Convergence of series of dilated functions and spectral norms of GCD matrices

Christoph Aistleitner; István Berkes; Kristian Seip; Michel Weber

Displaying similar documents to “Convergence of series of dilated functions and spectral norms of GCD matrices”

Almost everywhere convergence of generalized ergodic transforms for invertible power-bounded operators in $L^{p}$

Christophe Cuny (2011)

Colloquium Mathematicae

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We show that some results of Gaposhkin about a.e. convergence of series associated to a unitary operator U acting on L²(X,Σ,μ) (μ is a σ-finite measure) may be extended to the case where U is an invertible power-bounded operator acting on $L^{p} (X, Σ, μ)$ , p > 1. The proofs make use of the spectral integration initiated by Berkson-Gillespie and, more specifically, of recent results of the author.

On a devil’s staircase associated to the joint spectral radii of a family of pairs of matrices

Ian D. Morris, Nikita Sidorov (2013)

Journal of the European Mathematical Society

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The joint spectral radius of a finite set of real $d \times d$ matrices is defined to be the maximum possible exponential rate of growth of products of matrices drawn from that set. In previous work with K. G. Hare and J. Theys we showed that for a certain one-parameter family of pairs of matrices, this maximum possible rate of growth is attained along Sturmian sequences with a certain characteristic ratio which depends continuously upon the parameter. In this note we answer some open questions...

Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs

Kinkar Ch. Das, Muhuo Liu (2016)

Czechoslovak Mathematical Journal

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In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with $n$ vertices and clique number $ω$ $(2 \leq ω \leq n)$ are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.

Necessary and sufficient conditions for the $L^{1}$ -convergence of double Fourier series

Péter Kórus (2018)

Czechoslovak Mathematical Journal

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We extend the results of paper of F. Móricz (2010), where necessary conditions were given for the $L^{1}$ -convergence of double Fourier series. We also give necessary and sufficient conditions for the $L^{1}$ -convergence under appropriate assumptions.

Spectral Asymptotics for the $\bar{\partial}$ -Neumann Problem

Guy Métivier (1981-1982)

Publications mathématiques et informatique de Rennes

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Growth and smooth spectral synthesis in the Fourier algebras of Lie groups

Jean Ludwig, Lyudmila Turowska (2006)

Studia Mathematica

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Let G be a Lie group and A(G) the Fourier algebra of G. We describe sufficient conditions for complex-valued functions to operate on elements u ∈ A(G) of certain differentiability classes in terms of the dimension of the group G. Furthermore, generalizing a result of Kirsch and Müller [Ark. Mat. 18 (1980), 145-155] we prove that closed subsets E of a smooth m-dimensional submanifold of a Lie group G having a certain cone property are sets of smooth spectral synthesis. For such sets we...

Absolute continuity for Jacobi matrices with power-like weights

Wojciech Motyka (2007)

Colloquium Mathematicae

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This work deals with a class of Jacobi matrices with power-like weights. The main theme is spectral analysis of matrices with zero diagonal and weights $λ ₙ : = n^{α} (1 + Δ ₙ)$ where α ∈ (0,1]. Asymptotic formulas for generalized eigenvectors are given and absolute continuity of the matrices considered is proved. The last section is devoted to spectral analysis of Jacobi matrices with qₙ = n + 1 + (-1)ⁿ and $λ ₙ = \sqrt (q_{n - 1} q ₙ)$ .

Formulae for joint spectral radii of sets of operators

Victor S. Shulman, Yuriĭ V. Turovskii (2002)

Studia Mathematica

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The formula $ϱ (M) = m a x ϱ_{χ} (M), r (M)$ is proved for precompact sets M of weakly compact operators on a Banach space. Here ϱ(M) is the joint spectral radius (the Rota-Strang radius), $ϱ_{χ} (M)$ is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.

Strong spectral gaps for compact quotients of products of $PSL (2, ℝ)$ )

Dubi Kelmer, Peter Sarnak (2009)

Journal of the European Mathematical Society

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The existence of a strong spectral gap for quotients $Γ ∖ G$ of noncompact connected semisimple Lie groups is crucial in many applications. For congruence lattices there are uniform and very good bounds for the spectral gap coming from the known bounds towards the Ramanujan–Selberg conjectures. If $G$ has no compact factors then for general lattices a spectral gap can still be established, but there is no uniformity and no effective bounds are known. This note is concerned with the spectral...

