Random processes in Sobolev-Orlicz spaces

We establish conditions under which the trajectories of random processes from Orlicz spaces of random variables belong with probability one to Sobolev-Orlicz functional spaces, in particular to the classical Sobolev spaces defined on the entire real axis. This enables us to estimate the rate of convergence of wavelet expansions of random processes from the spaces L _p (Ω) and L ₂ (Ω) in the norm of the space L _q (ℝ). __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1340–1356, October, 2006.     

ON CONVERGENCE OF WAVELET PACKET EXPANSIONS

It is well known that the-Walsh-Fourier expansion of a function from the block spaceB _q([0,1]), 1B _q in certain periodized smooth periodic non-stationary wavelet packets bases based on the Haar filters. We also consider wavelet packets based on the Shannon filters and show that the expansion of L^p-functions, 1相似文献   

Mean size formula of wavelet subdivision tree on Heisenberg group

The purpose of this paper is to investigate the mean size formula of wavelet packets （wavelet subdivision tree） on Heisenberg group. The formula is given in terms of the p-norm joint spectral radius. The vector refinement equations on Heisenberg group and the subdivision tree on the Heisenberg group are discussed. The mean size formula of wavelet packets can be used to describe the asymptotic behavior of norm of the subdivision tree.     

Triangular Dirichlet Kernels and Growth of L p Lebesgue Constants

Let P be a polygon in ℤ² and consider the mapping of an L¹(\mathbbT²)L^{1}(\mathbb{T}^{2}) function into the partial sum of its Fourier series determined by the dilate of P by the integer N. If the image space is endowed with the L ^p norm, 1<p<∞, then the operator norm will be given by the L ^p norm of ∑_(m,n)∈NP e ^{2π i(mx+ny)}. The size of this operator norm is shown to be O(N ^2(1−1/p)) when the polygon is a triangle. The estimate is independent of the shape of the triangle. For a k sided polygon the corresponding estimate is O(kN ^2(1−1/p)).     

The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.     

Weighted Polynomials on Discrete Sets

 For a real interval I of positive length, we prove a necessary and sufficient condition which ensures that the continuous L _p (0 < p ⩽ ∞) norm of a weighted polynomial, P _n w ⁿ , deg P _n  ⩽ n, n ⩾ 1 is in an nth root sense, controlled by its corresponding discrete H?lder norm on a very general class of discrete subsets of I. As a by product of our main result, we establish inequalities and theorems dealing with zero distribution, zero location and sup and L _p infinite–finite range inequalities.     

Weighted Polynomials on Discrete Sets

 For a real interval I of positive length, we prove a necessary and sufficient condition which ensures that the continuous L _p (0 < p ⩽ ∞) norm of a weighted polynomial, P _n w ⁿ , deg P _n  ⩽ n, n ⩾ 1 is in an nth root sense, controlled by its corresponding discrete H?lder norm on a very general class of discrete subsets of I. As a by product of our main result, we establish inequalities and theorems dealing with zero distribution, zero location and sup and L _p infinite–finite range inequalities. Received April 4, 2001; in final form June 21, 2002     

Subspaces of L p , p > 2, with unconditional bases have equivalent partition and weight norms

In this note we give a simple proof that every subspace of L_p, 2 < p < ∞, with an unconditional basis has an equivalent norm determined by partitions and weights. Consequently L_p has a norm determined by partitions and weights. Received: 31 January 2005     

Cardinal Interpolation with Gaussian Kernels

In this paper, interpolation by scaled multi-integer translates of Gaussian kernels is studied. The main result establishes L _p Sobolev error estimates and shows that the error is controlled by the L _p multiplier norm of a Fourier multiplier closely related to the cardinal interpolant, and comparable to the Hilbert transform. Consequently, its multiplier norm is bounded independent of the grid spacing when 1<p<∞, and involves a logarithmic term when p=1 or ∞.     

Inversion formulas for the short-time Fourier transform

The inversion formula for the short-time Fourier transform is usually considered in the weak sense, or only for specific combinations of window functions and function spaces such as L₂. In the present article the so-called θ-summability (with a function parameter θ) is considered which induces norm convergence for a large class of function spaces. Under some conditions on θ we prove that the summation of the short-time Fourier transform of ƒ converges to ƒ in Wiener amalgam norms, hence also in the L_p sense for L_p functions, and pointwise almost everywhere.     

Multiwavelet packets and frame packets ofL 2(ℝ d )

The orthonormal basis generated by a wavelet ofL ²(ℝ) has poor frequency localization. To overcome this disadvantage Coifman, Meyer, and Wickerhauser constructed wavelet packets. We extend this concept to the higher dimensions where we consider arbitrary dilation matrices. The resulting basis ofL ²(ℝ ^d ) is called the multiwavelet packet basis. The concept of wavelet frame packet is also generalized to this setting. Further, we show how to construct various orthonormal bases ofL ²(ℝ ^d ) from the multiwavelet packets.     

Inversion formulas for the continuous wavelet transform

The inversion formula for the continuous wavelet transform is usually considered in the weak sense. In the present note we investigate the norm and a.e. convergence of the inversion formula in L _p and Wiener amalgam spaces. The summability of the inversion formula is also considered.     

