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Bartosz Trojan

Publications and source records attributed to Bartosz Trojan.

At least 37 records · Page 2Linked to original sources
arXiv2020

Asymptotic behaviour of Christoffel-Darboux kernel via three-term recurrence relation II

We study orthogonal polynomials with periodically modulated Jacobi parameters in the case when $0$ lies on the soft edge of the spectrum of the corresponding periodic Jacobi matrix. We determine when the orthogonality measure is absolutely continuous and we provide a constructive formula for it in terms of the limit of Turán determinants. We next consider asymptotics of the solutions of associated second order difference equation. Finally, we study scaling limits of the Christoffel--Darboux kernel.

math.SP↗
arXiv2020

Asymptotic behaviour of Christoffel-Darboux kernel via three-term recurrence relation I

For Jacobi parameters belonging to one of the three classes: asymptotically periodic, periodically modulated and the blend of these two, we study the asymptotic behavior of the Christoffel functions and the scaling limits of the Christoffel-Darboux kernel. We assume regularity of Jacobi parameters in terms of the Stolz class. We emphasize that the first class only gives rise to measures with compact supports.

math.SP↗
arXiv2020

About essential spectra of unbounded Jacobi matrices

We study spectral properties of unbounded Jacobi matrices with periodically modulated or blended entries. Our approach is based on uniform asymptotic analysis of generalized eigenvectors. We determine when the studied operators are self-adjoint. We identify regions where the point spectrum has no accumulation points. This allows us to completely describe the essential spectrum of these operators.

math.SP↗
arXiv2019

Endpoint estimates for the maximal function over prime numbers

Given an ergodic dynamical system $(X, \mathcal{B}, μ, T)$, we prove that for each function $f$ belonging to the Orlicz space $L(\log L)^2(\log \log L)(X, μ)$, the ergodic averages \[ \frac{1}{π(N)} \sum_{p \in \mathbb{P}_N} f\big(T^p x\big), \] converge for $μ$-almost all $x \in X$, where $\mathbb{P}_N$ is the set of prime numbers not larger that $N$ and $π(N) = \# \mathbb{P}_N$.

math.DS↗
arXiv2019

Pointwise ergodic theorems for some thin subsets of primes

We establish pointwise ergodic theorems for operators of Radon type along subsets of prime numbers of the form $\big\{\{ φ_1(n)\} < ψ(n)\big\}$. We achieve this by proving $\ell^p(\mathbb{Z})$ boundedness of $r$-variations, where $p > 1$ and $r > 2$.

math.CA↗
arXiv2019

Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients

We study solutions of three-term recurrence relations whose $N$-step transfer matrices belong to the uniform Stolz class. In particular, we derive the first order of their uniform asymptotics. For orthonormal polynomials we show more. Namely, we find the constructive formula for the density of their orthogonality measure in terms of Turán determinants and we determine their exact asymptotic behavior. We treat both bounded and unbounded cases in a uniform manner.

math.CA↗
arXiv2018

$\ell^p\big(\mathbb Z^d\big)$-estimates for discrete operators of Radon type: Maximal functions and vector-valued estimates

We prove $\ell^p\big(\mathbb Z^d\big)$ bounds, for $p\in(1, \infty)$, of discrete maximal functions corresponding to averaging operators and truncated singular integrals of Radon type, and their applications to pointwise ergodic theory. Our new approach is based on a unified analysis of both types of operators, and also yields an extension to the vector-valued form of these results.

math.CA↗
arXiv2017

Littlewood-Paley theory for triangle buildings

For the natural two parameter filtration $(\mathcal{F}_λ: λ\in P)$ on the boundary of a triangle building we define a maximal function and a square function and show their boundedness on $L^p(Ω_0)$ for $p \in (1, \infty)$. At the end we consider $L^p(Ω_0)$ boundedness of martingale transforms. If the building is of $\text{GL}(3, \mathbb{Q}_p)$ then $Ω_0$ can be identified with $p$-adic Heisenberg group.

