Infinite dimensional Gegenbauer functionals

Abdessatar Barhoumi; Habib Ouerdiane; Anis Riahi

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Displaying similar documents to “Infinite dimensional Gegenbauer functionals”

The max norm in $R^{n}$ -isometries and measure

Regina Cohen, James W. Fickett (1982)

Colloquium Mathematicae

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On Peano derivatives in $L^{p} (E_{n})$

S. Lasher (1968)

Studia Mathematica

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Characteristic points, rectifiability and perimeter measure on stratified groups

Valentino Magnani (2006)

Journal of the European Mathematical Society

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We establish an explicit connection between the perimeter measure of an open set $E$ with $C^{1}$ boundary and the spherical Hausdorff measure $S^{Q - 1}$ restricted to $\partial E$ , when the ambient space is a stratified group endowed with a left invariant sub-Riemannian metric and $Q$ denotes the Hausdorff dimension of the group. Our formula implies that the perimeter measure of $E$ is less than or equal to $S^{Q - 1} (\partial E)$ up to a dimensional factor. The validity of this estimate positively answers a conjecture raised by Danielli,...

Asymptotic nature of higher Mahler measure

(2014)

Acta Arithmetica

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We consider Akatsuka’s zeta Mahler measure as a generating function of the higher Mahler measure $m_{k} (P)$ of a polynomial $P,$ where $m_{k} (P)$ is the integral of $l o g^{k} | P |$ over the complex unit circle. Restricting ourselves to P(x) = x - r with |r| = 1 we show some new asymptotic results regarding $m_{k} (P)$ , in particular $| m_{k} (P) | / k! \to 1 / π$ as k → ∞.

Properties of Wiener-Wintner dynamical systems

I. Assani, K. Nicolaou (2001)

Bulletin de la Société Mathématique de France

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In this paper we prove the following results. First, we show the existence of Wiener-Wintner dynamical system with continuous singular spectrum in the orthocomplement of their respective Kronecker factors. The second result states that if $f \in L^{p}$ , $p$ large enough, is a Wiener-Wintner function then, for all $γ \in (1 + \frac{1}{2 p} - \frac{β}{2}, 1]$ , there exists a set $X_{f}$ of full measure for which the series $\sum_{n = 1}^{\infty} \frac{f (T^{n} x) e^{2 π i n ϵ}}{n^{γ}}$ converges uniformly with respect to $ϵ$ .

Relations between Shy Sets and Sets of $ν_{p}$ -Measure Zero in Solovay’s Model

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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An example of a non-zero non-atomic translation-invariant Borel measure $ν_{p}$ on the Banach space $ℓ_{p} (1 \leq p \leq \infty)$ is constructed in Solovay’s model. It is established that, for 1 ≤ p < ∞, the condition " $ν_{p}$ -almost every element of $ℓ_{p}$ has a property P" implies that “almost every” element of $ℓ_{p}$ (in the sense of [4]) has the property P. It is also shown that the converse is not valid.

Unique Bernoulli $g$ -measures

Anders Johansson, Anders Öberg, Mark Pollicott (2012)

Journal of the European Mathematical Society

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We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a $g$ -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique $g$ -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the $g$ -measure.

On an Invariant Borel Measure in Hilbert Space

G. Pantsulaia (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold $_{μ}$ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift $μ^{(a)}$ of μ by a vector $a \in ℓ ₂ ∖_{μ}$ are neither equivalent nor orthogonal. This extends a result established in [7].

Polynomial relations amongst algebraic units of low measure

John Garza (2014)

Acta Arithmetica

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For an algebraic number field and a subset $α_{1}, . . ., α_{r} \subseteq_{}$ , we establish a lower bound for the average of the logarithmic heights that depends on the ideal of polynomials in $ℚ [x_{1}, . . ., x_{r}]$ vanishing at the point $(α_{1}, . . ., α_{r})$ .

Interpolating sequences, Carleson measures and Wirtinger inequality

Eric Amar (2008)

Annales Polonici Mathematici

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Let S be a sequence of points in the unit ball of ℂⁿ which is separated for the hyperbolic distance and contained in the zero set of a Nevanlinna function. We prove that the associated measure $μ_{S} : = \sum_{a \in S} (1 - | a | ²) ⁿ δ_{a}$ is bounded, by use of the Wirtinger inequality. Conversely, if X is an analytic subset of such that any δ -separated sequence S has its associated measure $μ_{S}$ bounded by C/δⁿ, then X is the zero set of a function in the Nevanlinna class of . As an easy consequence, we prove that if S is a dual...

Invariant subspaces for operators in a general II1-factor

Uffe Haagerup, Hanne Schultz (2009)

Publications Mathématiques de l'IHÉS

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Let ℳ be a von Neumann factor of type II1 with a normalized trace τ. In 1983 L. G. Brown showed that to every operator T∈ℳ one can in a natural way associate a spectral distribution measure μ T (now called the Brown measure of T), which is a probability measure in ℂ with support in the spectrum σ(T) of T. In this paper it is shown that for every T∈ℳ and every Borel set B in ℂ, there is a unique closed T-invariant subspace $𝒦 = 𝒦_{T} (B)$ affiliated with ℳ, such that the Brown measure of ${T |}_{𝒦}$ is concentrated...

