Lp Convergence Research Articles

Nonparametric density estimation for spatial data with wavelets

Nonparametric density estimators are studied for d-dimensional, strongly spatial mixing data which are defined on a general N-dimensional lattice structure. We consider linear and nonlinear hard thresholded wavelet estimators derived from a d-dimensional multi-resolution analysis. We give sufficient criteria for the consistency of these estimators and derive rates of convergence in Lp′ for p′∈[1,∞). For this reason, we study density functions which are elements of a d-dimensional Besov space Bp,qs(Rd). We also verify the analytic correctness of our results in numerical simulations.

Lp convergence and complete convergence for weighted sums of AANA random variables

ABSTRACTLet {Xn, n ⩾ 1} be a sequence of asymptotically almost negatively associated (AANA, for short) random variables which is stochastically dominated by a random variable X, and {dni, 1 ⩽ i ⩽ n, n ⩾ 1} be a sequence of real function, which is defined on a compact set E. Under some suitable conditions, we investigate some convergence properties for weighted sums of AANA random variables, especially the Lp convergence and the complete convergence. As an application, the Marcinkiewicz–Zygmund-type strong law of large numbers for AANA random variables is obtained.

Quaternionic B-splines

We introduce B-splines Bq of quaternionic order q, defined on the real line for the purposes of multi-channel signal analysis. The functions Bq are defined first by their Fourier transforms, then as the solutions of a distributional differential equation of quaternionic order. The equivalence of these definitions requires properties of quaternionic Gamma functions and binomial expansions, both of which we investigate. The relationship between Bq and a backwards difference operator is shown, leading to a recurrence formula. We show that the collection of integer shifts of Bq is a Riesz basis for its span, hence generating a multiresolution analysis. Finally, we demonstrate the pointwise and Lp convergence of the quaternionic B-splines to quaternionic Gaussian functions.

Some Strong Convergence Theorems for Asymptotically almost Negatively Associated Random Variables

In this work, the complete moment convergence and Lp convergence for asymptotically almost negatively associated (AANA, in short) random variables are investigated. As an application, the complete convergence theorem for weighted sums of AANA random variables is obtained. These theorems obtained extend and improve some earlier results.

Two-scale convergence in thin domains with locally periodic rapidly oscillating boundary

The aim of this paper is to adapt the notion of two-scale convergence in Lp to the case of a measure converging to a singular one. We present a specific case when a thin cylinder with locally perio ...

Genuine link Baskakov–Durrmeyer operators

Abstract In the present paper, we propose a sequence of generalized genuine Baskakov–Durrmeyer-type link operators. In terms of ordinary approximation, we estimate local and global direct results and also study the weighted approximation result. In terms of simultaneous approximation, we establish an asymptotic formula of Voronovskaja kind. In the last section, we prove convergence in L p {L_{p}} -norm.

Convergence in Lp for Feynman path integrals

We consider a class of Schrödinger equations with time-dependent smooth magnetic and electric potentials having a growth at infinity at most linear and quadratic, respectively. We study the convergence in Lp with loss of derivatives, 1<p<∞, of the time slicing approximations of the corresponding Feynman path integral. The results are completely sharp and hold for long time, where no smoothing effect is available. The techniques are based on the decomposition and reconstruction of functions and operators with respect to certain wave packets in phase space.

Uniform FIR approximation of causal Wiener filters, with applications to causal coherence

Leveraging the relationship between Wiener filtering and the coherence function, a version of coherence is defined that captures the causal relationship between WSS processes. This causal coherence is interpreted in a modeling context and used to demonstrate what a frequency dependent measure for causality both can and cannot represent. To understand how well frequency dependent causal coherence spectra can be estimated with finite order approximations, the convergence of FIR causal Wiener filters to the full IIR causal Wiener filter is investigated as filter length goes to infinity. The main results prove Lp convergence of the frequency responses for p=1,2,∞ under certain Hölder continuity conditions on the power spectra, as well as give asymptotic upper bounds for the convergence error. As a corollary, under the same conditions, the uniform convergence of the power spectra of AR approximations is shown as model order goes to infinity.

On best convergence of series in the systems ϕ(nx)

Sufficient conditions for the best convergence in Lp of series in systems ϕ(nx), where ϕ(x) are polynomials or series in trigonometric andWalsh systems, are obtained.

Boundary effects and large-time behaviour for quasilinear equations with nonlinear damping

This paper is concerned with the asymptotic behaviour of solutions to quasilinear hyperbolic equations with nonlinear damping on the quarter-plane (x, t) ∈ ℝ+ x ∈ ℝ+. We obtain the Lp (1 ≤ p ≤ +∞) convergence rates of the solution to the quasilinear hyperbolic equations without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

Lp approximation with rates by multivariate generalized discrete singular operators

Here we give the approximation properties with rates of multivariate generalized discrete versions of Picard, Gauss–Weierstrass, and Poisson–Cauchy singular operators over RN,N ≥ 1. We treat both the unitary and non-unitary cases of the operators above. We derive quantitatively Lp convergence of these operators to the unit operator by involving the Lp higher modulus of smoothness of an Lp function.

