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Nearstandardness on a finite set

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Lyantse V.

Instytut Matematyczny Polskiej Akademii Nauk

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

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1 Lyantse V.

1 1997

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Nearstandardness on a finite set