Operators commuting with translations, and systems of difference equations

Laczkovich, Miklós

Colloquium Mathematicae, Tome 79 (1999), p. 1-22 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $ℬ = f : ℝ \to ℝ : f i s b o u n d e d$ , and $ℳ = f : ℝ \to ℝ : f i s L e b e s g u e m e a s u r a b l e$ . We show that there is a linear operator $Φ : ℬ \to ℳ$ such that Φ(f)=f a.e. for every $f \in ℬ \cap ℳ$ , and Φ commutes with all translations. On the other hand, if $Φ : ℬ \to ℳ$ is a linear operator such that Φ(f)=f for every $f \in ℬ \cap ℳ$ , then the group $G_{Φ}$ = a ∈ ℝ:Φ commutes with the translation by a is of measure zero and, assuming Martin’s axiom, is of cardinality less than continuum. Let Φ be a linear operator from $ℂ^{ℝ}$ into the space of complex-valued measurable functions. We show that if Φ(f) is non-zero for every $f (x) = e^{c x}$ , then $G_{Φ}$ must be discrete. If Φ(f) is non-zero for a single polynomial-exponential f, then $G_{Φ}$ is countable, moreover, the elements of $G_{Φ}$ are commensurable. We construct a projection from $ℂ^{ℝ}$ onto the polynomials that commutes with rational translations. All these results are closely connected with the solvability of certain systems of difference equations.

Publié le : 1999-01-01

Zbl 0929.28003

EUDML-ID : urn:eudml:doc:210701

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     author = {Mikl\'os Laczkovich},
     title = {Operators commuting with translations, and systems of difference equations},
     journal = {Colloquium Mathematicae},
     volume = {79},
     year = {1999},
     pages = {1-22},
     zbl = {0929.28003},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv80i1p1bwm}
}

Laczkovich, Miklós. Operators commuting with translations, and systems of difference equations. Colloquium Mathematicae, Tome 79 (1999) pp. 1-22. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv80i1p1bwm/

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