arXiv subjects
Publications and source records attributed to Denis Zelent.
We improve the currently best known bound for the norm of the embedding of $PW^2$ into $PW^q$ for all $q>2$. We show that our bound is asymptotically optimal as $q\to \infty$. Our approach combines the recently established theorem of optimal Paley-Wiener concentration with a Hardy-Littlewood-P\'olya majorization argument.
We characterize the extreme and exposed points of the unit ball (with respect to the $L^1$-norm) in the shift-invariant space generated by the Gaussian function, as well as in the quasi shift-invariant space generated by the hyperbolic secant.
We study concentration operators acting on the Fourier symmetric Sobolev space $H$ consisting of functions $f$ such that $\int_{\mathbb{R}} |f(x)|^2(1+x^2) dx + \int_{\mathbb{R}} |\hat{f}(\xi)|^2(1+\xi^2) d\xi < \infty $. We find that the Bargmann transform is a unitary operator from $H$ to a weighted Fock space. After identifying the reproducing kernel of $H$, we discover an unexpected phenomenon about the decay of the eigenvalues of a two-sided concentration operator, namely that the plunge region is of the same order of magnitude as the region where the eigenvalues are close to 1, contrasting the classical case of Paley--Wiener spaces.
We prove that every implicative aBE algebra satisfies the transitivity property. This means that every implicative aBE algebra is a Tarski algebra, and thus is also a commutative BCK algebra.
BCK-sequences and n-commutative BCK-algebras were introduced by T. Traczyk, together with two related problems. The first one, whether BCK-sequences are always prolongable. The second one, if the class of all n-commutative BCK-algebras is characterised by one identity. W. A. Dudek proved that the answer to the former question is positive in some special cases, e.g. when BCK-algebra is linearly ordered. T. Traczyk showed that the answer to the latter is affirmative for n = 1, 2. Nonetheless, by providing counterexamples, we proved that the answers to both those open problems are negative.