On the Carleson Measure Criterion in Linear Systems Theory

In Ho and Russell (SIAM J Control Optim 21(4):614–640, 1983), and Weiss (Syst Control Lett 10(1): 79–82, 1988), a Carleson measure criterion for admissibility of one-dimensional input elements with respect to diagonal semigroups is given. We extend their results from the Hilbert space situation (L ²-admissibility on the state space ℓ ₂) to the more general situation of L ^p -admissibility on the state space ℓ _q . For analytic diagonal semigroups we present a new result that does not rely on Laplace transform methods. A comparison of both criteria leads to a result on L ^p -admissibility for reciprocal systems in the sense of Curtain (Syst Control Lett 49(2):81–89, 2003).     

On the means of Fourier integrals and Bernstein inequality in the two-weighted setting

The norm estimation problem for Cesaro and Abel–Poisson operators acting from L_w^p(\mathbbR){L_{w}^{p}(\mathbb{R})} to L_v^q (\mathbbR){L_{v}^{q} (\mathbb{R})} where 1 < p ≤ q < ∞ was investigated. These results were generalized to the multidimensional case and applied to obtain generalizations of the Bernstein inequality for integral functions of finite degree of one and several variables.     

An example concerning the nonlinear pointwise ergodic theorem

It is shown that the pointwise ergodic theorem for Markov operators inL ₁, having a finite invariant measure, fails to extend to the case of nonlinear operators. Recall thatT is called nonexpansive inL _p if ‖Tf – Tg ‖ _p ≦‖f – g‖ _p holds for allf andg.     

Martingale transforms and L p -norm estimates of Riesz transforms on complete Riemannian manifolds

Under the condition that the Bakry–Emery Ricci curvature is bounded from below, we prove a probabilistic representation formula of the Riesz transforms associated with a symmetric diffusion operator on a complete Riemannian manifold. Using the Burkholder sharp L ^p -inequality for martingale transforms, we obtain an explicit and dimension-free upper bound of the L ^p -norm of the Riesz transforms on such complete Riemannian manifolds for all 1 < p < ∞. In the Euclidean and the Gaussian cases, our upper bound is asymptotically sharp when p→ 1 and when p→ ∞. Research partially supported by a Delegation in CNRS at the University of Paris-Sud during the 2005–2006 academic year.     

On a question of Brezis and Marcus

Motivated by a question of Brezis and Marcus, we show that the L^p–Hardy inequality involving the distance to the boundary of a convex domain, can be improved by adding an L^q norm q ≥ p, with a constant depending on the interior diameter of Ω.     

Large deviations of theL p-norm of a wiener process with drift

In this paper we calculate the exact asymptotics of the probability P{‖w(t)+uct‖ _p >u},u→∞, wherew(t) is the standard Wiener process and ‖x‖ _p is the ordinary norm in the spaceL ^p[0,1],p≥2. The result is obtained on the basis of a general theorem due to the author on the asymptotics of the Gaussian measureP(uD),u→∞, for a Borel setD belonging to a separable Banach space. Translated fromMatematicheskie Zametki, Vol. 65, No. 3, pp. 429–436, March, 1999.     

On zeta functions and families of Siegel modular forms

Let p be a prime, and let G = \textS\textp_g( \mathbbZ ) \Gamma = {\text{S}}{{\text{p}}_g}\left( \mathbb{Z} \right) be the Siegel modular group of genus g. This paper is concerned with p-adic families of zeta functions and L-functions of Siegel modular forms; the latter are described in terms of motivic L-functions attached to Sp _g ; their analytic properties are given. Critical values for the spinor L-functions are discussed in relation to p-adic constructions. Rankin’s lemma of higher genus is established. A general conjecture on a lifting of modular forms from GSp_2m   × GSp_2m to GSp_4m (of genus g = 4 m) is formulated. Constructions of p-adic families of Siegel modular forms are given using Ikeda–Miyawaki constructions.     

? 1-Summability of d-Dimensional Fourier Transforms

A general summability method of more-dimensional Fourier transforms is given with the help of a continuous function θ. Under some weak conditions on θ we show that the maximal operator of the ℓ ₁–θ-means of a tempered distribution is bounded from H _p (ℝ ^d ) to L _p (ℝ ^d ) for all d/(d+α)<p≤∞ and, consequently, is of weak type (1,1), where 0<α≤1 depends only on θ. As a consequence we obtain a generalization of the one-dimensional summability result due to Lebesgue, more exactly, the ℓ ₁–θ-means of a function f∈L ₁(ℝ ^d ) converge a.e. to f. Moreover, we prove that the ℓ ₁–θ-means are uniformly bounded on the spaces H _p (ℝ ^d ), and so they converge in norm (d/(d+α)<p<∞). Similar results are shown for conjugate functions. Some special cases of the ℓ ₁–θ-summation are considered, such as the Weierstrass, Picar, Bessel, Fejér, de La Vallée-Poussin, Rogosinski, and Riesz summations.