The Lévy-Khintchine formula and Nica-Speicher property for deformations of the free convolution

Łukasz Jan Wojakowski

Displaying similar documents to “The Lévy-Khintchine formula and Nica-Speicher property for deformations of the free convolution”

Probability measures corresponding to Aval numbers

Wojciech Młotkowski (2012)

Colloquium Mathematicae

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We describe the class of probability measures whose moments are given in terms of the Aval numbers. They are expressed as the multiplicative free convolution of measures corresponding to the ballot numbers $(m - k) / (m + k) (\binom{m + k}{m})$ .

The $V_{a}$ -deformation of the classical convolution

Anna Dorota Krystek (2007)

Banach Center Publications

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We study deformations of the classical convolution. For every invertible transformation T:μ ↦ Tμ, we are able to define a new associative convolution of measures by $μ *_{T} ν = T^{- 1} (T μ * T ν)$ . We deal with the $V_{a}$ -deformation of the classical convolution. We prove the analogue of the classical Lévy-Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the $V_{a}$ -deformed classical convolution and give the orthogonal...

On some generalization of the t-transformation

Anna Dorota Krystek (2010)

Banach Center Publications

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Using the Nevanlinna representation of the reciprocal of the Cauchy transform of probability measures, we introduce a two-parameter transformation $U^{}$ of probability measures on the real line ℝ, which is another possible generalization of the t-transformation. Using that deformation we define a new convolution by deformation of the free convolution. The central limit measure with respect to the -deformed free convolutions is still a Kesten measure, but the Poisson limit depends on the...

A convolution property of some measures with self-similar fractal support

Denise Szecsei (2007)

Colloquium Mathematicae

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We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube $I^{M} = {[0, 1)}^{M}$ , with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on $I^{M}$ ; (3) the measures have the convolution property that $μ * L^{p} \subseteq L^{p + ε}$ for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then $μ * L^{p} \subseteq L^{q}$ for any measure μ in our...

Some singular measures on the circle which improve $L^{p}$ spaces

David L. Ritter (1987)

Colloquium Mathematicae

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A comparison on the commutative neutrix convolution of distributions and the exchange formula

Adem Kiliçman (2001)

Czechoslovak Mathematical Journal

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Let $\tilde{f}$ , $\tilde{g}$ be ultradistributions in $𝒵^{'}$ and let ${\tilde{f}}_{n} = \tilde{f} * δ_{n}$ and ${\tilde{g}}_{n} = \tilde{g} * σ_{n}$ where ${δ_{n}}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $δ$ . Then the neutrix product $\tilde{f} ⋄ \tilde{g}$ is defined on the space of ultradistributions $𝒵^{'}$ as the neutrix limit of the sequence ${\frac{1}{2} ({\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n})}$ provided the limit $\tilde{h}$ exist in the sense that $\underset{n \to \infty}{N - l i m} \frac{1}{2} 〈 {\tilde{f}}_{n} \tilde{g} + \tilde{f} {\tilde{g}}_{n}, ψ 〉 = 〈 \tilde{h}, ψ 〉$ for all $ψ$ in $𝒵$ . We also prove that the neutrix convolution product $f ♦ * g$ exist in $𝒟^{'}$ , if and only if the neutrix product $\tilde{f} ⋄ \tilde{g}$ exist in $𝒵^{'}$ and the exchange formula $F (f ♦ * g) = \tilde{f} ⋄ \tilde{g}$ is then satisfied.

Two-level t-deformation

Łukasz Jan Wojakowski (2010)

Banach Center Publications

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In the present paper we define and study the properties of a deformation of measures and convolutions that works in a similar way to the $U_{t}$ deformation of Bożejko and Wysoczański, but in its definition operates on two levels of Jacobi coefficients of a measure, rather than on one.

On the product formula on noncompact Grassmannians

Piotr Graczyk, Patrice Sawyer (2013)

Colloquium Mathematicae

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We study the absolute continuity of the convolution $δ_{e^{X}}^{♮} * δ_{e^{Y}}^{♮}$ of two orbital measures on the symmetric space SO₀(p,q)/SO(p)×SO(q), q > p. We prove sharp conditions on X,Y ∈ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions. We show that the sharp criterion developed for SO₀(p,q)/SO(p)×SO(q) also serves for the spaces SU(p,q)/S(U(p)×U(q)) and Sp(p,q)/Sp(p)×Sp(q), q > p. We moreover apply our results to...

Symmetrization of probability measures, pushforward of order 2 and the Boolean convolution

Wojciech Młotkowski, Noriyoshi Sakuma (2011)

Banach Center Publications

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We study relations between the Boolean convolution and the symmetrization and the pushforward of order 2. In particular we prove that if μ₁,μ₂ are probability measures on [0,∞) then ${(μ ₁ ⨄ μ ₂)}^{s} = μ ₁^{s} ⨄ μ ₂^{s}$ and if ν₁,ν₂ are symmetric then ${(ν ₁ ⨄ ν ₂)}^{(2)} = ν ₁^{(2)} ⨄ ν ₂^{(2)}$ . Finally we investigate necessary and sufficient conditions under which the latter equality holds.

A convolution property of the Cantor-Lebesgue measure, II

Daniel M. Oberlin (2003)

Colloquium Mathematicae

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For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from $L^{p} ()$ to $L^{q} ()$ . We also give a condition on p which is necessary if this operator maps $L^{p} ()$ into L²().

Translation-invariant operators on Lorentz spaces L(1,q) with 0 < q < 1

Leonardo Colzani, Peter Sjögren (1999)

Studia Mathematica

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We study convolution operators bounded on the non-normable Lorentz spaces $L^{1, q}$ of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on $L^{1, q}$ . In particular, when the positions of the atoms of a discrete measure are linearly independent over...

