Foundations of Superposition Theory vol

David Carfi

Description

Foundations of Superposition Theory vol. 1 Superposition Algebra in the Space of Tempered Distributions and Applications to Economics and Physics David Carfı Edizioni Il Gabbiano 2010

• fourier transforms


• esummable families


• superposition


• slinear hulls


• relationships among


• frechet spaces


• topological supplements


• dirac basis


• dieudonne-schwartz theorem


• sclosed subsets

Sujets

• Fourier transform
• Superposition
• Fréchet space

Informations

• Publié par: pefav
• Nombre de lectures: 57
• Langue: English

Extrait

Foundations of Superposition Theory
vol. 1
Superposition Algebra in the Space of Tempered Distributions
and Applications to Economics and Physics
David Carf
Edizioni Il Gabbiano 20102Contents
I Introductions and preliminaries 11
0.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
0.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1 Preliminaries 17
1.1 Topological homomorphisms . . . . . . . . . . . . . . . . . . 18
1.2 Direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Topological supplements . . . . . . . . . . . . . . . . . . . . . 20
1.4 Right and Left inverses . . . . . . . . . . . . . . . . . . . . . . 21
1.5 Homomorphisms among Frechet spaces . . . . . . . . . . . 21
1.6 Dieudonne-Schwartz theorem . . . . . . . . . . . . . . . . . . 22
1.7 Banach-Steinhaus in barreled spaces . . . . . . . . . . . . . 23
1.8 Tempered distributions . . . . . . . . . . . . . . . . . . . . . 24
1.8.1 Test functions . . . . . . . . . . . . . . . . . . . . . . . 24
1.8.2 Tempered distributions . . . . . . . . . . . . . . . . . 25
1.9 Fourier transforms onS . . . . . . . . . . . . . . . . . . . . 25n
01.10 F onS . . . . . . . . . . . . . . . . . . . . 26n
II Superpositions 27
2 Summable families 29
2.1 Families of distributions . . . . . . . . . . . . . . . . . . . . . 29
S2.2 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
S2.3 Family generated by an operator . . . . . . . . . . . . . . . 31
S2.4 The operator generated by an family . . . . . . . . . . . . 32
S2.5 Characterizations of families . . . . . . . . . . . . . . . . . 33
2.6 Characterization of transposability . . . . . . . . . . . . . . 36
D2.7 of families (*) . . . . . . . . . . . . . . . 37
E E2.8 Families and summable families . . . . . . . . . . . . . . 37
3 Superpositions 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.1.1 The wonderful Dirac basis . . . . . . . . . . . . . . . 41
3.1.2 A dangerous expression . . . . . . . . . . . . . . . . . 42
34 CONTENTS
3.1.3 Toward a possible solution . . . . . . . . . . . . . . . 43
3.1.4 Inadequacy of convolutions . . . . . . . . . . . . . . . 44
3.1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 45
S 03.2 Superpositions of families inS . . . . . . . . . . . . . . . 45n
3.3 An alternative de nition of superposition . . . . . . . . . . 46
3.4 Superpositions of an E-family (*) . . . . . . . . . . . . . . . 48
3.5 Algebraic properties of superpositions . . . . . . . . . . . . 49
3.5.1 Bilinearity of superposition operator . . . . . . . . . 49
3.5.2 Selection property of the Dirac distributions . . . . 50
S3.5.3 Linear combination of an family . . . . . . . . . . . 51
S4 Linearity 53
4.1 Continuity of superposition operators . . . . . . . . . . . . 53
4.2 Superposition operator of a distribution . . . . . . . . . . . 54
S4.3 Linearity of superpositions . . . . . . . . . . . . . . . . . . . 56
4.4 Generalized distributive laws (*) . . . . . . . . . . . . . . . . 58
S5 Families inS 61n
5.1 Families inS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61n
S5.2 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
S5.3 Transpose of an Family . . . . . . . . . . . . . . . . . . . . . 64
S5.4 Operator of an family . . . . . . . . . . . . . . . . . . . . . . 65
S5.5 Continuity of operators of families . . . . . . . . . . . . . 66
6 Superpositions inS 69n
S6.1 Linear combinations . . . . . . . . . . . . . . . . . . . . . . . 69
S6.2 Superposition operator of families . . . . . . . . . . . . . 70
S6.3 Summability of families . . . . . . . . . . . . . . . . . . . . 71
6.4 Transpose family . . . . . . . . . . . . . . . . . . . . . . . . . . 72
S6.5 Linear functional . . . . . . . . . . . . . . . . . . . . . . . . . 72
S 06.6 Superpositions of families inS . . . . . . . . . . . . . . . 73n
S6.7 Linear superpositions of operators . . . . . . . . . . . . . . 75
7 First applications 77
7.1 The Fourier expansion theorem . . . . . . . . . . . . . . . . 77
7.2 Convolution as superpositions . . . . . . . . . . . . . . . . . 80
7.3 Some expressions of Dirac Calculus . . . . . . . . . . . . . . 82
7.3.1 The expansion of a vector in the Dirac basis . . . . 83
7.3.2 Fourier expansions and the momentum operator . 84
