Three results in Dunkl analysis

Béchir Amri; Jean-Philippe Anker; Mohamed Sifi

Displaying similar documents to “Three results in Dunkl analysis”

$ℰ_{B}$ -dimension function of bitopological spaces

M. Jelić (1987)

Matematički Vesnik

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On the Separation Dimension of $K_{ω}$

Yasunao Hattori, Jan van Mill (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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We prove that $t r t K_{ω} > ω + 1$ , where trt stands for the transfinite extension of Steinke’s separation dimension. This answers a question of Chatyrko and Hattori.

Dunkl-Gabor transform and time-frequency concentration

Saifallah Ghobber (2015)

Czechoslovak Mathematical Journal

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The aim of this paper is to prove two new uncertainty principles for the Dunkl-Gabor transform. The first of these results is a new version of Heisenberg’s uncertainty inequality which states that the Dunkl-Gabor transform of a nonzero function with respect to a nonzero radial window function cannot be time and frequency concentrated around zero. The second result is an analogue of Benedicks’ uncertainty principle which states that the Dunkl-Gabor transform of a nonzero function with...

Distributions of truncations of the heat kernel on the complex projective space

Nizar Demni (2014)

Annales mathématiques Blaise Pascal

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Let ${(U_{t})}_{t \geq 0}$ be a Brownian motion valued in the complex projective space $ℂ P^{N - 1}$ . Using unitary spherical harmonics of homogeneous degree zero, we derive the densities of $| U_{t}^{1} |^{2}$ and of $(| U_{t}^{1} |^{2}, | U_{t}^{2} |^{2})$ , and express them through Jacobi polynomials in the simplices of $ℝ$ and $ℝ^{2}$ respectively. More generally, the distribution of $(| U_{t}^{1} |^{2}, \dots, | U_{t}^{k} |^{2}), 2 \leq k \leq N - 1$ may be derived using the decomposition of the unitary spherical harmonics under the action of the unitary group $𝒰 (N - k + 1)$ yet computations become tedious. We also revisit the approach initiated in [] and...

The Hausdorff operators on the real Hardy spaces $H^{p} (ℝ)$

Yuichi Kanjin (2001)

Studia Mathematica

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We prove that the Hausdorff operator generated by a function ϕ is bounded on the real Hardy space $H^{p} (ℝ)$ , 0 < p ≤ 1, if the Fourier transform ϕ̂ of ϕ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on $H^{p} (ℝ)$ , 2/(2α+1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of $H^{p} (ℝ)$ .

Optimality of the range for which equivalence between certain measures of smoothness holds

Z. Ditzian (2010)

Studia Mathematica

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Recently it was proved for 1 < p < ∞ that $ω^{m} {(f, t)}_{p}$ , a modulus of smoothness on the unit sphere, and $K ̃ ₘ {(f, t^{m})}_{p}$ , a K-functional involving the Laplace-Beltrami operator, are equivalent. It will be shown that the range 1 < p < ∞ is optimal; that is, the equivalence $ω^{m} {(f, t)}_{p} \approx K ̃ ₘ {(f, t^{r})}_{p}$ does not hold either for p = ∞ or for p = 1.

An obstruction to $ℓ^{p}$ -dimension

Nicolas Monod, Henrik Densing Petersen (2014)

Annales de l’institut Fourier

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Let $G$ be any group containing an infinite elementary amenable subgroup and let $2 < p < \infty$ . We construct an exhaustion of $ℓ^{p} G$ by closed invariant subspaces which all intersect trivially a fixed non-trivial closed invariant subspace. This is an obstacle to $ℓ^{p}$ -dimension and gives an answer to a question of Gaboriau.

Polar wavelets and associated Littlewood-Paley theory

Epperson Jay, Frazier Michael

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Abstract We develop an almost orthogonal wavelet-type expansion in ℝ² which is adapted to polar coordinates. We start by defining a product Fourier-Hankel transform f̂ and proving a sampling formula for f such that f̂ is compactly supported. For general f, the sampling formula and a partition of unity lead to an identity of the form $f = \sum_{μ, k, m} ⟨ f, φ_{μ k m} ⟩ ψ_{μ k m}$ , in which each function $φ_{μ k m}$ and $ψ_{μ k m}$ is concentrated near a certain annular sector, has compactly supported product Fourier-Hankel transform, and is smooth...

