Distributions: generalized functions with applications in Sobolev spaces
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| Format: | Book |
| Language: | English |
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Berlin [u.a.]
De Gruyter
2012
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| Series: | De Gruyter textbook
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| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XXXVIII, 833 S. graph. Darst. 240 mm x 170 mm |
| ISBN: | 3110269279 9783110269277 |
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MARC
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| 020 | |a 9783110269277 |9 978-3-11-026927-7 | ||
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| 084 | |a SK 600 |0 (DE-625)143248: |2 rvk | ||
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| 100 | 1 | |a Bhattacharyya, Pulin K. |e Verfasser |0 (DE-588)1025884841 |4 aut | |
| 245 | 1 | 0 | |a Distributions |b generalized functions with applications in Sobolev spaces |c Pulin Kumar Bhattacharyya |
| 264 | 1 | |a Berlin [u.a.] |b De Gruyter |c 2012 | |
| 300 | |a XXXVIII, 833 S. |b graph. Darst. |c 240 mm x 170 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a De Gruyter textbook | |
| 650 | 0 | 7 | |a Distribution |g Funktionalanalysis |0 (DE-588)4070505-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Sobolev-Raum |0 (DE-588)4055345-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Distribution |g Funktionalanalysis |0 (DE-588)4070505-5 |D s |
| 689 | 0 | 1 | |a Sobolev-Raum |0 (DE-588)4055345-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-11-026929-1 |
| 856 | 4 | 2 | |m X:MVB |q text/html |u http://deposit.dnb.de/cgi-bin/dokserv?id=3950792&prov=M&dok%5Fvar=1&dok%5Fext=htm |3 Inhaltstext |
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IMAGE 1
CONTENTS
PREFACE VII
HOW TO USE THIS BOOK IN COURSES XXI
ACKNOWLEDGMENT XXV
NOTATION XXVII
1 SCHWARTZ DISTRIBUTIONS 1
1.1 INTRODUCTION: DIRAC'S DELTA FUNCTION S ( X ) AND ITS PROPERTIES 1
1.2 TEST SPACE OF SCHWARTZ 6
1.2.1 SUPPORT O F A CONTINUOUS FUNCTION 6
1.2.2 SPACE ) ( Q ) 9
1.2.3 SPACE ) M ( Q ) 13
1.2.4 SPACE D X ( & ) 13
1.2.5 PROPERTIES O F 3)(Q) 14
1.3 SPACE .)'(2) O F (SCHWARTZ) DISTRIBUTIONS 25
1.3.1 ALGEBRAIC DUAL SPACE *(2) 25
1.3.2 DISTRIBUTIONS AND THE SPACE )'(Q) O F DISTRIBUTIONS ON Q, . . .
26 1.3.3 CHARACTERIZATION, ORDER AND EXTENSION O F A DISTRIBUTION . . .
