AbstractLet T be a finite set for which card T is a natural nonstandard number. The linear space of complex-valued functions on T is nonstandard. For the analysis on we need a concept of nearstandardness in this space. A version how to introduce such a concept is proposed. Some elementary examples are given.
CONTENTSIntroduction.................................................................................................................50. 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.............................................................111. 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.....................................................................................192. Standardness on .......................................................................................20 2.1. The embedding .........................................................................20 2.2. The inductor ...........................................................................21 2.3. Standard and nearstandard functions on ; 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.........................263. 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.........................................................................304. 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.......................................................................................355. Discrete Fourier transform...................................................................................39 5.1. The shift ................................................................................................39 5.2. The operator .........................................................................................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 +∞.....................................................................................................526. 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...............................................61References...............................................................................................................63
1991 Mathematics Subject Classification: 03H05, 28E05, 47S20.
@book{bwmeta1.element.zamlynska-f087cba7-2c16-4839-8fcb-2dfaa841a8c5, author = {Lyantse V.}, title = {Nearstandardness on a finite set}, series = {GDML\_Books}, publisher = {Instytut Matematyczny Polskiej Akademi Nauk}, address = {Warszawa}, year = {1997}, zbl = {0890.03038}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.zamlynska-f087cba7-2c16-4839-8fcb-2dfaa841a8c5} }
Lyantse V. Nearstandardness on a finite set. GDML_Books (1997), http://gdmltest.u-ga.fr/item/bwmeta1.element.zamlynska-f087cba7-2c16-4839-8fcb-2dfaa841a8c5/