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Abstract
Let T be a finite set for which card T is a natural nonstandard number. The linear space $ℂ^T$ of complex-valued functions on T is nonstandard. For the analysis on $ℂ^T$ we need a concept of nearstandardness in this space. A version how to introduce such a concept is proposed. Some elementary examples are given.
Let T be a finite set for which card T is a natural nonstandard number. The linear space $ℂ^T$ of complex-valued functions on T is nonstandard. For the analysis on $ℂ^T$ we need a concept of nearstandardness in this space. A version how to introduce such a concept is proposed. Some elementary examples are given.
CONTENTS
Introduction.................................................................................................................5
0. Preliminary notes....................................................................................................7
0.1. Definitions...........................................................................................................7
0.2. ⟨nst⟩-condition....................................................................................................8
0.3. ⟨nst⟩-condition for linear operators.....................................................................9
0.4. Nearstandardness on ℬ(X;Y)............................................................................10
0.5. Strong and uniform nearstandardness.............................................................11
1. Standard filling.....................................................................................................13
1.1. Definition of a standard filling...........................................................................14
1.2. Charge spaces.................................................................................................15
1.3. Discrete interval...............................................................................................16
1.4. Exact inductors.................................................................................................18
1.5. Standard measure filling...................................................................................18
1.6. The embedding N → M.....................................................................................19
2. Standardness on $ℂ^𝕋$.......................................................................................20
2.1. The embedding $ℂ^𝕋 → L(T)$.........................................................................20
2.2. The inductor $Π:L(T) → ℂ^𝕋$...........................................................................21
2.3. Standard and nearstandard functions on $ℂ^𝕋$; standardized image.............23
2.4. Absolute continuity, integrability........................................................................23
2.5. Some "classical theorems"................................................................................25
2.6. Relation between the "discrete integral" and the ordinary one.........................26
3. The spaces ℍ and H............................................................................................26
3.1. Embedding and inductor...................................................................................27
3.2. Quasi-unity and the orthoprojector P................................................................28
3.3. Weak nearstandardness on ℍ.........................................................................30
4. Nearstandardness on ℬ(ℍ)..................................................................................31
4.1. The embedding Q and the inductor P...............................................................31
4.2. Exactness of P..................................................................................................31
4.3. Strong and uniform nearstandardness.............................................................32
4.4. Graph-nearstandardness.................................................................................34
4.5. ℬ₂-nearstandardness.......................................................................................35
5. Discrete Fourier transform...................................................................................39
5.1. The shift $U_θ$................................................................................................39
5.2. The operator $D_θ$.........................................................................................42
5.3. Discrete Riemann-Lebesgue lemma.................................................................44
5.4. A nearstandardness criterion...........................................................................46
5.5. Nearstandardness of the shift..........................................................................47
5.6. Nearstandardness of discrete differentiation....................................................49
5.7. Case a ~ +∞.....................................................................................................52
6. Application of equipment......................................................................................55
6.1. Induced equipment...........................................................................................56
6.2. H₋-nearstandardness.......................................................................................57
6.3. Example of equipment......................................................................................58
6.4. H₋-nearstandard operators..............................................................................59
6.5. H₋-nearstandardness of discrete differentiation...............................................61
References...............................................................................................................63
Introduction.................................................................................................................5
0. Preliminary notes....................................................................................................7
0.1. Definitions...........................................................................................................7
0.2. ⟨nst⟩-condition....................................................................................................8
0.3. ⟨nst⟩-condition for linear operators.....................................................................9
0.4. Nearstandardness on ℬ(X;Y)............................................................................10