. 27
1.3.4 EXAMPLES O F DISTRIBUTIONS 29
1.3.5 DISTRIBUTION DEFINED ON TEST SPACE )(2) O F COMPLEX-VALUED
FUNCTIONS 40
1.4 SOME MORE EXAMPLES O F INTERESTING DISTRIBUTIONS 41
1.5 MULTIPLICATION O F DISTRIBUTIONS BY C 00 -FUNCTIONS 51
1.6 PROBLEM O F DIVISION O F DISTRIBUTIONS 54
1.7 EVEN, ODD AND POSITIVE DISTRIBUTIONS 57
1.8 CONVERGENCE O F SEQUENCES O F DISTRIBUTIONS IN D'(2) 59
1.9 CONVERGENCE O F SERIES O F DISTRIBUTIONS IN $ ) ' ( Q ) 67
1.10 IMAGES O F DISTRIBUTIONS DUE TO CHANGE O F VARIABLES, HOMOGENEOUS,
INVARIANT, SPHERICALLY SYMMETRIC, CONSTANT DISTRIBUTIONS 68
1.10.1 PERIODIC DISTRIBUTIONS 75
1.11 PHYSICAL DISTRIBUTIONS VERSUS MATHEMATICAL DISTRIBUTIONS 84
1.11.1 PHYSICAL INTERPRETATION O F MATHEMATICAL DISTRIBUTIONS 84 1.11.2
LOAD INTENSITY 85
1.11.3 ELECTRICAL CHARGE DISTRIBUTION 88
1.11.4 SIMPLE LAYER AND DOUBLE LAYER DISTRIBUTIONS 90
1.11.5 RELATION WITH PROBABILITY DISTRIBUTION [7] 94
HTTP://D-NB.INFO/1018441549
IMAGE 2
XIV
CONTENTS
2 DIFFERENTIATION O F DISTRIBUTIONS AND APPLICATION O F DISTRIBUTIONAL
DERIVATIVES 96
2.1 INTRODUCTION: AN INTEGRAL DEFINITION O F DERIVATIVES O F C 1
-FUNCTIONS . . . 96 2.2 DERIVATIVES O F DISTRIBUTIONS 100
2.2.1 HIGHER-ORDER DERIVATIVES OF DISTRIBUTIONS T 101
2.3 DERIVATIVES O F FUNCTIONS IN THE SENSE O F DISTRIBUTION 102
2.4 CONDITIONS UNDER WHICH THE TWO NOTIONS O F DERIVATIVES OF FUNCTIONS
COINCIDE 119
2.5 DERIVATIVE O F PRODUCT A T WITH T 6 ,)'(2) AND A E C(F2) 121
2.6 PROBLEM O F DIVISION O F DISTRIBUTION REVISITED 125
2.7 PRIMITIVES O F A DISTRIBUTION AND DIFFERENTIAL EQUATIONS 131
2.8 PROPERTIES O F DISTRIBUTIONS WHOSE DISTRIBUTIONAL DERIVATIVES ARE
KNOWN 141 2.9 CONTINUITY O F DIFFERENTIAL OPERATOR D A : S)'(Q) -
)'(Q) 142
2.10 DELTA-CONVERGENT SEQUENCES O F FUNCTIONS IN 2)'(M") 149
2.11 TERM-BY-TERM DIFFERENTIATION O F SERIES O F DISTRIBUTIONS 154
2.12 CONVERGENCE O F SEQUENCES O F C K ( Q ) (RESP. C*"*(2)) IN )'(Q)
. . . 173 2.13 CONVERGENCE O F SEQUENCES O F L P (2), 1 P OO, IN
S)'(Q ) 173
2.14 TRANSPOSE (OR FORMAL ADJOINT) O F A LINEAR PARTIAL DIFFERENTIAL
OPERATOR . 175 2.15 APPLICATIONS: SOBOLEV SPACES H M ( Q ) , W M ' P ( Q
) 177
2.15.1 SOBOLEV SPACES 177
2.15.2 SPACE H M ( Q . ) 178
2.15.3 EXAMPLES O F FUNCTIONS BELONGING TO OR NOT BELONGING TO H M ( Q )
182 2.15.4 SEPARABILITY O F H M { S L ) 184
2.15.5 GENERALIZED POINCARE INEQUALITY IN 186
2.15.6 SPACE H*(2) 187
2.15.7 SPACE 191
2.15.8 QUOTIENT SPACE H M ( Q ) / M 191
2.15.9 QUOTIENT SPACE H M ( Q ) / P M - \ 193
2.15.10 OTHER EQUIVALENT NORMS IN H M ( Q ) 194
2.15.11 DENSITY RESULTS 195
2.15.12 ALGEBRAIC INCLUSIONS (C) AND IMBEDDING ( ^ - ) RESULTS 195
2.15.13 SPACE W M , P ( Q ) WITH M E N , I P OO 196
2.15.14 SPACE W*' P (SL), 1 P OO 200
2.15.15 SPACE W ~ M , Q { L ) 203