math.FA↗
arXiv2017

Limit theorems for random walks

We consider a random walk $S_τ$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner's subordination. We prove that under certain conditions on the subordinator $τ$ appropriately scaled random walk $S_τ$ converges in the Skorohod space to the symmetric $α$-stable process $B^α$. We also prove asymptotic formula for the transition function of $S_τ$ similar to the Pólya's asymptotic formula for $B^α$.

math.PR↗
arXiv2016

Asymptotic behaviour and estimates of slowly varying convolution semigroups

We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R} ^d$ under the assumption that its Lévy--Khintchine exponent varies slowly. We also derive some new estimates of the transition densities and Green functions.

math.PR↗
arXiv2015

Long time behaviour of random walks on the integer lattice

We consider an irreducible finite range random walk on the $d$-dimensional integer lattice and study asymptotic behaviour of its transition function $p(n; x)$. In particular, for simple random walk our asymptotic formula is valid as long as $n (n - |x|_1)^{-2}$ tends to zero.

math.PR↗
arXiv2015

L^p(Z^d)-estimates for discrete operators of Radon type: Variational estimates

We prove $\ell^p\big(\mathbb Z^d\big)$ bounds for $p\in(1, \infty)$, of $r$-variations $r\in(2, \infty)$, for discrete averaging operators and truncated singular integrals of Radon type. We shall present a new powerful method which allows us to deal with these operators in a unified way and obtain the range of parameters of $p$ and $r$ which coincide with the ranges of their continuous counterparts.

math.CA↗
arXiv2015

The Maximal Function and Square Function Control the Variation: An Elementary Proof

In this note we prove the following good-$λ$ inequality, for $r>2$, all $λ> 0$, $δ\in \big(0, \frac{1}{2} \big)$ \[ ν\big\{ V_r(f) > 3 λ; \mathcal{M}(f) \leq δλ\big\} \leq 4 ν\{s(f) > δλ\} + {δ^2 \left(1+\frac{16}{r-2}\right)^2} \cdot ν\big\{ V_r(f) > λ\big\}, \] where $\mathcal{M}(f)$ is the martingale maximal function, $s(f)$ is the conditional martingale square function. This immediately proves that $V_r(f)$ is bounded on $L^p$, $1 < p <\infty$ and moreover is integrable when the maximal function is.

math.CA↗
arXiv2015

On the Hardy--Littlewood majorant problem for arithmetic sets

The aim of this paper is to exhibit a wide class of sparse deterministic sets, $\mathbf B \subseteq \mathbb{N}$, so that \[ \limsup_{N \to \infty} N^{-1}|\mathbf B \cap [1,N]|= 0, \] for which the Hardy--Littlewood majorant property holds: \[ \sup_{|a_n|\le 1} \Big\| \sum_{n\in\mathbf B\cap[1, N]} a_n e^{2 πi n ξ}\Big \|_{L^p(\mathbb{T}, {\mathrm d} ξ)} \leq \mathbf{C}_p \Big\| \sum_{n\in\mathbf B\cap[1, N]} e^{2 πi n ξ} \Big\|_{L^p(\mathbb{T}, {\mathrm d} ξ)}, \] where $p \geq p_{\mathbf{B}}$ is sufficiently large, the implicit constant $\mathbf{C}_p$ is independent of $N$, and the supremum is taken over all complex sequences $ (a_n : n \in \mathbb{N})$ such that $|a_n| \leq 1$.

math.CA↗
arXiv2015

Two-parameter version of Bourgain's inequality: Rational frequencies

Our aim is to establish the first two-parameter version of Bourgain's maximal logarithmic inequality on $L^2(\mathbb R^2)$ for the rational frequencies. We achieve this by introducing a variant of a two-parameter Rademacher--Menschov inequality. The method allows us to control an oscillation seminorm as well.

math.CA↗