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Nikolai Nikolski

Publications and source records attributed to Nikolai Nikolski.

5 recordsLinked to original sources
arXiv2025

On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras

The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice $X$, including weighted $\ell^p$ spaces. In particular, general multiplicative and completely multiplicative sequences are treated. After the Fourier--Bohr transformation, we deal with the cyclicity property in function spaces on the corresponding infinite dimensional Reinhardt domain $\mathbb{D}^\infty_{X'}$. Functions with (weakly) dominating free term and (in particular) linearly factorable functions are considered. The most attention is paid to the cases of the polydiscs $\mathbb{D}^\infty_{X'}|\mathbb{C}^N=\mathbb{D}^N$ and the $\ell^p$-unit balls $\mathbb{D}^\infty_{X'}|\mathbb{C}^N=\mathbb{B}_p^N$, in particular to Dirichlet-type and Dirichlet--Drury--Arveson-type spaces and algebras, as $X=\ell^p(\mathbb{Z}_+^N,(1+α)^s)$, $s=(s_1,s_2,\dots)$ and $X=\ell^p(\mathbb{Z}_+^N,(\frac{α!}{|α|!})^t(1+|α|)^s)$, $s,t\geq 0$, as well as to their infinite variables analogues. We privileged the largest possible scale of spaces and the most elementary instruments used.

math.FA↗
arXiv2021

Sign intermixing for Riesz bases and frames measured in the Kantorovich-Rubinstein norm

We measure a sign interlacing phenomenon for Bessel sequences $ (u_{k})$ in $ L^{2}$ spaces in terms of the Kantorovich--Rubinstein mass moving norm $ \Vert u_{k}\Vert_{KR}$. Our main observation shows that, quantitatively, the rate of the decreasing $ \Vert u_{k}\Vert_{KR}\longrightarrow 0$ havily depends on S. Bernstein $ n$-widths of a compact of Lipschitz functions. In particular, it depends on the dimension of the measure space. We have sharp results on the worst and the best rate of convergence of Kantorovich--Rubinstein norms of frames on $d$-dimensional cube. Those rates are sharp.

math.CA↗
arXiv2018

On the Sign Distributions of Hilbert Space Frames

We show that the positive and negative parts $ u_{k}^{\pm }$ of any frame in a real $ L^{2}$ space with respect to a continuous measure have both "infinite $ l^{2}$ masses": 1) always, $ \sum _{k}u_{k}^{\pm }(x)^{2}=\infty $ almost everywhere (in particular, there exist no positive frames, nor Riesz bases), but 2) $ \sum _{k=1}^{n}(u_{k}^{+}(x)-u_{k}^{-}(x))^{2}$ can grow "locally" as slow as we wish (for $ n\longrightarrow \infty $), and 3) it can happen that $ \sum _{k=1}^{n}u_{k}^{-}(x)^{2}=\, o(\sum _{k=1}^{n}u_{k}^{+}(x)^{2})$, and vice versa, as $ n\longrightarrow \infty $ on a set of positive measure. Property 1) for the case of an orthonormal basis in $ L^{2}(0,1)$ was settled earlier (V. Ya. Kozlov, 1948) using completely different (and more involved) arguments. Our elementary treatment includes also the case of unconditional bases in a variety of Banach spaces. For property 2), we show that, moreover, whatever is a monotone sequence $ ε_{k}>0$ satisfying $ \sum _{k}ε^{2}_{k}=\, \infty $ there exists an orthonormal basis $ (u_{k})_{k\, }$in $ L^{2}$ such that $ \vert u_{k}(x)\vert \leq \, A(x)ε_{k}$, $ 0<A(x)<\, \infty $.

math.FA↗
arXiv2010

Invertibility threshold for $H^\infty$ trace algebras, and effective matrix inversions

For a given $δ$, $0<δ<1$, a Blaschke sequence $σ=\{λ_j\}$ is constructed such that every function $f$, $f\in H^\infty$, having $δ<δ_f=\inf_{λ\inσ}|f(λ)|\le\|f\|_\infty\le1$ is invertible in the trace algebra $H^\infty|σ$ (with a norm estimate of the inverse depending on $δ_f$ only), but there exists $f$ with $δ=δ_f\le\|f\|_\infty\le1$, which does not. As an application, a counterexample to a stronger form of the Bourgain--Tzafriri restricted invertibility conjecture for bounded operators is exhibited, where an ``orthogonal (or unconditional) basis'' is replaced by a ``summation block orthogonal basis''.

math.FA↗