Norm Estimates Research Articles

Two-sided estimates of norms of a class of matrix operators

Two-sided estimates of norms of a class of matrix operators

Best kernel approximation in Bergman spaces

Let H be a reproducing kernel Hilbert space of analytic functions on the unit disk D. The best kernel approximation problem for H is the following: given any positive integer n and any function f∈H find the best norm approximation of f by a linear combination of no more than n kernel functions K(z,zk), 1≤k≤n. The purpose of this paper is to prove the existence of best kernel approximation for weighted Bergman spaces with standard weights.

Finite-iteration learning tracking of multi-agent systems via the distributed optimization method

In this paper, the finite-iteration tracking issue of multi-agent systems is investigated with the distributed optimization method. Since the infinite-iteration strategy is unrealistic, the finite-iteration approach is adopted to study the problem. Moreover, the tracking performance is undesired for the traditional iterative learning schemes. Then optimization technique is used to cover the drawback. The tracking error is regarded as the optimization objective, and the consensus tracking issue is considered as the distributed optimization problem. An appropriate learning control strategy is designed on the basis of the gradient of the tracking error. Further, the optimization issue can be solved by means of the finite-iteration approach, and the learning algorithm can improve the tracking performance. Theoretical results are proposed by using the norm estimation and some inequalities skills. At last, a numerical simulation is given to illustrate the effectiveness of the main results.

Characterization of the non-homogenous Dirac-harmonic equation

We introduce the non-homogeneous Dirac-harmonic equation for differential forms and characterize the basic properties of solutions to this new type of differential equations, including the norm estimates and the convergency of sequences of the solutions. As applications, we prove the existence and uniqueness of the solutions to a special non-homogeneous Dirac-harmonic equation and its corresponding reverse Hölder inequality.

High accuracy analysis of Galerkin finite element method for Klein–Gordon–Zakharov equations

The main aim of this paper is to propose a Galerkin finite element method (FEM) for solving the Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity, and to give the error estimations of approximate solutions about the electronic fast time scale component q and the ion density deviation r. In which, the bilinear element is used for spatial discretization, and a second order difference scheme is implemented for temporal discretization. Moreover, by use of the combination of the interpolation and Ritz projection technique, and the interpolated postprocessing approach, for weaker regularity requirements of the exact solution, the superclose and global superconvergence estimations of q in H1−norm are deduced. At the same time, the superconvergence of the auxiliary variable φ (−Δφ=rt) in H1−norm and optimal error estimation of r in L2−norm are derived. Meanwhile, we also discuss the extensions of our scheme to more general finite elements. Finally, the numerical experiments are provided to confirm the validity of the theoretical analysis.

A unified framework of high order structure-preserving B-splines Galerkin methods for coupled nonlinear Schrödinger systems

Using a general computational framework, we derive an optimal error estimate in the L2 norm for a semi discrete method based on high order B-splines Galerkin spatial discretizations, applied to a coupled nonlinear Schrödinger system with cubic nonlinearity. A fully discrete method based on a conservative nonlinear splitting Crank-Nicolson time step is then proposed; and conservation of the mass and the energy is theoretically proven. To validate its accuracy in space and time, and its conservation properties, several numerical experiments are carried out with B-splines up to order 7.

FUNCTIONAL CALCULI FOR SECTORIAL OPERATORS AND RELATED FUNCTION THEORY

AbstractWe construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.

Compact Hankel Operators Between Distinct Hardy Spaces and Commutators

The paper is devoted to the study of compactness of Hankel operators acting between distinct Hardy spaces generated by Banach function lattices. We prove an analogue of Hartman’s theorem characterizing compact Hankel operators in terms of properties of their symbols. As a byproduct we give an estimation of the essential norm of such operators. Furthermore, compactness of commutators and semicommutators of Toeplitz operators for unbounded symbols is discussed.

A Bound Based Method for Beamformer Design in Downlink Cloud Radio Access Networks

Cloud Radio Access Network is a candidate solution in 5G and beyond where we intend to serve as many users as possible with minimum transmit power and minimum fronthaul data transmission. In this study, we aim to maximize a mixed expression of the number of users served, the number of fronthaul links used, and the total transmitted power by beamforming optimization under imperfect channel state information. We find a theoretical upper bound and propose a method based on the bound derivation. We compare its performance with heuristic search and norm approximation methods solving a series of convex sub-problems. The detailed simulations show that the proposed method outperforms other baseline techniques in terms of the number of users served, fronthaul data rate, power consumption, and computational complexity.

Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation

In this paper, we propose L^2(J;H^1_0(Omega )) and L^2(J;L^2(Omega )) norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.

Robust Hybrid High-Order Method on Polytopal Meshes with Small Faces

Abstract We design a Hybrid High-Order (HHO) scheme for the Poisson problem that is fully robust on polytopal meshes in the presence of small edges/faces. We state general assumptions on the stabilisation terms involved in the scheme, under which optimal error estimates (in discrete and continuous energy norms, as well as L 2 L^{2} -norm) are established with multiplicative constants that do not depend on the maximum number of faces in each element, or the relative size between an element and its faces. We illustrate the error estimates through numerical simulations in 2D and 3D on meshes designed by agglomeration techniques (such meshes naturally have elements with a very large numbers of faces, and very small faces).