On the spectral radius in $L_{1} (G)$

E. Porada (1971)

Colloquium Mathematicae

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A spectral gap property for subgroups of finite covolume in Lie groups

Bachir Bekka, Yves Cornulier (2010)

Colloquium Mathematicae

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Let G be a real Lie group and H a lattice or, more generally, a closed subgroup of finite covolume in G. We show that the unitary representation $λ_{G / H}$ of G on L²(G/H) has a spectral gap, that is, the restriction of $λ_{G / H}$ to the orthogonal complement of the constants in L²(G/H) does not have almost invariant vectors. This answers a question of G. Margulis. We give an application to the spectral geometry of locally symmetric Riemannian spaces of infinite volume.

Spectral synthesis and operator synthesis

K. Parthasarathy, R. Prakash (2006)

Studia Mathematica

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Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{\infty} (G)$ -submodule X̂ of ℬ(L²(G)) (where $V^{\infty} (G)$ is the weak-* Haagerup tensor product $L^{\infty} (G) \otimes_{w * h} L^{\infty} (G)$ ), define the concept of X̂-operator synthesis and prove that a...

A correction to “Some mean convergence and complete convergence theorems for sequences of $m$ -linearly negative quadrant dependent random variables”

Yongfeng Wu, Andrew Rosalsky, Andrei Volodin (2017)

Applications of Mathematics

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The authors provide a correction to “Some mean convergence and complete convergence theorems for sequences of $m$ -linearly negative quadrant dependent random variables”.

Absolutely continuous and singular spectral shift functions

Nurulla Azamov

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Given a self-adjoint operator H₀, a self-adjoint trace-class operator V and a fixed Hilbert-Schmidt operator F with trivial kernel and cokernel, using the limiting absorption principle an explicit set Λ(H₀;F) ⊂ ℝ of full Lebesgue measure is defined, such that for all λ ∈ Λ(H₀+rV;F) ∩ Λ(H₀;F), where r ∈ ℝ, the wave $w_{\pm} (λ; H ₀ + r V, H ₀)$ and the scattering matrices S(λ;H₀+rV,H₀) can be defined unambiguously. Many well-known properties of the wave and scattering matrices and operators are proved, including...

A spectral gap theorem in SU $(d)$

Jean Bourgain, Alex Gamburd (2012)

Journal of the European Mathematical Society

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We establish the spectral gap property for dense subgroups of SU $(d)$ $(d \geq 2)$ , generated by finitely many elements with algebraic entries; this result was announced in [BG3]. The method of proof differs, in several crucial aspects, from that used in [BG] in the case of SU $(2)$ .

Uniform convergence of N-dimensional Walsh-Fourier series

U. Goginava (2005)

Studia Mathematica

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We establish conditions on the partial moduli of continuity which guarantee uniform convergence of the N-dimensional Walsh-Fourier series of functions f from the class $C_{W} (I^{N}) \cap ⋂_{i = 1}^{N} B V_{i, p (n)}$ , where p(n)↑ ∞ as n → ∞.

On the zero set of the Kobayashi-Royden pseudometric of the spectral unit ball

Nikolai Nikolov, Pascal J. Thomas (2008)

Annales Polonici Mathematici

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Given A∈ Ωₙ, the n²-dimensional spectral unit ball, we show that if B is an n×n complex matrix, then B is a “generalized” tangent vector at A to an entire curve in Ωₙ if and only if B is in the tangent cone $C_{A}$ to the isospectral variety at A. In the case of Ω₃, the zero set of the Kobayashi-Royden pseudometric is completely described.

GCD sums from Poisson integrals and systems of dilated functions

Christoph Aistleitner, István Berkes, Kristian Seip (2015)

Journal of the European Mathematical Society

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Upper bounds for GCD sums of the form $\sum_{k, ℓ = 1}^{N} \frac{{(\gcd (n_{k}, n_{ℓ}))}^{2 α}}{{(n_{k} n_{ℓ})}^{α}}$ are established, where ${(n_{k})}_{1 \leq k \leq N}$ is any sequence of distinct positive integers and $0 < α \leq 1$ ; the estimate for $α = 1 / 2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $α = 1 / 2$ . The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to...

On the convergence theory of double $K$ -weak splittings of type II

Vaibhav Shekhar, Nachiketa Mishra, Debasisha Mishra (2022)

Applications of Mathematics

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Recently, Wang (2017) has introduced the $K$ -nonnegative double splitting using the notion of matrices that leave a cone $K \subseteq ℝ^{n}$ invariant and studied its convergence theory by generalizing the corresponding results for the nonnegative double splitting by Song and Song (2011). However, the convergence theory for $K$ -weak regular and $K$ -nonnegative double splittings of type II is not yet studied. In this article, we first introduce this class of splittings and then discuss the convergence theory...

