The relation between the dual and the adjoint Radon transforms

Cnops, J.

Displaying similar documents to “The relation between the dual and the adjoint Radon transforms”

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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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...

On the dual space of $H_{B}^{1, \infty}$

Oscar Blasco (1988)

Colloquium Mathematicae

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A note on almost strong liftings

C. Ionescu-Tulcea, R. Maher (1971)

Annales de l'institut Fourier

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Let $X$ be a locally compact space. A lifting $ρ$ of $M_{R}^{\infty} (X, μ)$ where $μ$ is a positive measure on $X$ , is almost strong if for each bounded, continuous function $f$ , $ρ (f)$ and $f$ coincide locally almost everywhere. We prove here that the set of all measures $μ$ on $X$ such that there exists an almost strong lifting of $M_{R}^{\infty} (X, | μ |)$ is a band.

Best constants for some operators associated with the Fourier and Hilbert transforms

B. Hollenbeck, N. J. Kalton, I. E. Verbitsky (2003)

Studia Mathematica

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We determine the norm in $L^{p} (ℝ ₊)$ , 1 < p < ∞, of the operator $I - ℱ_{s} ℱ_{c}$ , where $ℱ_{c}$ and $ℱ_{s}$ are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the $L^{p}$ -norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real...

Noncommutative extensions of the Fourier transform and its logarithm

Romuald Lenczewski (2002)

Studia Mathematica

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We introduce noncommutative extensions of the Fourier transform of probability measures and its logarithm to the algebra (S) of complex-valued functions on the free semigroup S = FS(z,w) on two generators. First, to given probability measures μ, ν with all moments finite, we associate states μ̂, ν̂ on the unital free *-bialgebra (ℬ,ε,Δ) on two self-adjoint generators X,X’ and a projection P. Then we introduce and study cumulants which are additive under the convolution μ̂* ν̂ = μ̂ ⊗...

Convex Corson compacta and Radon measures

Grzegorz Plebanek (2002)

Fundamenta Mathematicae

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Assuming the continuum hypothesis, we show that (i) there is a compact convex subset L of $Σ (ℝ^{ω ₁})$ , and a probability Radon measure on L which has no separable support; (ii) there is a Corson compact space K, and a convex weak*-compact set M of Radon probability measures on K which has no $G_{δ}$ -points.

Uncertainty principles for integral operators

Saifallah Ghobber, Philippe Jaming (2014)

Studia Mathematica

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The aim of this paper is to prove new uncertainty principles for integral operators with bounded kernel for which there is a Plancherel Theorem. The first of these results is an extension of Faris’s local uncertainty principle which states that if a nonzero function $f \in L ² (ℝ^{d}, μ)$ is highly localized near a single point then (f) cannot be concentrated in a set of finite measure. The second result extends the Benedicks-Amrein-Berthier uncertainty principle and states that a nonzero function $f \in L ² (ℝ^{d}, μ)$ and...

General Dirichlet series, arithmetic convolution equations and Laplace transforms

Helge Glöckner, Lutz G. Lucht, Štefan Porubský (2009)

Studia Mathematica

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In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g: ℕ → ℂ to convolution equations of the form $a_{d} * g^{* d} + a_{d - 1} * g^{* (d - 1)} + \dots + a ₁ * g + a ₀ = 0$ , where $a ₀, . . ., a_{d} : ℕ \to ℂ$ are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form $\sum_{x \in X} f (x) e^{- s x}$ ( $s \in ℂ^{k}$ ), where $X \subseteq {[0, \infty)}^{k}$ is an additive subsemigroup. If X is discrete and a certain solvability criterion...

Variation for the Riesz transform and uniform rectifiability

Albert Mas, Xavier Tolsa (2014)

Journal of the European Mathematical Society

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For $1 \leq n < d$ integers and $ρ > 2$ , we prove that an $n$ -dimensional Ahlfors-David regular measure $μ$ in $ℝ^{d}$ is uniformly $n$ -rectifiable if and only if the $ρ$ -variation for the Riesz transform with respect to $μ$ is a bounded operator in $L^{2} (μ)$ . This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the $L^{2} (μ)$ boundedness of the Riesz transform to the uniform rectifiability of $μ$ .

Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³

E. Ferreyra, T. Godoy, M. Urciuolo (2004)

Studia Mathematica

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Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by $σ (A) = \int_{B} χ_{A} (x, φ (x)) d x$ where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from $L^{p} (ℝ ³)$ to $L^{q} (Σ, d σ)$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.

The punctured plane: alternating projections and $L^{2}$ -angles

M. Skwarczyński (1991)

Annales Polonici Mathematici

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Sidon sets and $I^{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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Existence of solutions to the nonstationary Stokes system in $H_{- μ}^{2, 1}$ , μ ∈ (0,1), in a domain with a distinguished axis. Part 2. Estimate in the 3d case

W. M. Zajączkowski (2007)

Applicationes Mathematicae

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We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity v₀ and the external force f belong to some weighted Sobolev spaces. It is assumed that the weight is the (-μ )th power of the distance to the axis. Let $f \in L_{2, - μ}$ , $v ₀ \in H_{- μ} ¹$ , μ ∈ (0,1). We prove an estimate of the velocity in the $H_{- μ}^{2, 1}$ norm and of the gradient of the pressure in the norm of $L_{2, - μ}$ . We apply the Fourier transform with respect to the variable along...

Covariant differential operators and Green's functions

Miroslav Engliš, Jaak Peetre (1997)

Annales Polonici Mathematici

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The basic idea of this paper is to use the covariance of a partial differential operator under a suitable group action to determine suitable associated Green’s functions. For instance, we offer a new proof of a formula for Green’s function of the mth power $Δ^{m}$ of the ordinary Laplace’s operator Δ in the unit disk found in a recent paper (Hayman-Korenblum, J. Anal. Math. 60 (1993), 113-133). We also study Green’s functions associated with mth powers of the Poincaré invariant Laplace operator...

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...

Common zero sets of equivalent singular inner functions

Keiji Izuchi (2004)

Studia Mathematica

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Let μ and λ be bounded positive singular measures on the unit circle such that μ ⊥ λ. It is proved that there exist positive measures μ₀ and λ₀ such that μ₀ ∼ μ, λ₀ ∼ λ, and $| ψ_{μ ₀} | < 1 \cap | ψ_{λ ₀} | < 1 = \emptyset$ , where $ψ_{μ}$ is the associated singular inner function of μ. Let $(μ) = ⋂_{ν; ν \sim μ} Z (ψ_{ν})$ be the common zeros of equivalent singular inner functions of $ψ_{μ}$ . Then (μ) ≠ ∅ and (μ) ∩ (λ) = ∅. It follows that μ ≪ λ if and only if (μ) ⊂ (λ). Hence (μ) is the set in the maximal ideal space of $H^{\infty}$ which relates naturally to the set of measures equivalent...

$L^{p} - L^{q}$ 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 $φ (w) = {| w |}^{2 m}$ .