Operators commuting with translations, and systems of difference equations

Laczkovich, Miklós. "Operators commuting with translations, and systems of difference equations." Colloquium Mathematicae 80.1 (1999): 1-22. <http://eudml.org/doc/210701>.

@article{Laczkovich1999,
abstract = {Let $\{\mathcal \{B\}\} =\{f:ℝ → ℝ: f is bounded\}$, and $\{\mathcal \{M\}\} =\{f:ℝ → ℝ: f is Lebesgue measurable\}$. We show that there is a linear operator $Φ :\{\mathcal \{B\}\} → \{\mathcal \{M\}\}$ such that Φ(f)=f a.e. for every $f ∈ \{\mathcal \{B\}\} ∩ \{\mathcal \{M\}\}$, and Φ commutes with all translations. On the other hand, if $Φ : \{\mathcal \{B\}\} → \{\mathcal \{M\}\}$ is a linear operator such that Φ(f)=f for every $f ∈ \{\mathcal \{B\}\} ∩ \{\mathcal \{M\}\}$, 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 $f(x)=e^\{cx\}$, 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.},
author = {Laczkovich, Miklós},
journal = {Colloquium Mathematicae},
keywords = {operators commuting with translation; systems of difference equations},
language = {eng},
number = {1},
pages = {1-22},
title = {Operators commuting with translations, and systems of difference equations},
url = {http://eudml.org/doc/210701},
volume = {80},
year = {1999},
}

TY - JOUR
AU - Laczkovich, Miklós
TI - Operators commuting with translations, and systems of difference equations
JO - Colloquium Mathematicae
PY - 1999
VL - 80
IS - 1
SP - 1
EP - 22
AB - Let ${\mathcal {B}} ={f:ℝ → ℝ: f is bounded}$, and ${\mathcal {M}} ={f:ℝ → ℝ: f is Lebesgue measurable}$. We show that there is a linear operator $Φ :{\mathcal {B}} → {\mathcal {M}}$ such that Φ(f)=f a.e. for every $f ∈ {\mathcal {B}} ∩ {\mathcal {M}}$, and Φ commutes with all translations. On the other hand, if $Φ : {\mathcal {B}} → {\mathcal {M}}$ is a linear operator such that Φ(f)=f for every $f ∈ {\mathcal {B}} ∩ {\mathcal {M}}$, 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 $f(x)=e^{cx}$, 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.
LA - eng
KW - operators commuting with translation; systems of difference equations
UR - http://eudml.org/doc/210701
ER -