Extensions of generic measure-preserving actions

Julien Melleray (2014)

Annales de l’institut Fourier

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We show that, whenever $Γ$ is a countable abelian group and $Δ$ is a finitely-generated subgroup of $Γ$ , a generic measure-preserving action of $Δ$ on a standard atomless probability space $(X, μ)$ extends to a free measure-preserving action of $Γ$ on $(X, μ)$ . This extends a result of Ageev, corresponding to the case when $Δ$ is infinite cyclic.

Duality of measures of non-𝒜-compactness

Juan Manuel Delgado, Cándido Piñeiro (2015)

Studia Mathematica

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Let be a Banach operator ideal. Based on the notion of -compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non–compactness of an operator. We consider a map $χ_{}$ (respectively, $n_{}$ ) acting on the operators of the surjective (respectively, injective) hull of such that $χ_{} (T) = 0$ (respectively, $n_{} (T) = 0$ ) if and only if the operator T is -compact (respectively, injectively -compact). Under certain conditions on the ideal , we prove an equivalence inequality involving...

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

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In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X, d, μ)$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B (x, r)$ converge to $f (x)$ when $r$ converges to $0$ . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

On Ordinary and Standard Lebesgue Measures on $ℝ^{\infty}$

Gogi Pantsulaia (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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New concepts of Lebesgue measure on $ℝ^{\infty}$ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on $ℝ^{\infty}$ for α = (1,1,...).

Stable random fields and geometry

Shigeo Takenaka (2010)

Banach Center Publications

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Let (M,d) be a metric space with a fixed origin O. P. Lévy defined Brownian motion X(a); a ∈ M as 0. X(O) = 0. 1. X(a) - X(b) is subject to the Gaussian law of mean 0 and variance d(a,b). He gave an example for $M = S^{m}$ , the m-dimensional sphere. Let $Y (B); B \in ℬ (S^{m})$ be the Gaussian random measure on $S^{m}$ , that is, 1. Y(B) is a centered Gaussian system, 2. the variance of Y(B) is equal of μ(B), where μ is the uniform measure on $S^{m}$ , 3. if B₁ ∩ B₂ = ∅ then Y(B₁) is independent of Y(B₂). 4. for $B_{i}$ , i = 1,2,..., $B_{i} \cap B_{j} = \emptyset$ ,...

Note on the variance of the sum of Gaussian functionals

Marek Beśka (2010)

Applicationes Mathematicae

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Let $(X_{i}, i = 1, 2, . . .)$ be a Gaussian sequence with $X_{i} \in N (0, 1)$ for each i and suppose its correlation matrix $R = {(ρ_{i j})}_{i, j \geq 1}$ is the matrix of some linear operator R:l₂→ l₂. Then for $f_{i} \in L ² (μ)$ , i=1,2,..., where μ is the standard normal distribution, we estimate the variation of the sum of the Gaussian functionals $f_{i} (X_{i})$ , i=1,2,... .

Osgood type conditions for an m th-order differential equation

Stanisaw Szufla (1998)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We present a new theorem on the differential inequality $u^{(m)} \leq w (u)$ . Next, we apply this result to obtain existence theorems for the equation $x^{(m)} = f (t, x)$ .

Some results on derangement polynomials

Mehdi Hassani, Hossein Moshtagh, Mohammad Ghorbani (2022)

Commentationes Mathematicae Universitatis Carolinae

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We study moments of the difference $D_{n} (x) - x^{n} n! e^{- 1 / x}$ concerning derangement polynomials $D_{n} (x)$ . For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for $x > 0$ . For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

On the characterization of harmonic functions with initial data in Morrey space

Bo Li, Jinxia Li, Bolin Ma, Tianjun Shen (2024)

Czechoslovak Mathematical Journal

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Let $(X, d, μ)$ be a metric measure space satisfying the doubling condition and an $L^{2}$ -Poincaré inequality. Consider the nonnegative operator $ℒ$ generalized by a Dirichlet form on $X$ . We will show that a solution $u$ to $(- \partial_{t}^{2} + ℒ) u = 0$ on $X \times ℝ_{+}$ satisfies an $α$ -Carleson condition if and only if $u$ can be represented as the Poisson integral of the operator $ℒ$ with the trace in the generalized Morrey space $L^{2, α} (X)$ , where $α$ is a nonnegative function defined on a class of balls in $X$ . This result extends the analogous characterization...

A Green's function for θ-incomplete polynomials

Joe Callaghan (2007)

Annales Polonici Mathematici

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Let K be any subset of $ℂ^{N}$ . We define a pluricomplex Green’s function $V_{K, θ}$ for θ-incomplete polynomials. We establish properties of $V_{K, θ}$ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of $ℝ^{N}$ , we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on $K ∖ s u p p {(d d^{c} V_{K, θ})}^{N}$ . We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute $s u p p {(d d^{c} V_{K, θ})}^{N}$ when K is a compact...