Weighted L1 Spaces Research Articles

Noncommutative pointwise convergence of orthogonal expansions of several variables

AbstractIn this paper, the boundedness of the maximal function for an operator‐valued weighted L1 space on the unit sphere of , in which the weight functions are invariant under finite reflection groups, is established. We use it to prove the boundedness of orthogonal expansions in h‐harmonics and this result applies to various methods of summability. Furthermore, we obtain the corresponding pointwise convergence theorems. At last, we give some results in the special reflection group .

A new quadrature scheme based on an Extended Lagrange Interpolation process

Let w(x)=e−xβxα, w¯(x)=xw(x) and let {pm(w)}m, {pm(w¯)}m be the corresponding sequences of orthonormal polynomials. Since the zeros of pm+1(w) interlace those of pm(w¯), it makes sense to construct an interpolation process essentially based on the zeros of Q2m+1:=pm+1(w)pm(w¯), which is called “Extended Lagrange Interpolation”. In this paper the convergence of this interpolation process is studied in suitable weighted L1 spaces, in a general framework which completes the results given by the same authors in weighted Lup((0,+∞)), 1≤p≤∞ (see [31], [28]). As an application of the theoretical results, an extended product integration rule, based on the aforesaid Lagrange process, is proposed in order to compute integrals of the type∫0+∞f(x)k(x,y)u(x)dx,u(x)=e−xβxγ(1+x)λ,γ>−1,λ∈R+, where the kernel k(x,y) can be of different kinds. The rule, which is stable and fast convergent, is used in order to construct a computational scheme involving the single product integration rule studied in [23]. It is shown that the “compound quadrature sequence” represents an efficient proposal for saving 1/3 of the evaluations of the function f, under unchanged speed of convergence.

Weighted [formula omitted] approximation on [formula omitted] via discrete de la Vallée Poussin means

We consider some discrete approximation polynomials, namely discrete de la Vallée Poussin means, which have been recently deduced from certain delayed arithmetic means of the Fourier–Jacobi partial sums, in order to get a near–best approximation in suitable spaces of continuous functions equipped with the weighted uniform norm. By the present paper we aim to analyze the behavior of such discrete de la Vallée means in weighted L1 spaces, where we provide error bounds for several classes of functions, included functions of bounded variation. In all the cases, under simple conditions on the involved Jacobi weights, we get the best approximation order. During our investigations, a weighted L1 Marcinkiewicz type inequality has been also stated.

Some error bounds for Gauss–Jacobi quadrature rules

We estimate the error of Gauss–Jacobi quadrature rule applied to a function f, which is supposed locally absolutely continuous in some Besov type spaces, or of bounded variation on [−1,1]. In the first case the error bound concerns the weighted main part φ-modulus of smoothness of f introduced by Z. Ditzian and V. Totik, while in the second case we deal with a Stieltjes integral with respect to f. The stated estimates generalize several error bounds from literature and, in both the cases, they assure the same convergence rate of the error of best polynomial approximation in weighted L1 space.

Global stability for the prion equation with general incidence.

We consider the so-called prion equation with the general incidence term introduced in [14], and we investigate the stability of the steady states. The method is based on the reduction technique introduced in [11]. The argument combines a recent spectral gap result for the growth-fragmentation equation in weighted L1 spaces and the analysis of a nonlinear system of three ordinary differential equations.

The singular kernel coagulation equation with multifragmentation

In this article, we prove the existence of solutions to singular coagulation equations with multifragmentation. We use weighted L1 spaces to deal with the singularities and to obtain regular solutions. The Smoluchowski kernel is covered by our proof. The weak L1 compactness methods are applied to suitably chosen approximating equations as a base of our proof. A more restrictive uniqueness result is also given. Copyright © 2014 John Wiley & Sons, Ltd.