2.15.16 QUOTIENT SPACE W M ' P ( Q ) / M F O R / N E N , 1 / ? O O .
. . . 203
2.15.17 DENSITY RESULTS 207
2.15.18 A NON-DENSITY RESULT 208
2.15.19 ALGEBRAIC INCLUSION C AND IMBEDDING ( ^ - ) RESULTS 209
2.15.20 SPACE W S ' P ( S L ) FOR ARBITRARY S S L 209
IMAGE 3
CONTENTS X V
3 DERIVATIVES O F PIECEWISE SMOOTH FUNCTIONS, GREEN'S FORMULA,
ELEMENTARY SOLUTIONS, APPLICATIONS TO SOBOLEV SPACES 211
3.1 DISTRIBUTIONAL DERIVATIVES O F PIECEWISE SMOOTH FUNCTIONS 211
3.1.1 CASE O F SINGLE VARIABLE (N = 1) 211
3.1.2 CASE O F TWO VARIABLES (N = 2) 215
3.1.3 CASE O F THREE VARIABLES (N = 3) 230
3.2 UNBOUNDED DOMAIN 2 C M", GREEN'S FORMULA 235
3.3 ELEMENTARY SOLUTIONS 238
3.4 APPLICATIONS 257
4 ADDITIONAL PROPERTIES O F )'(S2) 263
4.1 REFLEXIVITY O F )(2) AND DENSITY O F 3)(2) IN *)'(}) 263
4.2 CONTINUOUS IMBEDDING O F DUAL SPACES O F BANACH SPACES IN )'(&) . .
265 4.3 APPLICATIONS: SOBOLEV SPACES W ~ M ' Q ( Q . ) 269
4.3.1 SPACE W ~ M , Q ( Q ) , I Q OO, M * N 273
5 LOCAL PROPERTIES, RESTRICTIONS, UNIFICATION PRINCIPLE, SPACE 6'(M") O
F DISTRIBUTIONS WITH COMPACT SUPPORT 280
5.1 NULL DISTRIBUTION IN AN OPEN SET 280
5.2 EQUALITY O F DISTRIBUTIONS IN AN OPEN SET 280
5.3 RESTRICTION O F A DISTRIBUTION TO AN OPEN SET 280
5.4 UNIFICATION PRINCIPLE 283
5.5 SUPPORT O F A DISTRIBUTION 285
5.6 DISTRIBUTIONS WITH COMPACT SUPPORT 286
5.7 SPACE 6 ' ( R " ) O F DISTRIBUTIONS WITH COMPACT SUPPORT 287
5.7.1 SPACE S(M") 287
5.7.2 SPACE S ' ( R N ) 288
5.8 DEFINITION O F ( T , J ) FOR (P E C ( R " ) AND T E WITH
NON-COMPACT SUPPORT 296
6 CONVOLUTION O F DISTRIBUTIONS 298
6.1 TENSOR PRODUCT 298
6.2 CONVOLUTION O F FUNCTIONS 303
6.3 CONVOLUTION O F TWO DISTRIBUTIONS 315
6.4 REGULARIZATION O F DISTRIBUTIONS BY CONVOLUTION 327
6.5 APPROXIMATION O F DISTRIBUTIONS BY C-FUNCTIONS 329
6.6 CONVOLUTION O F SEVERAL DISTRIBUTIONS 331
6.7 DERIVATIVES O F CONVOLUTIONS, CONVOLUTION O F DISTRIBUTIONS ON A
CIRCLE T AND THEIR FOURIER SERIES REPRESENTATIONS ON T 333
6.8 APPLICATIONS 349
6.9 CONVOLUTION EQUATIONS (SEE ALSO SECTION 8.7, CHAPTER 8) . . . . T. .
. 364
IMAGE 4
X V I CONTENTS
6.10 APPLICATION O F CONVOLUTIONS IN ELECTRICAL CIRCUIT ANALYSIS AND
HEAT FLOW
PROBLEMS 375
6.10.1 ELECTRIC CIRCUIT ANALYSIS PROBLEM [7] 375
6.10.2 EXCITATIONS AND RESPONSES DEFINED BY SEVERAL FUNCTIONS OR
DISTRIBUTIONS [7] 380
7 FOURIER TRANSFORMS O F FUNCTIONS O F L 1 (R") AND S(M") 383
7.1 FOURIER TRANSFORMS O F INTEGRABLE FUNCTIONS IN L 1 (IR' ! ) 383
7.2 SPACE O F INFINITELY DIFFERENTIABLE FUNCTIONS WITH RAPID DECAY AT
INFINITY 405
7.2.1 SPACE S ( R " ) 407
7.3 CONTINUITY O F LINEAR MAPPING FROM S ( R " ) INTO S ( R " ) 412
7.4 IMBEDDING RESULTS 413
7.5 DENSITY RESULTS 415
7.6 FOURIER TRANSFORM O F FUNCTIONS O F S ( R " ) 417