The Webster scalar curvature flow on CR sphere. Part I

This is the first of two papers, in which we prove some properties of the Webster scalar curvature flow. More precisely, we establish the long-time existence, Lp convergence and the blow-up analysis for the solution of the flow. As a by-product, we prove the convergence of the CR Yamabe flow on the CR sphere. The results in this paper will be used to prove a result of prescribing Webster scalar curvature on the CR sphere, which is the main result of the second paper.

QuickSelect Tree Process Convergence, With an Application to Distributional Convergence for the Number of Symbol Comparisons Used by Worst-Case Find

We define a sequence of tree-indexed processes closely related to the operation of the QuickSelect search algorithm (also known as Find) for all the various values of n (the number of input keys) and m (the rank of the desired order statistic among the keys). As a ‘master theorem’ we establish convergence of these processes in a certain Banach space, from which known distributional convergence results as n → ∞ about (1)the number of key comparisons requiredare easily recovered (a)when m/n → α ∈ [0, 1], and(b)in the worst case over the choice of m. From the master theorem it is also easy, for distributional convergence of(2)the number of symbol comparisons required, both to recover the known result in the case (a) of fixed quantile α and to establish our main new result in the case (b) of worst-case Find.Our techniques allow us to unify the treatment of cases (1) and (2) and indeed to consider many other cost functions as well. Further, all our results provide a stronger mode of convergence (namely, convergence in Lp or almost surely) than convergence in distribution. Extensions to MultipleQuickSelect are discussed briefly.

Asymptotic properties of supercritical branching processes in random environments

We consider a supercritical branching process (Z n ) in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale W n = Z n / $\mathbb{E}$ [Z n |ξ], the convergence rates of W − W n (by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in L p , and the convergence in probability), and limit theorems (such as central limit theorems, moderate and large deviations principles) on (log Z n ).

L p -convergence rates to nonlinear diffusion waves for quasilinear equations with nonlinear damping

This paper is concerned with the asymptotic behavior of the solution for quasilinear hyperbolic equations with nonlinear damping. The main novelty in this paper is that we obtain the Lp(2 ≤ p ≤ +∞) convergence rates of the solution to the quasilinear hyperbolic equations, and we need none of the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton (Q Appl Math 61:295–313, 2003).

Convergence in L p ([0, T]) of Wavelet Expansions of φ-Sub-Gaussian Random Processes

The article presents new results on convergence in L p ([0,T]) of wavelet expansions of φ-sub-Gaussian random processes. The convergence rate of the expansions is obtained. Specifications of the obtained results are discussed.

Wong–Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces

In this paper, we study the Wong–Zakai approximation of the solution to the stochastic differential equation on a domain D in a Euclidean space with normal reflection at the boundary. We prove the Lp convergence of the approximation in C([0,T]→D̄) under some general conditions on D.

Band-limited scaling and wavelet expansions

Operators Qjf=∑n∈Z〈f,φ˜jn〉φjn are studied for a class of band-limited functions φ and a wide class of tempered distributions φ˜. Convergence of Qjf to f as j→+∞ in the L2-norm is proved under a very mild assumption on φ, φ˜, and the rate of convergence is equal to the order of Strang–Fix condition for φ. To study convergence in Lp, p>1, we assume that there exists δ∈(0,1/2) such that φˆ¯φ˜ˆ=1 a.e. on [−δ,δ], φˆ=0 a.e. on [l−δ,l+δ] for all l∈Z∖{0}. For appropriate band-limited or compactly supported functions φ˜, the estimate ‖f−Qjf‖p⩽Cωr(f,2−j)Lp, where ωr denotes the r-th modulus of continuity, is obtained for arbitrary r∈N. For tempered distributions φ˜, we proved that Qjf tends to f in the Lp-norm, p⩾2, with an arbitrary large approximation order. In particular, for some class of differential operators L, we consider φ˜ such that Qjf=∑n∈ZLf(2−j⋅)(n)φjn. The corresponding wavelet frame-type expansions are found.

Applications of Fourier analysis in homogenization of Dirichlet problem I. Pointwise estimates

In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex and smooth domain, and smooth operator and boundary data, we prove pointwise convergence results, namely|uε(x)−u0(x)|⩽Cκε(d−1)/21d(x)κ,∀x∈D,∀κ>d−1, where uε and u0 are solutions of respectively oscillating and homogenized Dirichlet problems, and d(x) is the distance of x from the boundary of D. As a corollary for all 1⩽p<∞ we obtain Lp convergence rate as well.

Formula omitted] and almost sure convergence of a Milstein scheme for stochastic partial differential equations

In this paper, Lp convergence and almost sure convergence of the Milstein approximation of a partial differential equation of advection–diffusion type driven by a multiplicative continuous martingale is proven. The (semidiscrete) approximation in space is a projection onto a finite dimensional function space. The considered space approximation has to have an order of convergence fitting to the order of convergence of the Milstein approximation and the regularity of the solution. The approximation of the driving noise process is realized by the truncation of the Karhunen–Loève expansion of the driving noise according to the overall order of convergence. Convergence results in Lp and almost sure convergence bounds for the semidiscrete approximation as well as for the fully discrete approximation are provided.