Uniform estimates for Stokes equations in a domain with a small hole and applications in homogenization problems

We consider the Dirichlet problem of the Stokes equations in a domain with a shrinking hole in $${\mathbb {R}}^d, \ d\ge 2$$ . A typical observation is that, the Lipschitz norm of the domain goes to infinity as the size of the hole goes to zero. Thus, if $$p\ne 2$$ , the classical results indicate that the $$W^{1,p}$$ estimate of the solution may go to infinity as the size of the hole tends to zero. With the presence of the shrinking hole in a fixed domain, we give a complete description for the uniform $$W^{1,p}$$ estimates of the solution for all $$1<p<\infty $$ . We show that the uniform $$W^{1,p}$$ estimate holds if and only if $$d'<p<d$$ ( $$p=2$$ when $$d=2$$ ). We then give two applications in the study of homogenization problems in fluid mechanics: a generalization of the restriction operator and a construction of Bogovskii type operator in perforated domains with a quantitative estimate of the operator norm.

Two-stage sampling in the estimation of growth parameters and percentile norms: sample weights versus auxiliary variable estimation

BackgroundThe use of auxiliary variables with maximum likelihood parameter estimation for surveys that miss data by design is not a widespread approach, despite its documented improved efficiency over traditional approaches that deploy sampling weights. Although efficiency gains from the use of Normally distributed auxiliary variables in a model have been recorded in the literature, little is known about the effects of non-Normal auxiliary variables in the parameter estimation.MethodsWe simulate growth data to mimic SCALES, a two-stage survey of language development with a screening phase (stage one) for which data are observed for the whole sample and an intensive assessments phase (stage two), for which data are observed for a sub-sample, selected using stratified random sampling. In the simulation, we allow a fully observed Poisson distributed stratification criterion to be correlated with the partially observed model responses and develop five generalised structural equation growth models that host the auxiliary information from this criterion. We compare these models with each other and with a weighted growth model in terms of bias, efficiency, and coverage. We finally apply our best performing model to SCALES data and show how to obtain growth parameters and population norms.ResultsParameter estimation from a model that incorporates a non-Normal auxiliary variable is unbiased and more efficient than its weighted counterpart. The auxiliary variable method is capable of producing efficient population percentile norms and velocities.ConclusionsThe deployment of a fully observed variable that dominates the selection of the sample and correlates strongly with the incomplete variable of interest appears beneficial for the estimation process.

Negative norm estimates and superconvergence results in Galerkin method for strongly nonlinear parabolic problems

The conforming finite element Galerkin method is applied to discretise in the spatial direction for a class of strongly nonlinear parabolic problems. Using elliptic projection of the associated linearised stationary problem with Gronwall type result, optimal error estimates are derived, when piecewise polynomials of degree r≥1 are used, which improve upon earlier results of Axelsson ((1977) [3]) requiring for 2d r≥2 and for 3d r≥3. Based on quasi-projection technique introduced by Douglas et al. ((1978) [11]), superconvergence result for the error between Galerkin approximation and approximation through quasi-projection is established for the semidiscrete Galerkin scheme. Further, a priori error estimates in Sobolev spaces of negative index are derived. Moreover, in a single space variable, nodal superconvergence results between the true solution and Galerkin approximation are established.

A weak Galerkin least squares finite element method of Cauchy problem for Poisson equation

In this paper, we introduce a weak Galerkin (WG) least squares finite element method for the Cauchy problem. This finite element method generates a symmetric, positive definite system and can work on general mesh. Optimal order of convergence for the WG approximation in an energy norm is established. The numerical examples confirm the theory.

On the Sharp-Weighted Norm Estimate for the Commutator of Littlewood–Paley Operators

On the Sharp-Weighted Norm Estimate for the Commutator of Littlewood–Paley Operators

Essential norm estimates for multiplication operators on $$L_p(\mu )$$ spaces

Essential norm estimates for multiplication operators on $$L_p(\mu )$$ spaces

Weighted composition operators between n-th α-weighted spaces

Let Vnα be the Banach space of holomorphic functions on the open unit disk 𝔻 in the complex plane consisting of those f such that ∥ f ∥Vnα:=∑i=0n−1 f i0+supz∈D1−|z|2α fnz< ∞ and Vn,0α be the closed subspace of Vnα consisting of those f for which lim|z|→1(1 − |z|2)α |f (n)(z)|=0, where n is any nonnegative integer and α>0. We give boundedness characterizations, norm estimates and essential norm estimates of weighted composition operators Wψ,φ : Vnα→Vmβ and Wψ,φ : Vn,0α→Vm,0β, respectively, where Wψ,φ f (z)=ψ(z) f (φ(z)). As a corollary, we characterize the compactness of Wψ,φ. Specifically, our characterizations involve not only the classical Julia–Carathéodory type condition, but also the powers φk. In addition, our results extend several well-known results in the literature.

Norm Resolvent Convergence of Discretized Fourier Multipliers

We prove norm estimates for the difference of resolvents of operators and their discrete counterparts, embedded into the continuum using biorthogonal Riesz sequences. The estimates are given in the operator norm for operators on square integrable functions, and depend explicitly on the mesh size for the discrete operators. The operators are a sum of a Fourier multiplier and a multiplicative potential. The Fourier multipliers include the fractional Laplacian and the pseudo-relativistic free Hamiltonian. The potentials are real, bounded, and Hölder continuous. As a side-product, the Hausdorff distance between the spectra of the resolvents of the continuous and discrete operators decays with the same rate in the mesh size as for the norm resolvent estimates. The same result holds for the spectra of the original operators in a local Hausdorff distance.

Integral operators for nonlocally compact group modules

The article is devoted to integral operators for nonlocally compact group modules. Operator valued functions for nonlocally compact groups and their normed spaces of different types are investigated. For this purpose quasi-invariant measures on nonlocally compact groups are used. Estimates of norms of integral operators are given. Moreover, convolutions of functions having operator values and values in Banach spaces are scrutinized. Examples are given.