A kinetic model for polytropic gases with internal energy

AbstractWe consider a mixture of N ideal, polytropic gases. Each species is described by a distribution function fi(t, x, v, I) ≥ 0, 1 ≤ i ≤ N, defined on ${\rm I\!R}_+\times{\rm I\!R}^3\times{\rm I\!R}^3\times{\rm I\!R}_+$, and its evolution is governed by a Boltzmann‐type equation. In order to recover the energy law of polytropic gases, the authors of [4] proposed a kinetic model in the framework of a weighted L1 space. Another approach has been developed in [3] in the context of polyatomic gases. Following this previous lead, our model provides a L2 framework in both variables v and I, to eventually perform a mathematical study of the diffusion asymptotics, as it was done in [2] for a model without energy exchange. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Sobolev type embeddings and an inhomogeneous quasilinear elliptic equation on [formula omitted] with singular weights

In this paper we build a Sobolev type embedding from a weighted Sobolev space into a weighted L1 space on the entire space RN. As applications, we study the existence and the uniqueness of radial solutions of the inhomogeneous quasilinear elliptic equation {−div(|∇u|p−2∇u)+V(|x|)|u|q−2u=Q(|x|),x∈RN,u(x)→0,|x|→∞ with singular radial weighted functions V and Q.

Exponential convergence to equilibrium for subcritical solutions of the Becker–Döring equations

We prove that any subcritical solution to the Becker–Döring equations converges exponentially fast to the unique steady state with same mass. Our convergence result is quantitative and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, for which several bounds are provided. This improves the known convergence result by Jabin and Niethammer (2003) [17]. Our approach is based on a careful spectral analysis of the linearized Becker–Döring equation (which is new to our knowledge) in both a Hilbert setting and in certain weighted ℓ1 spaces. This spectral analysis is then combined with uniform exponential moment bounds of solutions in order to obtain a convergence result for the nonlinear equation.

Infinite matrices bounded on weighted [formula omitted] spaces

This paper investigates conditions under which an infinite matrix will be bounded as a linear operator between two weighted ℓ1 spaces, and examines the relationship between the matrix and the weight vectors. It is shown that every infinite matrix is bounded as an operator between two weighted ℓ1 spaces, for suitable weights. Necessary conditions and separate sufficient conditions for an infinite matrix to be bounded on some weighted ℓ1 space (with the same weight for its domain and range) are given. We then show a connection between these results and the classical Schur Test which gives a sufficient condition for an infinite matrix to be bounded on the standard ℓ2 (Hilbert) space.

Strong fragmentation and coagulation with power-law rates

Existence of global classical solutions to fragmentation and coagulation equations with unbounded coagulation rates has been recently proved for initial conditions with finite higher order moments. These results cannot be directly generalized to the most natural space of solutions with finite mass and number of particles due to the lack of precise characterization of the domain of the generator of the fragmentation semigroup. In this paper we show that such a generalization is possible in the case when both fragmentation and coagulation are described by power-law rates which are commonly used in the engineering practice. This is achieved through direct estimates of the resolvent of the fragmentation operator, which in this case is explicitly known, proving that it is sectorial and carefully intertwining the corresponding intermediate spaces with appropriate weighted L1 spaces.

Approximation with polynomial kernels and SVM classifiers

This paper presents an error analysis for classification algorithms generated by regularization schemes with polynomial kernels. Explicit convergence rates are provided for support vector machine (SVM) soft margin classifiers. The misclassification error can be estimated by the sum of sample error and regularization error. The main difficulty for studying algorithms with polynomial kernels is the regularization error which involves deeply the degrees of the kernel polynomials. Here we overcome this difficulty by bounding the reproducing kernel Hilbert space norm of Durrmeyer operators, and estimating the rate of approximation by Durrmeyer operators in a weighted L1 space (the weight is a probability distribution). Our study shows that the regularization parameter should decrease exponentially fast with the sample size, which is a special feature of polynomial kernels.

Almost Everywhere Convergence of Orthogonal Expansions of Several Variables

For a weighted L1 space on the unit sphere of Rd+1, in which the weight functions are invariant under finite reflection groups, a maximal function is introduced and used to prove the almost everywhere convergence of orthogonal expansions in h-harmonics. The result applies to various methods of summability, including the de La Vallee Poussin means and the Cesaro means. Similar results are also established for weighted orthogonal expansions on the unit ball and on the simplex of Rd.

Cyclic elements under translation in weighted L1 spaces on R+

Cyclic elements under translation in weighted L1 spaces on R+

Spectral analysis in weighted L1 spaces on R

Spectral analysis in weighted L1 spaces on R