7.4 Some extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 85
SIII Linear Algebra and Geometry 89
S S8 Linear hulls of families 91
S8.1 Linear hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91CONTENTS 5
S8.2 Algebraic properties of linear hulls . . . . . . . . . . . . . 92
S8.3 Systems of generators . . . . . . . . . . . . . . . . . . . . . . 94
8.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
S8.4 Topological properties of linear hulls . . . . . . . . . . . . 96
S8.5 Closedness of linear hulls . . . . . . . . . . . . . . . . . . . 97
S8.5.1 Examples of systems of generators . . . . . . . . . 98
S8.6 Kernel of an family . . . . . . . . . . . . . . . . . . . . . . . 99
8.6.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . 100
S8.7 Linear hull of a subset . . . . . . . . . . . . . . . . . . . . . 101
9 Bases 103
S9.1 Linear independence . . . . . . . . . . . . . . . . . . . . . . 103
S9.2 Topology and linear independence . . . . . . . . . . . . . . 104
9.3 Uniqueness of representation . . . . . . . . . . . . . . . . . . 106
S9.4 Characterizations of linear independence . . . . . . . . . 106
S9.5 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
S9.6 Algebraic characterizations of bases . . . . . . . . . . . . 108
9.6.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
S9.7 Totality of bases . . . . . . . . . . . . . . . . . . . . . . . . . 109
S9.8 Topological characterizations of bases . . . . . . . . . . . 110
S9.9 Equivalent families . . . . . . . . . . . . . . . . . . . . . . . . 110
S10 Closedness 113
S10.1 Closed subsets . . . . . . . . . . . . . . . . . . . . . . . . . . 113
S10.2 hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.3 Relationships among di erent hulls . . . . . . . . . . . . . . 115
10.3.1 Relationships among linear hulls . . . . . . . . . . . 116
S10.3.2 Closure of subspaces . . . . . . . . . . . . . . . . . . 117
S10.3.3 among linear and closed hulls 117
S10.4 Unions of closed sets . . . . . . . . . . . . . . . . . . . . . . 119
S10.5 Closed linear hull of families . . . . . . . . . . . . . . . . . 119
S10.6 Closedness and topology . . . . . . . . . . . . . . . . . . . . 120
S11 Connectedness 123
S D11.1 Connected and connected sets . . . . . . . . . . . . . . . 123
S 011.1.1 Families inS containing a given distribution . . . 124n
S 011.1.2 F in starshaped subsets of S . . . . . . . . 124n
S 011.1.3 Connected subsets of S . . . . . . . . . . . . . . . . 126n
D 1L11.2 Closed sets (*) . . . . . . . . . . . . . . . . . . . . . . . . . 127
11.2.1 Preliminaries on the spaceD 1 . . . . . . . . . . . . . 127L
11.3 Sums of series as superpositions . . . . . . . . . . . . . . . . 1296 CONTENTS
SIV Linear operators 133
S12 Linear operators 135
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
S12.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
S 012.3 Op de ned on S . . . . . . . . . . . . . . . . . . . . 137n
S 012.4 Characterization of operators onS . . . . . . . . . . . . . 138n
S 012.5 Linear operators onS . . . . . . . . . . . . . . . . . . . . . 139n
S12.6 Examples of linear operators . . . . . . . . . . . . . . . . . 140
S12.6.1 The superposition operator of an family . . . . . . 140
12.6.2 Transpose operators . . . . . . . . . . . . . . . . . . . 141
S12.7 Characterization of linear operators . . . . . . . . . . . . 143
S13 Applications of linear operators 145
S13.1 Bases of subspaces . . . . . . . . . . . . . . . . . . . . . . . . 145
S13.2 of closed subspaces . . . . . . . . . . . . . . . . . . . 146
S13.3 Bases of barreled . . . . . . . . . . . . . . . . . . 148
S13.4 Superpositions of linear operators . . . . . . . . . . . . . 148
S S13.5 Linear operators and bases . . . . . . . . . . . . . . . . . . 150
S13.6 Invertibility of linear operators . . . . . . . . . . . . . . . . 151
S13.7 Linear operators on subspaces . . . . . . . . . . . . . . . . 152
S13.8 Compositions of linear operators . . . . . . . . . . . . . . . 153
S14 Homomorphisms 155
S14.1 . . . . . . . . . . . . . . . . . . . . . . . . . 156
S14.2 Injective linear homomorphisms . . . . . . . . . . . . . . . 157
S14.3 Surjective linear . . . . . . . . . . . . . . 158
S14.4 Stable families . . . . . . . . . . . . . . . . . . . . . . . . . . 159
S S14.5 Linear operators and closedness . . . . . . . . . . . . . . 160
S S14.6 Homomorphism and . . . . . . . . . . . . . . 162
S14.7 Invertibility of linear homomorphism . . . . . . . . . . . 162
S14.8 Left inverse of homomorphisms . . . . . . . . . . . 163
S15 Green’s families 165
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
15.1.1 Green’s functions . . . . . . . . . . . . . . . . . . . . . 165
15.1.2 Motivations for Green’s families . . . . . . . . . . . . 166
015.2 Green’s families inS . . . .