7.7 FOURIER INVERSION THEOREM IN S ( R " ) 418
8 FOURIER TRANSFORMS O F DISTRIBUTIONS AND SOBOLEV SPACES O F ARBITRARY
ORDER H S ( R " ) 423
8.1 MOTIVATION FOR A POSSIBLE DEFINITION O F THE FOURIER TRANSFORM OF A
DISTRIBUTION 423
8.2 SPACE S"(M") O F TEMPERED DISTRIBUTIONS 424
8.2.1 TEMPERED DISTRIBUTIONS 424
8.2.2 SPACE S'(R") 426
8.2.3 EXAMPLES O F TEMPERED DISTRIBUTIONS O F S ' ( R " ) 426
8.2.4 CONVERGENCE O F SEQUENCES IN S ' ( R " ) 429
8.2.5 DERIVATIVES O F TEMPERED DISTRIBUTIONS 432
8.3 FOURIER TRANSFORM O F TEMPERED DISTRIBUTIONS 435
8.3.1 FOURIER TRANSFORMS O F DIRAC DISTRIBUTIONS AND THEIR DERIVATIVES
438 8.3.2 INVERSION THEOREM FOR FOURIER TRANSFORMS ON S ' ( R " ) 440
8.3.3 FOURIER TRANSFORM O F EVEN AND ODD TEMPERED DISTRIBUTIONS . . .
441 8.4 FOURIER TRANSFORM O F DISTRIBUTIONS WITH COMPACT SUPPORT 445
8.5 FOURIER TRANSFORM O F CONVOLUTION O F DISTRIBUTIONS 450
8.5.1 FOURIER TRANSFORMS O F CONVOLUTIONS 451
8.6 DERIVATIVES O F FOURIER TRANSFORMS AND FOURIER TRANSFORMS OF
DERIVATIVES O F TEMPERED DISTRIBUTIONS 458
8.7 FOURIER TRANSFORM METHODS FOR DIFFERENTIAL EQUATIONS AND ELEMENTARY
SOLUTIONS IN S ' ( R " ) 476
8.8 LAPLACE TRANSFORM O F DISTRIBUTIONS ON R 492
8.8.1 SPACE '+ 492
8.8.2 DISTRIBUTION T ~ L E S ) ' + (SEE ALSO CONVOLUTION ALGEBRA A =
)'+ (6.9.15B)) 496
IMAGE 5
CONTENTS X V L L
8.8.3 INVERSE DC _1 O F LAPLACE TRANSFORM 497
8.9 APPLICATIONS 502
8.9.1 SOBOLEV SPACES H S ( R N ) 502
8.9.2 IMBEDDING RESULT 503
8.9.3 SOBOLEV SPACES H M ( M N ) O F INTEGRAL ORDER M ON R " 507
8.9.4 SOBOLEV'S IMBEDDING THEOREM (SEE ALSO IMBEDDING RESULTS IN SECTION
8.12) 512
8.9.5 IMBEDDING RESULT: S ( R " ) -* H S ( R " ) 521
8.9.6 DENSITY RESULTS H S ( R " ) 522
8.9.7 DUAL SPACE ( H S ( R N ) ) ' 523
8.9.8 TRACE PROPERTIES O F ELEMENTS O F H S ( R N ) 526
8.10 SOBOLEV SPACES ON 2 ^ R " REVISITED 546
8.10.1 SPACE H S ( Q ) WITH S E R, Q R " 546
8.10.2 M-EXTENSION PROPERTY O F 2 550
8.10.3 W-EXTENSION PROPERTY O F R^_ 558
8.10.4 M-EXTENSION PROPERTY O F C M -REGULAR DOMAINS 2 569
8.10.5 SPACE H S ( Q ) WITH S E R + , C L " 573
8.10.6 DENSITY RESULTS IN H S ( Q ) 578
8.10.7 DUAL SPACE H ~ S ( Q ) 579
8.10.8 SPACE H Q ( Q ) WITH S 0 579
8.10.9 SPACE H ~ S ( Q ) WITH S 0 580
8.10.10 SPACE W /,J,/ '(^) FOR REAL S 0 AND L P O O 580
8.10.11 SPACE // 0 (J2) WITH S 0 585
8.10.12 DUAL SPACE ( H Q 0 ( Q ) ) ' FOR 5 0 591
8.10.13 SPACE H / QQ /'(^) FOR S 0, 1 P OO 591
8.10.14 RESTRICTIONS O F DISTRIBUTIONS IN SOBOLEV SPACES 593
8.10.15 DIFFERENTIATION O F DISTRIBUTIONS IN H S (2) WITH J E R 598
8.10.16 DIFFERENTIATION O F DISTRIBUTIONS U * H S (2) WITH S 0 . . .
. 601 8.11 COMPACTNESS RESULTS IN SOBOLEV SPACES 605
8.11.1 COMPACT IMBEDDING RESULTS IN H S (Q.), H Q ( 2) AND // Q 0(2) .