The Fourier transform in Lebesgue spaces

Erik Talvila (2025)

Czechoslovak Mathematical Journal

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For each $f \in L^{p} (ℝ)$ ( $1 \leq p < \infty$ ) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each $p$ , a norm is defined so that the space of Fourier transforms is isometrically isomorphic to $L^{p} (ℝ)$ . There is an exchange theorem and inversion in norm.

Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces

S. Thangavelu (2002)

Colloquium Mathematicae

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Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis $Y_{δ, j} : δ \in K ̂ ₀, 1 \leq j \leq d_{δ}$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_{t}$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_{δ} (i λ + ϱ)$ be the Kostant polynomials. We establish the following...

Univoque sets for real numbers

Fan Lü, Bo Tan, Jun Wu (2014)

Fundamenta Mathematicae

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For x ∈ (0,1), the univoque set for x, denoted (x), is defined to be the set of β ∈ (1,2) such that x has only one representation of the form x = x₁/β + x₂/β² + ⋯ with $x_{i} \in 0, 1$ . We prove that for any x ∈ (0,1), (x) contains a sequence ${β_{k}}_{k \geq 1}$ increasing to 2. Moreover, (x) is a Lebesgue null set of Hausdorff dimension 1; both (x) and its closure $\bar{(x)}$ are nowhere dense.

Universal acyclic resolutions for arbitrary coefficient groups

Michael Levin (2003)

Fundamenta Mathematicae

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We prove that for every compactum X and every integer n ≥ 2 there are a compactum Z of dimension ≤ n+1 and a surjective $U V^{n - 1}$ -map r: Z → X such that for every abelian group G and every integer k ≥ 2 such that $d i m_{G} X \leq k \leq n$ we have $d i m_{G} Z \leq k$ and r is G-acyclic.

On the continuity of the Hausdorff dimension of the Julia-Lavaurs sets

Ludwik Jaksztas (2011)

Fundamenta Mathematicae

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Let f₀(z) = z²+1/4. We denote by ₀ the set of parameters σ ∈ ℂ for which the critical point 0 escapes from the filled-in Julia set K(f₀) in one step by the Lavaurs map $g_{σ}$ . We prove that if σ₀ ∈ ∂₀, then the Hausdorff dimension of the Julia-Lavaurs set $J_{0, σ}$ is continuous at σ₀ as the function of the parameter $σ \in \bar{₀}$ if and only if $H D (J_{0, σ ₀}) \geq 4 / 3$ . Since $H D (J_{0, σ}) > 4 / 3$ on a dense set of parameters which correspond to preparabolic points, the lower semicontinuity implies the continuity of $H D (J_{0, σ})$ on an open and dense subset of...

$L^{p} - L^{q}$ estimates for some convolution operators with singular measures on the Heisenberg group

T. Godoy, P. Rocha (2013)

Colloquium Mathematicae

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We consider the Heisenberg group ℍⁿ = ℂⁿ × ℝ. Let ν be the Borel measure on ℍⁿ defined by $ν (E) = \int_{ℂ ⁿ} χ_{E} (w, φ (w)) η (w) d w$ , where $φ (w) = \sum_{j = 1}^{n} a_{j} | w_{j} | ²$ , w = (w₁,...,wₙ) ∈ ℂⁿ, $a_{j} \in ℝ$ , and η(w) = η₀(|w|²) with $η ₀ \in C_{c}^{\infty} (ℝ)$ . We characterize the set of pairs (p,q) such that the convolution operator with ν is $L^{p} (ℍ ⁿ) - L^{q} (ℍ ⁿ)$ bounded. We also obtain $L^{p}$ -improving properties of measures supported on the graph of the function $φ (w) = {| w |}^{2 m}$ .

