Universal Consistency of Local Polynomial Kernel Regression Estimates

Regression function estimation from independent and identically distributed data is considered. The L ₂ error with integration with respect to the design measure is used as an error criterion. It is shown that suitably defined local polynomial kernel estimates are weakly and strongly universally consistent, i.e., it is shown that the L ₂ errors of these estimates converge to zero almost surely and in L ₁ for all distributions.     

Rates of convergence for partitioning and nearest neighbor regression estimates with unbounded data

Estimation of regression functions from independent and identically distributed data is considered. The L₂ error with integration with respect to the design measure is used as an error criterion. Usually in the analysis of the rate of convergence of estimates besides smoothness assumptions on the regression function and moment conditions on Y also boundedness assumptions on X are made. In this article we consider partitioning and nearest neighbor estimates and show that by replacing the boundedness assumption on X by a proper moment condition the same rate of convergence can be shown as for bounded data.     

Polyharmonic Approximation on the Sphere

The purpose of this article is to provide new error estimates for a popular type of spherical basis function (SBF) approximation on the sphere: approximating by linear combinations of Green’s functions of polyharmonic differential operators. We show that the L _p approximation order for this kind of approximation is σ for functions having L _p smoothness σ (for σ up to the order of the underlying differential operator, just as in univariate spline theory). This improves previous error estimates, which penalized the approximation order when measuring error in L _p , p>2 and held only in a restrictive setting when measuring error in L _p , p<2.     

Analysis of finite element methods for second order boundary value problems using mesh dependent norms

This paper presents a new approach to the analysis of finite element methods based onC ⁰-finite elements for the approximate solution of 2nd order boundary value problems in which error estimates are derived directly in terms of two mesh dependent norms that are closely ralated to theL ₂ norm and to the 2nd order Sobolev norm, respectively, and in which there is no assumption of quasi-uniformity on the mesh family. This is in contrast to the usual analysis in which error estimates are first derived in the 1st order Sobolev norm and subsequently are derived in theL ₂ norm and in the 2nd order Sobolev norm — the 2nd order Sobolev norm estimates being obtained under the assumption that the functions in the underlying approximating subspaces lie in the 2nd order Sobolev space and that the mesh family is quasi-uniform.     

Extension Theorems for Spaces Arising from Approximation by Translates of a Basic Function

We establish extension theorems for functions in spaces which arise naturally in studying interpolation by radial basic functions. These spaces are akin in some way to the non-integer-valued Sobolev spaces, although they are considerably more general. Such extensions allow us to establish local error estimates in a way which we make precise in the introductory section of our paper. There are many other applications of these fundamental results, including improved L_p error estimates for interpolation by shifts of a single basic function, but these applications have been left to a later paper.     

A Class of Periodic Function Spaces and Interpolation on Sparse Grids

We present a unified approach to error estimates of periodic interpolation on equidistant, full, and sparse grids for functions from a scale of function spaces which includes L ₂-Sobolev spaces, the Wiener algebra and the Korobov spaces.     

A posteriori error estimates for a coupled wave system with a local damping

We study a finite element method applied to a system of coupled wave equations in a bounded smooth domain in \mathbbR^d {\mathbb{R}^d} , d = 1, 2, 3, associated with a locally distributed damping function. We start with a spatially continuous finite element formulation allowing jump discontinuities in time. This approach yields, L ₂(L ₂) and L _∞(L ₂), a posteriori error estimates in terms of weighted residuals of the system. The proof of the a posteriori error estimates is based on the strong stability estimates for the corresponding adjoint equations. Optimal convergence rates are derived upon the maximal available regularity of the exact solution and justified through numerical examples. Bibliography: 14 titles. Illustrations: 4 figures.     

PiecewiseL-splines

Leta=x ₀<x ₁<...<x _N =b be a partition of the interval [a, b] and letL be a normalm-th order linear differential operator. The purpose of this note is to point out that spline functions in one variable need not be excluded to piecewise fits of functions belonging to the null spaceN(L ^* L) on each closed subinterval [x _i,x _i+1], 0in-1 but may be extended to piecewise fits of functions belonging toN(L _i ^* L _i) on each subinterval [x _i,x _i+1] provided theL _i's are selected from a uniformly bounded family of normal linear differential operators. Furthermore when theL _i's are so selected one obtains the usual integral relations and error estimates obtained for splines [2, 8 and 9] including the extended error estimates obtained by Swartz and Varga [10].     

The best approximation of Laplace operator by linear bounded operators in the space <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We obtain close two-sided estimates for the best approximation of Laplace operator by linear bounded operators on the class of functions for which the square of the Laplace operator belongs to the L _p -space. We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class defined with an error. In a particular case (p = 2) we solve all three problems exactly.     

On Jackson-Type Inequalities for Functions Defined on a Sphere

We obtain exact estimates for the approximation of functions defined on a sphere in the metrics of C and L ₂ by linear methods of summation of Fourier series in spherical harmonics in the case where differential and difference properties of these functions are defined in the space L ₂. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 291–304, March, 2005.     

