraic dual space D*(]). 1.3.2 Distributions and the space D'(]) of distributions on ]1.3.3 Characterization, order and extension of a distribution ; 1.3.4 Examples of distributions ; 1.3.5 Distribution defined on test space D(]) of complex-valued functions ; 1.4 Some more examples of interesting distributions ; 1.5 Multiplication of distributions by C -functions ; 1.6 Problem of division of distributions. 1.7 Even, odd and positive distributions 1.8 Convergence of sequences of distributions in D'(]); 1.9 Convergence of series of distributions in D'(]) ; 1.10 Images of distributions due to change of variables, homogeneous, invariant, spherically symmetric, constant distributions ; 1.10.1 Periodic distributions. 1.11 Physical distributions versus mathematical distributions 1.11.1 Physical interpretation of mathematical distributions ; 1.11.2 Load intensity ; 1.11.3 Electrical charge distribution ; 1.11.4 Simple layer and double layer distributions. 1.11.5 Relation with probability distribution [7] 2 Differentiation of distributions and application of distributional derivatives ; 2.1 Introduction: an integral definition of derivatives of C1-functions ; 2.2 Derivatives of distributions. Theory of distributions (Functional analysis) Textbooks. Sobolev spaces Textbooks. MATHEMATICS Functional Analysis. bisacsh Sobolev spaces fast Theory of distributions (Functional analysis) fast Distribution. gnd Sobolev-Raum gnd http://d-nb.info/gnd/4055345-0 |
| subject_GND | http://d-nb.info/gnd/4055345-0 |
| title | Distributions : generalized functions with applications in Sobolev spaces / |
| title_auth | Distributions : generalized functions with applications in Sobolev spaces / |
| title_exact_search | Distributions : generalized functions with applications in Sobolev spaces / |
| title_full | Distributions : generalized functions with applications in Sobolev spaces / Pulin Kumar Bhattacharyya. |
| title_fullStr | Distributions : generalized functions with applications in Sobolev spaces / Pulin Kumar Bhattacharyya. |
| title_full_unstemmed | Distributions : generalized functions with applications in Sobolev spaces / Pulin Kumar Bhattacharyya. |
| title_short | Distributions : |
| title_sort | distributions generalized functions with applications in sobolev spaces |
| title_sub | generalized functions with applications in Sobolev spaces / |
| topic | Theory of distributions (Functional analysis) Textbooks. Sobolev spaces Textbooks. MATHEMATICS Functional Analysis. bisacsh Sobolev spaces fast Theory of distributions (Functional analysis) fast Distribution. gnd Sobolev-Raum gnd http://d-nb.info/gnd/4055345-0 |
| topic_facet | Theory of distributions (Functional analysis) Textbooks. Sobolev spaces Textbooks. MATHEMATICS Functional Analysis. Sobolev spaces Theory of distributions (Functional analysis) Distribution. Sobolev-Raum Textbooks |
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| work_keys_str_mv | AT bhattacharyyapulink distributionsgeneralizedfunctionswithapplicationsinsobolevspaces |