Spectral projections for the twisted Laplacian

Herbert Koch, Fulvio Ricci (2007)

Studia Mathematica

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Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L = - 1 / 2 \sum_{j = 1}^{n} [(\partial_{x_{j}} + i y_{j}) ² + (\partial_{y_{j}} - i x_{j}) ²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{λ}$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $| | P_{λ} {u | |}_{L^{p} (ℝ^{d})} ≲ λ^{ϱ (p)} {| | u | |}_{L ² (ℝ^{d})}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p...

Complete $f$ -moment convergence for weighted sums of WOD arrays with statistical applications

Xi Chen, Xinran Tao, Xuejun Wang (2023)

Kybernetika

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Complete $f$ -moment convergence is much more general than complete convergence and complete moment convergence. In this work, we mainly investigate the complete $f$ -moment convergence for weighted sums of widely orthant dependent (WOD, for short) arrays. A general result on Complete $f$ -moment convergence is obtained under some suitable conditions, which generalizes the corresponding one in the literature. As an application, we establish the complete consistency for the weighted linear estimator...

Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series

Earl Berkson (2014)

Studia Mathematica

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Let $f \in V_{r} () \cup_{r} ()$ , where, for 1 ≤ r < ∞, $V_{r} ()$ (resp., $_{r} ()$ ) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values,...

On the joint spectral radius

Vladimír Müller (1997)

Annales Polonici Mathematici

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We prove the $_{p}$ -spectral radius formula for n-tuples of commuting Banach algebra elements

Absolute convergence of multiple Fourier integrals

Yurii Kolomoitsev, Elijah Liflyand (2013)

Studia Mathematica

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Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of $L_{p}$ integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular cases. Sharpness and applications of the results obtained are also discussed.

A note on necessary and sufficient conditions for convergence of the finite element method

Kučera, Václav

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In this short note, we present several ideas and observations concerning finite element convergence and the role of the maximum angle condition. Based on previous work, we formulate a hypothesis concerning a necessary condition for $O (h)$ convergence and show a simple relation to classical problems in measure theory and differential geometry which could lead to new insights in the area.

A.e. convergence of anisotropic partial Fourier integrals on Euclidean spaces and Heisenberg groups

D. Müller, E. Prestini (2010)

Colloquium Mathematicae

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We define partial spectral integrals $S_{R}$ on the Heisenberg group by means of localizations to isotropic or anisotropic dilates of suitable star-shaped subsets V containing the joint spectrum of the partial sub-Laplacians and the central derivative. Under the assumption that an L²-function f lies in the logarithmic Sobolev space given by $l o g (2 + L_{α}) f \in L ²$ , where $L_{α}$ is a suitable “generalized” sub-Laplacian associated to the dilation structure, we show that $S_{R} f (x)$ converges a.e. to f(x) as R → ∞.

Norm convergence of some power series of operators in $L^{p}$ with applications in ergodic theory

Christophe Cuny (2010)

Studia Mathematica

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Let X be a closed subspace of $L^{p} (μ)$ , where μ is an arbitrary measure and 1 < p < ∞. Let U be an invertible operator on X such that $s u p_{n \in ℤ} | | U ⁿ | | < \infty$ . Motivated by applications in ergodic theory, we obtain (optimal) conditions for the convergence of series like $\sum_{n \geq 1} (U ⁿ f) / n^{1 - α}$ , 0 ≤ α < 1, in terms of $| | f + \dots + U^{n - 1} f {| |}_{p}$ , generalizing results for unitary (or normal) operators in L²(μ). The proofs make use of the spectral integration initiated by Berkson and Gillespie and, more particularly, of results from a paper by Berkson-Bourgain-Gillespie. ...

Regular statistical convergence of double sequences

Ferenc Móricz (2005)

Colloquium Mathematicae

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The concepts of statistical convergence of single and double sequences of complex numbers were introduced in [1] and [7], respectively. In this paper, we introduce the concept indicated in the title. A double sequence $x_{j k} : (j, k) \in ℕ ²$ is said to be regularly statistically convergent if (i) the double sequence $x_{j k}$ is statistically convergent to some ξ ∈ ℂ, (ii) the single sequence $x_{j k} : k \in ℕ$ is statistically convergent to some $ξ_{j} \in ℂ$ for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence $x_{j k} : j \in ℕ$ is statistically convergent...