On the relationship between functions with the same optimal knots in spline and piecewise polynomial approximation

The relationship between functions with the same optimal knots for L₂[0, 1] approximation by kth order splines or piecewise polynomials is investigated. It is shown that if two functions have positive continuous kth derivatives they will have the same optimal knots if and only if they differ by a polynomial of order k. An application to design selection for continuous time regression is considered and extensions to L_p approximation are also provided.     

L-Splines

In this paper, we study the problem of unique interpolation and approximation by a class of spline functions,L-splines, containing as special cases the deficient and generalized spline functions ofAhlberg, Nilson, andWalsh [3, 5, 6], the Chebyshevian spline functions ofKarlin andZiegler [27], and the piecewise Hermite polynomial functions, as considered in [17]. We first give sufficient conditions for unique interpolation byL-spline functions in Section 2. Then, we obtain newL andL ² error estimates for interpolation byL-splines in Section 4, and show that these error estimates are, in a certain sense, sharp. In addition, we make a similar study for theg-splines ofSchoenberg, cf. [44, 3], in Section 5. In Section 6, an application of these new error estimates is made to the analysis of the error made in the use of finite dimensional subspaces ofL-splines andg-splines. in the Rayleigh-Ritz procedure for the class of nonlinear two-point boundary value problems studied in [17].Because of the rapid growth of the number of papers devoted to or connected with the topic of splines, we believe that a compilation of papers on splines for the reader's use is desirable, and such a list is found in the References at the end of this paper.This research was supported in part by NSF Grant GP-5553Papers not specifically concerned with splines are referred to in the text by [1, 2], etc.     

Semidiscrete Galerkin Approximation for a Linear Stochastic Parabolic Partial Differential Equation Driven by an Additive Noise

We study the semidiscrete Galerkin approximation of a stochastic parabolic partial differential equation forced by an additive space-time noise. The discretization in space is done by a piecewise linear finite element method. The space-time noise is approximated by using the generalized L₂ projection operator. Optimal strong convergence error estimates in the L₂ and norms with respect to the spatial variable are obtained. The proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem. The error estimates are applicable in the multi-dimensional case. AMS subject classification (2000) 65M, 60H15, 65C30, 65M65.Received April 2004. Revised September 2004. Communicated by Anders Szepessy.     

Estimates of difference norms for functions in anisotropic Sobolev spaces

We investigate the spaces of functions on ?ⁿ for which the generalized partial derivatives Dequation/tex2gif-sup-2.gif_kf exist and belong to different Lorentz spaces Lequation/tex2gif-sup-3.gif . For the functions in these spaces, the sharp estimates of the Besov type norms are found. The methods used in the paper are based on estimates of non‐increasing rearrangements. These methods enable us to cover also the case when some of the p_k's are equal to 1. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Near Best Tree Approximation

Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in the approximant are required to align themselves in a certain connected tree structure. This makes their positions easy to encode. Previous work [4,6] has established upper bounds for the error of tree approximation for certain (Besov) classes of functions. This paper, in contrast, studies tree approximation of individual functions with the aim of characterizing those functions with a prescribed approximation error. We accomplish this in the case that the approximation error is measured in L ₂, or in the case p2, in the Besov spaces B _p ⁰(L _p ), which are close to (but not the same as) L _p . Our characterization of functions with a prescribed approximation order in these cases is given in terms of a certain maximal function applied to the wavelet coefficients.     

The P-version of the mixed-finite element method for nonlinear second-order elliptic problems

The p-version of the mixed finite element method is considered for nonlinear second-order elliptic problems. Existence and uniqueness of the approximation are demonstrated and optimal order error estimates in L² are derived for the three relevant functions. Error estimates for the scalar function are also given in L^q, 2 ? q ? + ∞. © 1996 John Wiley & Sons, Inc.     

Error estimates of cubature formulas exact for Haar polynomials in classes of two-variable functions satisfying a general Lipschitz condition

Upper error estimates are obtained for cubature formulas with the Haar d-property in the classes Lip(L ₁, L ₂) of two-variable functions satisfying a general Lipschitz condition. It is shown that the error of minimal cubature formulas possessing the Haar d-property have the best order of convergence to zero in the indicated classes.     

Applications of discrete maximal <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> regularity for finite element operators

In this paper, we present applications of discrete maximal L _p regularity for finite element operators. More precisely, we show error estimates of order h ² for linear and certain semilinear problems in various L _p (Ω)-norms. Discrete maximal regularity allows us to prove error estimates in a very easy and efficient way. Moreover, we also develop interpolation theory for (fractional powers of) finite element operators and extend the results on discrete maximal L _p regularity formerly proved by the author. The author was supported by the DFG-Graduiertenkolleg 853.     

Numerical study using explicit multistep Galerkin finite element method for the MRLW equation

In this article, an explicit multistep Galerkin finite element method for the modified regularized long wave equation is studied. The discretization of this equation in space is by linear finite elements, and the time discretization is based on explicit multistep schemes. Stability analysis and error estimates of our numerical scheme are derived. Numerical experiments indicate the validation of the scheme by L₂– and L_∞– error norms and three invariants of motion.4 © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1875–1889, 2015     

and Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L ²(Ω) and H ¹(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.