Simple fractions and linear decomposition of some convolutions of measures

Jolanta K. Misiewicz, Roger Cooke (2001)

Discussiones Mathematicae Probability and Statistics

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Every characteristic function φ can be written in the following way: φ(ξ) = 1/(h(ξ) + 1), where h(ξ) = ⎧ 1/φ(ξ) - 1 if φ(ξ) ≠ 0 ⎨ ⎩ ∞ if φ(ξ) = 0 This simple remark implies that every characteristic function can be treated as a simple fraction of the function h(ξ). In the paper, we consider a class C(φ) of all characteristic functions of the form $φ_{a} (ξ) = [a / (h (ξ) + a)]$ , where φ(ξ) is a fixed characteristic function. Using the well known theorem on simple fraction decomposition of rational functions we obtain...

Hankel determinant for a class of analytic functions of complex order defined by convolution

S. M. El-Deeb, M. K. Aouf (2015)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we obtain the Fekete-Szego inequalities for the functions of complex order defined by convolution. Also, we find upper bounds for the second Hankel determinant $| a_{2} a_{4} - a_{3}^{2} |$ for functions belonging to the class $S_{γ}^{b} (g (z); A, B)$ .

One-parameter semigroups in the convolution algebra of rapidly decreasing distributions

(2012)

Colloquium Mathematicae

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The paper is devoted to infinitely differentiable one-parameter convolution semigroups in the convolution algebra $_{C}^{'} (ℝ ⁿ; M_{m \times m})$ of matrix valued rapidly decreasing distributions on ℝⁿ. It is proved that $G \in_{C}^{'} (ℝ ⁿ; M_{m \times m})$ is the generating distribution of an i.d.c.s. if and only if the operator $\partial_{t} \otimes_{m \times m} - G *$ on $ℝ^{1 + n}$ satisfies the Petrovskiĭ condition for forward evolution. Some consequences are discussed.

Compact convolution operators between $L_{p} (G)$ -spaces

G. Crombez, W. Govaerts (1978)

Colloquium Mathematicae

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The type set for some measures on $ℝ^{2 n}$ with $n$ -dimensional support

E. Ferreyra, T. Godoy, Marta Urciuolo (2002)

Czechoslovak Mathematical Journal

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Let $ϕ_{1}, \dots, ϕ_{n}$ be real homogeneous functions in $C^{\infty} (ℝ^{n} - {0})$ of degree $k \geq 2$ , let $ϕ (x) = (ϕ_{1} (x), \dots, ϕ_{n} (x))$ and let $μ$ be the Borel measure on $ℝ^{2 n}$ given by $μ (E) = \int_{ℝ^{n}} χ_{E} (x, ϕ (x)) {| x |}^{γ - n} d x$ where $d x$ denotes the Lebesgue measure on $ℝ^{n}$ and $γ > 0$ . Let $T_{μ}$ be the convolution operator $T_{μ} f (x) = (μ * f) (x)$ and let $E_{μ} = {(1 / p, 1 / q) ∥ T_{μ} ∥_{p, q} < \infty, 1 \leq p, q \leq \infty} .$ Assume that, for $x \neq 0$ , the following two conditions hold: $det (d^{2} ϕ (x) h)$ vanishes only at $h = 0$ and $det (d ϕ (x)) \neq 0$ . In this paper we show that if $γ > n (k + 1) / 3$ then $E_{μ}$ is the empty set and if $γ \leq n (k + 1) / 3$ then $E_{μ}$ is the closed segment with endpoints $D = (1 - \frac{γ}{n (k + 1)}, 1 - \frac{2 γ}{n (k + 1)})$ and $D^{'} = (\frac{2 γ}{n (1 + k)}, \frac{γ}{n (1 + k)})$ . Also, we give some examples.

On a theorem of Saeki concerning convolution squares of singular measures

Thomas Körner (2008)

Bulletin de la Société Mathématique de France

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If $1 > α > 1 / 2$ , then there exists a probability measure $μ$ such that the Hausdorff dimension of the support of $μ$ is $α$ and $μ * μ$ is a Lipschitz function of class $α - \frac{1}{2}$ .

Computation of some examples of Brown's spectral measure in free probability

Philippe Biane, Franz Lehner (2001)

Colloquium Mathematicae

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We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include $u ₙ + u_{\infty}$ where uₙ and $u_{\infty}$ are the generators of ℤₙ and ℤ respectively, in the free product ℤₙ*ℤ, or elliptic elements of the form $S_{α} + i S_{β}$ where $S_{α}$ and $S_{β}$ are free semicircular elements of variance α and β.

Operators which commute with convolutions on subspaces of $L_{\infty} (G)$

Anthony To-Ming Lau (1978)

Colloquium Mathematicae

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

Sets of $β$ -expansions and the Hausdorff measure of slices through fractals

Tom Kempton (2016)

Journal of the European Mathematical Society

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We study natural measures on sets of $β$ -expansions and on slices through self similar sets. In the setting of $β$ -expansions, these allow us to better understand the measure of maximal entropy for the random $β$ -transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing,...

Multiplicative free square of the free Poisson measure and examples of free symmetrization

Melanie Hinz, Wojciech Młotkowski (2010)

Colloquium Mathematicae

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We compute the moments and free cumulants of the measure $ρ_{t} : = π_{t} ⊠ π_{t}$ , where $π_{t}$ denotes the free Poisson law with parameter t > 0. We also compute free cumulants of the symmetrization of $ρ_{t}$ . Finally, we introduce the free symmetrization of a probability measure on ℝ and provide some examples.

Limit theorems for random fields

Nguyen van Thu

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CONTENTSIntroduction............................................................................................................................................................................ 51. Notation and preliminaries............................................................................................................................................ 52. Statement of the problem..................................................................................................................................................