The inverse Laplace transform of the product of two modified Bessel functions $K_{n v} [a^{1 / 2 n} p^{1 / 2 n}] K_{n μ} [a^{1 / 2 n} p^{1 / 2 n}]$ where n=1, 2, 3,...

F. M. Ragab (1963)

Annales Polonici Mathematici

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Birational positivity in dimension $4$

Behrouz Taji (2014)

Annales de l’institut Fourier

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In this paper we prove that for a nonsingular projective variety of dimension at most 4 and with non-negative Kodaira dimension, the Kodaira dimension of coherent subsheaves of $Ω^{p}$ is bounded from above by the Kodaira dimension of the variety. This implies the finiteness of the fundamental group for such an $X$ provided that $X$ has vanishing Kodaira dimension and non-trivial holomorphic Euler characteristic.

Infinite Iterated Function Systems Depending on a Parameter

Ludwik Jaksztas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper is motivated by the problem of dependence of the Hausdorff dimension of the Julia-Lavaurs sets $J_{0, σ}$ for the map f₀(z) = z²+1/4 on the parameter σ. Using homographies, we imitate the construction of the iterated function system (IFS) whose limit set is a subset of $J_{0, σ}$ , given by Urbański and Zinsmeister. The closure of the limit set of our IFS $ϕ_{σ, α}^{n, k}$ is the closure of some family of circles, and if the parameter σ varies, then the behavior of the limit set is similar to the behavior of...

On a fixobject property for $l^{c} β$ -semigroupoids

M. Đurić (1973)

Matematički Vesnik

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Index of the Dirac operator in $R^{n}$

Jan A. Rempała (1985)

Annales Polonici Mathematici

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Equivalence of measures of smoothness in $L_{p} (S^{d - 1})$ , 1 < p < ∞

F. Dai, Z. Ditzian, Hongwei Huang (2010)

Studia Mathematica

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Suppose Δ̃ is the Laplace-Beltrami operator on the sphere $S^{d - 1}, Δ_{ρ}^{k} f (x) = Δ_{ρ} Δ_{ρ}^{k - 1} f (x)$ and $Δ_{ρ} f (x) = f (ρ x) - f (x)$ where ρ ∈ SO(d). Then $ω^{m} {(f, t)}_{L_{p} (S^{d - 1})} \equiv s u p ∥ Δ_{ρ}^{m} f ∥_{L_{p} (S^{d - 1})} : ρ \in S O (d), m a x_{x \in S^{d - 1}} ρ x \cdot x \geq c o s t$ and $K ̃ ₘ {(f, t^{m})}_{p} \equiv i n f ∥ f - g ∥_{L_{p} (S^{d - 1})} + t^{m} ∥ {(- Δ ̃)}^{m / 2} g ∥_{L_{p} (S^{d - 1})} : g \in ({(- Δ ̃)}^{m / 2})$ are equivalent for 1 < p < ∞. We note that for even m the relation was recently investigated by the second author. The equivalence yields an extension of the results on sharp Jackson inequalities on the sphere. A new strong converse inequality for $L_{p} (S^{d - 1})$ given in this paper plays a significant role in the proof.

Embedding in $T_{3}$ - $F_{σ}$ spaces

D. W. Hajek, I. Irizarry (1981)

Matematički Vesnik

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Pointwise Fourier inversion of distributions on spheres

Francisco Javier González Vieli (2017)

Czechoslovak Mathematical Journal

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Given a distribution $T$ on the sphere we define, in analogy to the work of Łojasiewicz, the value of $T$ at a point $ξ$ of the sphere and we show that if $T$ has the value $τ$ at $ξ$ , then the Fourier-Laplace series of $T$ at $ξ$ is Abel-summable to $τ$ .

On the fundamental solution of the equation $∆^{p} u (X) + k (r) u (X) = 0$

J. Musiałek (1969)

Annales Polonici Mathematici

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The spectrum of the Laplace operator for a spherical space form

Gr. Tsagas (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si determina lo spettro di un operatore di Laplace di una «spherical space form» $(M,g)$ e si studia l’influenza di tale spettro su $(M,g)$ .