0.5. Strong and uniform nearstandardness.............................................................11
1. Standard filling.....................................................................................................13
1.1. Definition of a standard filling...........................................................................14
1.2. Charge spaces.................................................................................................15
1.3. Discrete interval...............................................................................................16
1.4. Exact inductors.................................................................................................18
1.5. Standard measure filling...................................................................................18
1.6. The embedding N → M.....................................................................................19
2. Standardness on $ℂ^𝕋$.......................................................................................20
2.1. The embedding $ℂ^𝕋 → L(T)$.........................................................................20
2.2. The inductor $Π:L(T) → ℂ^𝕋$...........................................................................21
2.3. Standard and nearstandard functions on $ℂ^𝕋$; standardized image.............23
2.4. Absolute continuity, integrability........................................................................23
2.5. Some "classical theorems"................................................................................25
2.6. Relation between the "discrete integral" and the ordinary one.........................26
3. The spaces ℍ and H............................................................................................26
3.1. Embedding and inductor...................................................................................27
3.2. Quasi-unity and the orthoprojector P................................................................28
3.3. Weak nearstandardness on ℍ.........................................................................30
4. Nearstandardness on ℬ(ℍ)..................................................................................31
4.1. The embedding Q and the inductor P...............................................................31
4.2. Exactness of P..................................................................................................31
4.3. Strong and uniform nearstandardness.............................................................32
4.4. Graph-nearstandardness.................................................................................34
4.5. ℬ₂-nearstandardness.......................................................................................35
5. Discrete Fourier transform...................................................................................39
5.1. The shift $U_θ$................................................................................................39
5.2. The operator $D_θ$.........................................................................................42
5.3. Discrete Riemann-Lebesgue lemma.................................................................44
5.4. A nearstandardness criterion...........................................................................46
5.5. Nearstandardness of the shift..........................................................................47
5.6. Nearstandardness of discrete differentiation....................................................49
5.7. Case a ~ +∞.....................................................................................................52
6. Application of equipment......................................................................................55
6.1. Induced equipment...........................................................................................56
6.2. H₋-nearstandardness.......................................................................................57
6.3. Example of equipment......................................................................................58
6.4. H₋-nearstandard operators..............................................................................59
6.5. H₋-nearstandardness of discrete differentiation...............................................61
References...............................................................................................................63
Słowa kluczowe
Tematy
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne
tom/nr w serii:
369
Liczba stron
63
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXIX
Daty
wydano
1997
otrzymano
1995-06-06
poprawiono
1996-11-22
Twórcy
autor
- Department of Mathematics, University of Lviv, 1 Universytetska str., 290602 Lviv, Ukraine , wlanc@litech.lviv.ua
Bibliografia
- [1] S. Albeverio, J. Fenstad, R. Hoegh-Krohn and T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York, 1986.
- [2] Yu. M. Berezanskiĭ, G. F. Us and Z. G. Sheftel, Functional Analysis, Vyshcha shkola, Kiev, 1990 (in Russian).
- [3] P. Cartier et Y. Feneyrol-Perrin, Méthodes Infinitésimales en Analyse, Lecture Notes in Math. (to appear).
- [4] N. J. Cutland et al. (eds.), Nonstandard Analysis and its Applications, Cambridge Univ. Press, 1988.
- [5] N. J. Cutland et al. (eds.), Developments in Nonstandard Mathematics, Proc. CIMNS 1994, Pitman Res. Notes 336, Longman, 1995.
- [6] F. Diener et G. Reeb, Analyse Nonstandard, Hermann, Paris, 1989.
- [7] T. S. Kudryk, V. E. Lyantse and G. I. Chuiko, Nearstandardness on finite set, Mat. Studii 2 (1993), 25-34.
- [8] T. S. Kudryk, V. E. Lyantse and G. I. Chuiko, Nearstandard operators, Mat. Studii 3 (1994), 29-40.
- [9] A. G. Kusraev and S. S. Kutateladze, Nonstandard Methods of Analysis, Nauka, Novosibirsk, 1990 (in Russian).
- [10] P. A. Loeb, Conversion from nonstandard to standard measure spaces and applications to probability theory, Trans. Amer. Math. Soc. 211 (1975), 113-122.
- [11] R. Lutz and M. Goze, Nonstandard Analysis: a Practical Guide with Applications, Lecture Notes in Math. 881, Springer, 1981.
- [12] E. Nelson, Radically Elementary Probability Theory, Princeton Univ. Press, Princeton, N.J., 1987.
- [13] E. Nelson, Internal set theory: a new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (1977), 1165-1198.
- [14] A. Robert, Nonstandard Analysis, Wiley, 1988.
Języki publikacji
| EN |
Uwagi
1991 Mathematics Subject Classification: 03H05, 28E05, 47S20.
Identyfikator YADDA
bwmeta1.element.zamlynska-f087cba7-2c16-4839-8fcb-2dfaa841a8c5
Identyfikatory
ISSN
0012-3862
Kolekcja
DML-PL
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