Unicity of best one-sidedL 1-approximations

Summary This paper deals with the problem of uniqueness in one-sidedL ₁-approximation. The chief purpose is to characterize finite dimensional subspacesG of the space of continuous or differentiable functions which have a unique best one-sidedL ₁-approximation. In addition, we study a related problem in moment theory. These considerations have an important application to the uniqueness of quadrature formulae of highest possible degree of precision.     

The convergence of the best discrete linear Lp approximation as p → 1

It is well known that the best discrete linear L_p approximation converges to a special best Chebyshev approximation as p → ∞. In this paper it is shown that the corresponding result for the case p → 1 is also true. Furthermore, the special best L₁ approximation obtained as the limit is characterized as the unique solution of a nonlinear programming problem on the set of all L₁ solutions.     

Error estimates for a single phase quasilinear stefan problem with a forcing term

In this paper, we apply finite element Galerkin method to a singlephase quasi-linear Stefan problem with a forcing term. We consider the existence and uniqueness of a semidiscrete approximation and optimal error estimates inL ₂, L_∞,H ₁ andH ₂ norms for semidiscrete Galerkin approximations are derived.     

Problem of uniqueness of an element of the best nonsymmetric <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>-approximation of continuous functions with values in <Emphasis Type="Italic">KB</Emphasis>-spaces

We consider the problem of characterization of subspaces of uniqueness of an element of the best nonsymmetric L ₁-approximation of functions that are continuous on a metric compact set of functions with values in a KB-space. We find classes of test functions that characterize the uniqueness of an element of the best nonsymmetric approximation. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 7, pp. 867–878, July, 2008.     

Weak solutions of hessian equations

We consider the existence, uniqueness and Holder regularity of weak solutions of Hessian equations, determined by the elementary symmetric functions, with L_p inhomogeneous terms. The notion of weak solution is defined by approximation and our treatment draws on the classical theory, together with recent L_p estimates resulting from our isoperimetric inequalities for quermassintegrals on non-convex domains.     

Property A in an Infinite-Dimensional Space. I

   Abstract. We prove that an infinite-dimensional space of piecewise polynomial functions of degree at most n-1 with infinitely many simple knots, n ≥ 2 , satisfies Property A. Apart from its independent interest, this result allows us to solve an open classical problem (n ≥ 3 ) in theory of best approximation: the uniqueness of best L ₁ -approximation by n -convex functions to an integrable, continuous function defined on a bounded interval. In this first part of the paper we prove the case n=2 and give key results in order to complete the general proof in the second part.     

On strong perturbations of the haar system inL 1(0,1)-space

In the paper we consider a Haar system perturbed in the sense of theL ₁(0,1)-metric. We prove that this perturbation is stronger than perturbations in the case of basis stability or stability of complete systems; moreover, the systems obtained as the result of a perturbation are complete inL ₁(0,1). An approximation algorithm inL ₁(0,1) for these systems is given. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 596–602, October, 1999.     

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.     

Best one-sided L 1-approximation by harmonic functions

Let , where B is the open unit ball in (), and let denote the collection of functions h in which are harmonic on B and satisfy on . A function h ^* in is called a best harmonic one-sided L ¹-approximant to f if for all h in . This paper characterizes such approximants and discusses questions of existence and uniqueness. Corresponding results for approximation on the cylinder are also established, but the proofs in this case are more difficult and rely on recent work concerning tangential harmonic approximation. The characterizations are quite different in nature from those recently obtained for harmonic L ¹-approximation without a one-sidedness condition. Received: 25 September 1997     

Multivariate approximation in the worst case setting over reproducing kernel Hilbert spaces

We study the worst case setting for approximation of d variate functions from a general reproducing kernel Hilbert space with the error measured in the L_∞ norm. We mainly consider algorithms that use n arbitrary continuous linear functionals. We look for algorithms with the minimal worst case errors and for their rates of convergence as n goes to infinity. Algorithms using n function values will be analyzed in a forthcoming paper.We show that the L_∞ approximation problem in the worst case setting is related to the weighted L₂ approximation problem in the average case setting with respect to a zero-mean Gaussian stochastic process whose covariance function is the same as the reproducing kernel of the Hilbert space. This relation enables us to find optimal algorithms and their rates of convergence for the weighted Korobov space with an arbitrary smoothness parameter α>1, and for the weighted Sobolev space whose reproducing kernel corresponds to the Wiener sheet measure. The optimal convergence rates are n^-(α-1)/2 and n^-1/2, respectively.We also study tractability of L_∞ approximation for the absolute and normalized error criteria, i.e., how the minimal worst case errors depend on the number of variables, d, especially when d is arbitrarily large. We provide necessary and sufficient conditions on tractability of L_∞ approximation in terms of tractability conditions of the weighted L₂ approximation in the average case setting. In particular, tractability holds in weighted Korobov and Sobolev spaces only for weights tending sufficiently fast to zero and does not hold for the classical unweighted spaces.     