616 8.12 SOBOLEV'S IMBEDDING RESULTS 617
8.12.1 COMPACT IMBEDDING RESULTS 632
8.13 SOBOLEV SPACES H S ( T ) , W S ' P { T ) ON A MANIFOLD BOUNDARY T
634
8.13.1 SURFACE INTEGRALS ON BOUNDARY T O F BOUNDED FLCL" 634
8.13.2 ALTERNATIVE DEFINITION O F H S ( T ) WITH T E C M -CLASS (RESP.
C-CLASS) 637
8.13.3 SPACE H S ( T ) (S 0) WITH T IN C M -CLASS (RESP. C-CLASS) .
638 8.13.4 SOBOLEV SPACES ON BOUNDARY CURVES F IN R 2 641
8.13.5 SPACES FOR POLYGONAL SIDES T,- E C-CLASS,
1 I N 651
IMAGE 6
XV111
CONTENTS
8.14 TRACE RESULTS IN SOBOLEV SPACES ON 2 M" 651
8.14.1 TRACE RESULTS IN 652
8.14.2 TRACE RESULTS IN H M ( Q . ) WITH BOUNDED DOMAIN C M " . . . 654
8.14.3 TRACE RESULTS IN H^'^-SPACES 670
8.14.4 TRACE RESULTS FOR POLYGONAL DOMAINS 2 C M 2 6 7 2
8.14.5 TRACE RESULTS FOR BOUNDED DOMAINS WITH CURVILINEAR POLYGONAL
BOUNDARY T IN M 2 685
8.14.6 TRACES O F NORMAL COMPONENTS IN L P ( D I V ; 2) 686
8.14.7 TRACE THEOREMS BASED ON GREEN'S FORMULA 691
8.14.8 TRACES ON TO C T 710
9 VECTOR-VALUED DISTRIBUTIONS 712
9.1 MOTIVATION 712
9.2 VECTOR-VALUED FUNCTIONS 712
9.3 SPACES O F VECTOR-VALUED FUNCTIONS 715
9.4 VECTOR-VALUED DISTRIBUTIONS 718
9.5 DERIVATIVES O F VECTOR-VALUED DISTRIBUTIONS 723
9.6 APPLICATIONS 724
9.6.1 SPACE E(0, T ; V , W ) 725
9.6.2 HILBERT SPACE WJ (0, T; V ) 725
9.6.3 HILBERT SPACE W 2 (0, T ; V ) 728
9.6.4 GREEN'S FORMULA 729
A FUNCTIONAL ANALYSIS (BASIC RESULTS) 731
A.O PRELIMINARY RESULTS 731
A.0.1 AN IMPORTANT RESULT ON LOGICAL IMPLICATION ( = ) AND
NON-IMPLICATION (=^4) 731
A.0.2 SUPREMUM (L.U.B.) AND INFIMUM (G.L.B.) 732
A.0.3 METRIC SPACES AND IMPORTANT RESULTS THEREIN 732
A.0.4 IMPORTANT SUBSETS O F A METRIC SPACE X = (X, D ) 735
A.0.5 COMPACT SETS IN M" WITH THE USUAL METRIC D.2 737
A.0.6 ELEMENTARY PROPERTIES O F FUNCTIONS O F REAL VARIABLES 738
A.0.7 LIMIT O F A FUNCTION AT A CLUSTER POINT XO E K " 738
A.O.8 LIMIT SUPERIOR AND LIMIT INFERIOR O F A SEQUENCE IN M 739
A.0.9 POINTWISE AND UNIFORM CONVERGENCE O F SEQUENCES O F FUNCTIONS 740
A.0.10 CONTINUITY AND UNIFORM CONTINUITY O F / E 3? (1) 740
A . L IMPORTANT PROPERTIES O F CONTINUOUS FUNCTIONS 741
A. 1.1 SOME REMARKABLE PROPERTIES ON COMPACT SETS IN R " 741
A.1.2 C^(2)-PARTITION O F UNITY ON COMPACT SET K C C 2 C R " . . 741
A.1.3 CONTINUOUS EXTENSION THEOREMS 741
A.2 FINITE AND INFINITE DIMENSIONAL LINEAR SPACES 743
A.2.1 LINEAR SPACES 743
IMAGE 7
CONTENTS X I X
A.2.2 LINEAR FUNCTIONALS 746
A.2.3 LINEAR OPERATORS 747
A.3 NORMED LINEAR SPACES 748
A.3.1 SEMI-NORM AND NORM 748
A.3.2 CLOSED SUBSPACE, DENSE SUBSPACE, BANACH SPACE AND ITS SEPARABILITY
750
A.4 BANACH SPACES O F CONTINUOUS FUNCTIONS 750
A.4.1 BANACH SPACES C(2), C K (2) 750
A.5 BANACH SPACES C 0 ' ^ ^ ) , 0 A 1, O F HOLDER CONTINUOUS
