arXiv subjects

K. S. Kazarian

Publications and source records attributed to K. S. Kazarian.

5 recordsLinked to original sources

arXiv2019

A uniformly bounded complete Euclidean system

A uniformly bounded complete orthonormal system of functions $Θ=\{ θ_n\}_{n=1}^{\infty},$ $ \|θ_n\|_{L^\infty_{[0,1]} } \leq M $ is constructed such that $\sum_{n=1}^{\infty} a_{n}θ_{n}$ converges almost everywhere on $[0,1]$ if $\{ a_n\}_{n=1}^{\infty} \in \, l^2$ and $\sum_{n=1}^{\infty} a_{n}θ_{n}$ diverges a. e. for any $\{ a_n\}_{n=1}^{\infty} \not\in \, l^2$. Thus Menshov's theorem on the representation of measurable, almost everywhere finite, functions by almost everywhere convergent trigonometric series cannot be extended to the class of uniformly bounded complete orthonormal systems.

math.CA↗

arXiv2019

On the structure of finitely generated shift-invariant subspaces

A characterization of finitely generated shift-invariant subspaces is given when generators are g-minimal. An algorithm is given for the determination of the coefficients in the well known representation of the Fourier transform of an element of the finitely generated shift-invariant subspace as a linear combination of Fourier transformations of generators. An estimate for the norms of those coefficients is derived. For the proof a sort of orthogonalization procedure for generators is used which reminds the well known Gram-Schmidt orthogonalization process.

math.CA↗

arXiv2019

Generalized Fourier series by double trigonometric system

Necessary and sufficient conditions are obtained on the function $M$ such that $\{ M(x,y) e^{i kx}e^{i my}: (k,m)\in Ω\}$ is complete and minimal in $L^{p}(\mathbb{T}^{2})$ when $Ω^{c}=\{(0,0)\}$ and $Ω^{c} = 0\times\mathbb{Z}$. If $Ω^{c} = 0\times\mathbb{Z}_{0},$ $\mathbb{Z}_{0} = \mathbb{Z}\setminus\{0\}$ it is proved that the system $\{ M(x,y) e^{i kx}e^{i my}: (k,m)\in Ω\}$ cannot be complete minimal in $L^{p}(\mathbb{T}^{2})$ for any $M\in L^{p}(\mathbb{T}^{2})$. In the case, $Ω^{c}=\{(0,0)\}$ necessary and conditions are found in terms of the one-dimensional case.

math.CA↗

arXiv2018

On the orthogonal democratic systems in the $L^p$ spaces

The concept of bidemocratic pair for a Banach space was introduced in \cite{KS:18}. We construct a family of orthonormal systems $\mathfrak{F}_{l},$ $l\in (0,\infty)$ of functions defined on $[-1,1]$ such that the pair $(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is bidemocratic for $L^{p}[-1,1]$ and for $L^{p'}[-1,1]$ if $l\in (0, \frac{p}{2(p-2)}]$, where $p>2$ and $p'= \frac{p}{p-1}$. The system $\mathfrak{F}_{l}$ is not democratic for $L^{p'}[-1,1]$ when $l\in (\frac{p}{2(p-2)}, \frac{p}{p-2}). $ When $l> \frac{p}{2(p-2)}$ the pair $(\mathfrak{F}_{l},\mathfrak{F}_{l})$ is not bidemocratic neither for $L^{p}[-1,1]$ nor for $L^{p'}[-1,1]$.

math.FA↗

arXiv2013

Generalized Fourier series and shift-invariant subspaces

A principal shift invariant subspace of $L^{2}(\IR)$ is isometric to a weighted norm space $L^{2}(\IT, w)$. Using results obtained earlier by the author on the basis properties of subsystems of the trigonometric system in the weighted norm spaces $L^{2}(\IT, w)$ we obtain corresponding results concerning generators of principal shift invariant subspaces.

math.CA↗