Best φ-approximation by n-convex functions

A n-convex function defined on a bounded open interval J ₀ n ≥2 is the (n?l)-st indefinite integral of a nondecreasing function. This fact and the simple structure of the latter enable to obtain concrete results about a n-convex best φ approximation g to a function f ? L _φ on J ₀, where φ: [0, ∞) → [0, ∞) is a convex function that generaJizes the p^th -power functions, 1 ≤ p < ∞. It is shown that g may also be a best generalized spline φ approximation to the restriction of f on the maximal subintervals of J₀ where g is a generalized spline. This is the situation in some cases, among which the L_p -approximation is includedp ≥ 1. For n = 2 it is proven that g is a polynomial of best φ-approximation to f ? L _φ on any maximal interval where g is a polynomial. If f is in addition continuous, then this fact implies the uniqueness of g Under the same assumption, it is shown that the best 3-convex L ₁-approximation is also unique whenever its derivative is bounded.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

Near-best multivariate approximation by Fourier series, Chebyshev series and Chebyshev interpolation

A set of results concerning goodness of approximation and convergence in norm is given for L_∞ and L₁ approximation of multivariate functions on hypercubes. Firstly the trigonometric polynomial formed by taking a partial sum of a multivariate Fourier series and the algebraic polynomials formed either by taking a partial sum of a multivariate Chebyshev series of the first kind or by interpolating at a tensor product of Chebyshev polynomial zeros are all shown to be near-best L_∞ approximations. Secondly the trigonometric and algebraic polynomials formed by taking, respectively, a partial sum of a multivariate Fourier series and a partial sum of a multivariate Chebyshev series of the second kind are both shown to be hear-best L₁ approximations. In all the cases considered, the relative distance of a near-best approximation from a corresponding best approximation is shown to be at most of the order of Π log n_j, where n_j (j = 1, 2,…, N) are the respective degrees of approximation in the N individual variables. Moreover, convergence in the relevant norm is established for all the sequences of near-best approximations under consideration, subject to appropriate restrictions on the function space.     

Uniform reciprocal approximation subject to coefficient constraints

In this paper best approximation by reciprocals of functions of a subspace U_n=span (u₁,...,u_n) satisfying coefficient constraints is considered. We present a characterization of best approximations. When (u₁,...,u_n) is a Descartes system an explicit characterization of best approximations by equioscillations is given. Existence and uniqueness results are shown. Moreover, the theory is applied to best approximaitons by reciprocals of polynomials.     

A Vlasov–Poisson plasma model with non‐L1 data

It is considered the Vlasov–Poisson equation for a plasma confined in an unbounded cylinder and it is proven an existence and uniqueness result for non‐L₁ (but almost L₁) initial charge distribution. Copyright © 2004 John Wiley & Sons, Ltd.     

Approximation in the mean by convex functions

We give elementary proofs for the existence and uniqueness of the best L₁-approximation to a continuous function from the class of convex functions on a closed interval, and describe thebest approximation in terms of certain piecewise linear functions.     

On the uniqueness of elements of the best approximation and the best one-sided approximation in the spaceL 1

The problem of the uniqueness of elements of the best approximations in the spaceL ₁ [a, b] is studied. We consider the problem of the best approximation and the best (, )-approximation of continuous functions and the problem of the best one-sided approximation of continuously differentiable functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 475–483, May, 1994.     

Boundary control of parabolic systems: Finite-element approximation

Finite-element approximation of a Dirichlet type boundary control problem for parabolic systems is considered. An approach based on the direct approximation of an input-output semigroup formula is applied. Error estimates inL ₂[OT; L ₂()] andL ₂[OT; L ₂()] norms are derived for optimal state and optimal control, respectively. It turns out that these estimates areoptimal with respect to the approximation theoretic properties.Research supported in part under Grant no. NSG 4015, National Aeronautics and Space Administration.     

The Gibbs phenomenon for bestL 1-trigonometric polynomial approximation

The classical Gibbs phenomenon for the Fourier sections (bestL ₂-trigonometric polynomial approximants) of a jump function asserts that, near the jump, these sections overshoot the function by an asymptotically constant factorg (theL ₂-Gibbs constant). In this paper we show that, for a class of one-jump discontinuous functions, a similar phenomenon holds for the trigonometric polynomials of bestL ₁-approximation. We determine theL ₁-Gibbs constant , which is substantially smaller thang. Furthermore, we prove that uniform convergence of bestL ₁-approximants takes place on intervals that avoid the jump. In the analysis we obtain some strong uniqueness theorems for bestL ₁-approximants.Communicated by Vladimir N. Temlyakov.     

Critical Points and Error Rank in Best H 2 Matrix Rational Approximation of Fixed McMillan Degree

This paper deals with best rational approximation of prescribed McMillan degree to matrix-valued functions in the real Hardy space of the complement of the unit disk endowed with the Frobenius L ₂ -norm. We describe the topological structure of the set of approximants in terms of inner-unstable factorizations. This allows us to establish a two-sided tangential interpolation equation for the critical points of the criterion, and to prove that the rank of the error F-H is at most k-n when F is rational of degree k , and H is critical of degree n . In the particular case where k=n , it follows that H=F is the unique critical point, and this entails a local uniqueness result when approximating near-rational functions. January 23, 1996. Date revised: September 16, 1996.