FUNCTIONS . 753 A.5.1 HOLDER CONTINUITY AND LIPSCHITZ CONTINUITY 753
A.5.2 HOLDER SPACE C"*(2) 754
A.5.3 SPACE C K ' X ( U ) , 0 A 1 754
A.6 QUOTIENT SPACE V / M 756
A.7 CONTINUOUS LINEAR FUNCTIONALS ON NORMED LINEAR SPACES 756
A.7.1 SPACE V ' 756
A.7.2 HAHN-BANACH EXTENSION O F LINEAR FUNCTIONALS IN ANALYTIC FORM 757
A.7.3 CONSEQUENCES O F THE HAHN-BANACH THEOREM IN NORMED LINEAR SPACES
758
A.8 CONTINUOUS LINEAR OPERATORS ON NORMED LINEAR SPACES 760
A.8.1 SPACE X ( V ; W ) 760
A.8.2 CONTINUOUS EXTENSION O F CONTINUOUS LINEAR OPERATORS BY DENSITY
761
A.8.3 ISOMORPHISMS AND ISOMETRIC ISOMORPHISMS 762
A.8.4 GRAPH OF AN OPERATOR A E .{V ; W ) AND GRAPH NORM 762
A.9 REFLEXIVITY O F BANACH SPACES 763
A. 10 STRONG, WEAK AND WEAK-* CONVERGENCE IN BANACH SPACE V 763
A. 10.1 STRONG CONVERGENCE - 763
A. 10.2 WEAK CONVERGENCE 764
A.10.3 WEAK-* CONVERGENCE - -* IN BANACH SPACE V ' 764
A. 11 COMPACT LINEAR OPERATORS IN BANACH SPACES 764
A. 12 HILBERT SPACE V 765
A . O DUAL SPACE V ' O F A HILBERT SPACE V, REFLEXIVITY O F V 768
A.14 STRONG, WEAK AND WEAK-* CONVERGENCES IN A HILBERT SPACE 769
A. 15 SELF-ADJOINT AND UNITARY OPERATORS IN HILBERT SPACE V 769
A. 16 COMPACT LINEAR OPERATORS IN HILBERT SPACES 769
B L P -SPACES 771
B.L LEBESGUE MEASURE JX ON M" 771
B . L . L LEBESGUE-MEASURABLE SETS IN M N 7 7 1
B.L.2 SETS WITH ZERO (LEBESGUE) MEASURE IN M" 772
B.L.3 PROPERTY P HOLDS ALMOST EVERYWHERE (A.E.) ON 2 775
IMAGE 8
X X
CONTENTS
B.2 SPACE M ( L ) O F LEBESGUE-MEASURABLE FUNCTIONS ON 2 776
B.2.1 MEASURABLE FUNCTIONS AND SPACE CM(2) 776
B.2.2 POINTWISE CONVERGENCE A.E. ON Q. 778
B.3 LEBESGUE INTEGRALS AND THEIR IMPORTANT PROPERTIES 778
B.3.1 LEBESGUE INTEGRAL OF A BOUNDED FUNCTION ON BOUNDED DOMAIN 2 778
B.3.2 IMPORTANT PROPERTIES O F LEBESGUE INTEGRALS (KOLMOGOROV AND FOMIN
[20]) 780
B.3.3 SOME IMPORTANT APPROXIMATION AND DENSITY RESULTS IN L 1 (2) . 784
B.4 SPACES L P ( I 2 ) , 1 P OO 788
B.4.1 BASIC PROPERTIES 788
B.4.2 DUAL SPACE (L / '(2)) / O F L P ( Q ) FOR 1 P OO 794
B.4.3 SPACE L 2 ( Q ) 797
B.4.4 SOME NEGATIVE PROPERTIES O F L (2) 798
B.4.5 SOME NICE PROPERTIES O F L (2) 799
B.4.6 SPACE LJ^ C (2) INCLUSION RESULTS 799
C OPEN COVER AND PARTITION O F UNITY 803
C.L C^(2)-PARTITION O F UNITY THEOREM FOR COMPACT SETS 803
D BOUNDARY GEOMETRY 808
D . L BOUNDARY GEOMETRY 808
D . L . L LOCALLY ONE-SIDED AND TWO-SIDED BOUNDED DOMAINS 2 808 D.L.2
STAR-SHAPED DOMAIN 2 808
D.L.3 CONE PROPERTY AND UNIFORM CONE PROPERTY 809
D.L.4 SEGMENT PROPERTY 811
D.2 CONTINUITY AND DIFFERENTIAL PROPERTIES O F A BOUNDARY 812
D.2.1 CONTINUITY AND DIFFERENTIAL PROPERTIES 812
D.2.2 OPEN COVER O F T, LOCAL COORDINATE SYSTEMS {/"}"_J
AND MAPPINGS { / R }^L I 813
D.2.3 PROPERTIES O F THE MAPPINGS / R : M " - 1 - R, 1 R N . .
