Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na https://bibliotekanauki.pl
Preferencje help
Polish version English version Język
Widoczny [Schowaj] Abstrakt
Liczba wyników

Znaleziono wyników: 5

Liczba wyników na stronie
first rewind previous Strona / 1 next fast forward last

Wyniki wyszukiwania

help Sortuj według:

help Ogranicz wyniki do:
first rewind previous Strona / 1 next fast forward last
1
Content available remote

Operators commuting with translations, and systems of difference equations

100%
Laczkovich M.
Colloquium Mathematicum
|
1999
|
tom 80
|
nr 1
1-22
EN
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.
2
Content available remote

On the linear Denjoy property of two-variable continuous functions

64%
Laczkovich M., Matszangosz Á.
Colloquium Mathematicum
|
2015
|
tom 141
|
nr 2
157-173
EN
The classical Denjoy-Young-Saks theorem gives a relation, here termed the Denjoy property, between the Dini derivatives of an arbitrary one-variable function that holds almost everywhere. Concerning the possible generalizations to higher dimensions, A. S. Besicovitch proved the following: there exists a continuous function of two variables such that at each point of a set of positive measure there exist continuum many directions, in each of which one Dini derivative is infinite and the other three are zero, thus violating the bilateral Denjoy property. Our aim is to show that for two-variable continuous functions it is possible that on a set of positive measure there exist directions in which even the one-sided Denjoy behaviour is violated. We construct continuous functions of two variables such that (i) both of its one-sided derivatives equal ∞ in continuum many directions on a set of positive measure, and (ii) all four directional Dini derivatives are finite and distinct in continuum many directions on a set of positive measure.
3
Content available remote

Ideal limits of sequences of continuous functions

64%
Laczkovich M., Recław I.
Fundamenta Mathematicae
|
2009
|
tom 203
|
nr 1
39-46
EN
We prove that for every Borel ideal, the ideal limits of sequences of continuous functions on a Polish space are of Baire class one if and only if the ideal does not contain a copy of Fin × Fin. In particular, this is true for $F_{σδ}$ ideals. In the proof we use Borel determinacy for a game introduced by C. Laflamme.
4
Content available remote

Strong Fubini properties for measure and category

64%
Ciesielski K., Laczkovich M.
Fundamenta Mathematicae
|
2003
|
tom 178
|
nr 2
171-188
EN
Let (FP) abbreviate the statement that $∫_{0}^{1} (∫_{0}^{1} fdy)dx = ∫_{0}^{1} (∫_{0}^{1} fdx)dy$ holds for every bounded function f: [0,1]² → ℝ whenever each of the integrals involved exists. We shall denote by (SFP) the statement that the equality above holds for every bounded function f: [0,1]² → ℝ having measurable vertical and horizontal sections. It follows from well-known results that both of (FP) and (SFP) are independent of the axioms of ZFC. We investigate the logical connections of these statements with several other strong Fubini type properties of the ideal of null sets. In particular, we establish the equivalence of (SFP) to the nonexistence of certain sets with paradoxical properties, a phenomenon that was already known for (FP). We also give the category analogues of these statements and, whenever possible, we try to put the statements in a setting of general ideals as initiated by Recław and Zakrzewski.
5
Content available remote

Lipschitz differences and Lipschitz functions

51%
Balcerzak M., Buczolich Z., Laczkovich M.
Colloquium Mathematicum
|
1997
|
tom 72
|
nr 2
319-324
first rewind previous Strona / 1 next fast forward last
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.