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