814
D.3 ALTERNATIVE DEFINITION O F LOCALLY ONE-SIDED DOMAIN 816
D.4 ALTERNATIVE DEFINITION O F CONTINUITY AND DIFFERENTIAL PROPERTIES O
F Q. AS A MANIFOLD IN R " 817
D.5 ATLAS/LOCAL CHARTS O F T 818
BIBLIOGRAPHY 819
INDEX 823 |
| any_adam_object | 1 |
| author | Bhattacharyya, Pulin K. |
| author_GND | (DE-588)1025884841 |
| author_facet | Bhattacharyya, Pulin K. |
| author_role | aut |
| author_sort | Bhattacharyya, Pulin K. |
| author_variant | p k b pk pkb |
| building | Verbundindex |
| bvnumber | BV040324684 |
| classification_rvk | SK 600 |
| ctrlnum | (OCoLC)797182098 (DE-599)DNB1018441549 |
| dewey-full | 510 515.782 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics 515 - Analysis |
| dewey-raw | 510 515.782 |
| dewey-search | 510 515.782 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV040324684 |
| illustrated | Illustrated |
| indexdate | 2024-08-21T00:02:03Z |
| institution | BVB |
| isbn | 3110269279 9783110269277 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025179245 |
| oclc_num | 797182098 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-83 DE-11 DE-384 DE-20 DE-824 DE-188 |
| owner_facet | DE-19 DE-BY-UBM DE-83 DE-11 DE-384 DE-20 DE-824 DE-188 |
| physical | XXXVIII, 833 S. graph. Darst. 240 mm x 170 mm |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | De Gruyter |
| record_format | marc |
| series2 | De Gruyter textbook |
| spelling | Bhattacharyya, Pulin K. Verfasser (DE-588)1025884841 aut Distributions generalized functions with applications in Sobolev spaces Pulin Kumar Bhattacharyya Berlin [u.a.] De Gruyter 2012 XXXVIII, 833 S. graph. Darst. 240 mm x 170 mm txt rdacontent n rdamedia nc rdacarrier De Gruyter textbook Distribution Funktionalanalysis (DE-588)4070505-5 gnd rswk-swf Sobolev-Raum (DE-588)4055345-0 gnd rswk-swf Distribution Funktionalanalysis (DE-588)4070505-5 s Sobolev-Raum (DE-588)4055345-0 s DE-604 Erscheint auch als Online-Ausgabe 978-3-11-026929-1 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3950792&prov=M&dok%5Fvar=1&dok%5Fext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025179245&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bhattacharyya, Pulin K. Distributions generalized functions with applications in Sobolev spaces Distribution Funktionalanalysis (DE-588)4070505-5 gnd Sobolev-Raum (DE-588)4055345-0 gnd |
| subject_GND | (DE-588)4070505-5 (DE-588)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 | Distribution Funktionalanalysis (DE-588)4070505-5 gnd Sobolev-Raum (DE-588)4055345-0 gnd |
| topic_facet | Distribution Funktionalanalysis Sobolev-Raum |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3950792&prov=M&dok%5Fvar=1&dok%5Fext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025179245&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bhattacharyyapulink distributionsgeneralizedfunctionswithapplicationsinsobolevspaces |