Projections on Convex Sets in the Relaxed Limit

The paper establishes the continuity of the best approximation, or the projection, of a function in L _p for p[1,), on a closed convex set in the space, when the set varies and converges to a limit set in the Young-measure relaxation of the space. To this end a strong-type convergence and a convexity structure are identified on the space of Young measures. The appropriate convergence of sets with respect to which the continuity holds is the Mosco convergence of sets associated with the strong-type convergence of functions.     

Whitney type inequalities for local anisotropic polynomial approximation

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in L_p(Q) with 1≤p≤∞. Here Q is a d-parallelepiped in R^d with sides parallel to the coordinate axes. We consider the error of best approximation of a function f by algebraic polynomials of fixed degree at most r_i−1 in variable , and relate it to a so-called total mixed modulus of smoothness appropriate to characterizing the convergence rate of the approximation error. This theorem is derived from a Johnen type theorem on equivalence between a certain K-functional and the total mixed modulus of smoothness which is proved in the present paper.     

Calculation of linear bestL p -approximations

In this paper we are concerned with finding theL _p -solution (i.e. minimizing theL _p -norm of the residual vector) to a linear approximation problem or, equivalenty, to an overdetermined system of linear equations. An embedding method is described in which the damped Newton iteration is applied to a series of perturbed problems in order to guarantee convergence and also increase the convergence rate.     

K-functional, weighted moduli of smoothness, and best weighted polynomial approximation on a simplex

An inverse theorem for the best weighted polynomial approximation of a function in (S) is established. We also investigate Besov spaces generated by Freud weight and their connection with algebraic polynomial approximation in , wherew _α is a Jacobi-type weight onS, 0<p ≤ ∞,S is a simplex andW _λ is a Freud weight. For Ditzian-TotikK-functionals onL _p(S), 1 ≤p ≤ ∞, we obtain a new equivalence expression.     

Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data

Summary In this paper we study the convergence properties of a fully discrete Galerkin approximation with a backwark Euler time discretization scheme. An approach based on semigroup theory is used to deal with the nonsmooth Dirichlet boundary data which cannot be handled by standard techniques. This approach gives rise to optimal rates of convergence inL _p[O,T;L ₂()] norms for boundary conditions inL _p[O,T;L ₂()], 1p.     

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.     

On the relation between the Jackson and Jung constants of the spacesL p

For any infinitely metrizable compact Abelian groupG; 1pq<,n , the following relations are proved: whereK _pq(G, n, G) is the largest Jackson constant in the approximation of the system of characters by polynomials of ordern, d _pq(G, n, G) is the best Jackson constant,J(L _p(G), L_q(G)) is the Jung constant of the pair of real spaces (L _p(G), L_q(G)), and.Translated fromMatematicheskie Zametki, Vol. 58, No. 6, pp. 828–836, December, 1995.This work was supported by the Russian Foundation for Basic Research under grant No. 95-01-00657.     

The Central Approximation Theorems for the Method of Left Gamma Quasi-Interpolants in L p spaces

The optimal degree of approximation of the method of Gammaoperators G _n in L _p spaces is O(n ^-1). In order to obtain much faster convergence, quasi-interpolants G _n ^(k) of G _n in the sense of Sablonnière are considered. We show that for fixed k the operator-norms G _n ^(k) _p are uniformly bounded in n. In addition to this, for the first time in the theory of quasi-interpolants, all central problems for approximation methods (direct theorem, inverse theorem, equivalence theorem) could be solved completely for the L _p metric. Left Gamma quasi-interpolants turn out to be as powerful as linear combinations of Gammaoperators [6].     

Fully discrete approximations of parabolic boundary-value problems with nonsmooth boundary data

We study a numerical scheme for the approximation of parabolic boundary-value problems with nonsmooth boundary data. This fully discrete scheme requires no boundary constraints on the approximating elements. Our principal result is the derivation of optimal convergence estimates in L_p[0,T; L₂()] norms for boundary data in L_p[0, T; L₂()], 1p . For the same algorithms, we also show that the convergence remains optimal even in higher norms. The techniques employed are based on the theory of analytic semigroups combined with singular integrals.This paper was written in 1990, when the author was in the Department of Mathematical Sciences, University of Cincinnati. A preliminary version of this research was presented at the SIAM Annual Meeting in July 1989.     

Approximation of sine-shaped functions by constants in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>, <Emphasis Type="Italic">p</Emphasis> < 1

We investigate the best approximations of sine-shaped functions by constants in the spaces L_p for p < 1. In particular, we find the best approximation of perfect Euler splines by constants in the spaces L_p for certain p(0,1).Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 745–762, June, 2004.     

Greedy expansions in Banach spaces

We study convergence and rate of convergence of expansions of elements in a Banach space X into series with regard to a given dictionary . For convenience we assume that is symmetric: implies . The primary goal of this paper is to study representations of an element f∈X by a series

On the convergence of double Fourier series of functions in L P[0, 1]2, where p=(p 1, p 2)

Pointwise convergence of double trigonometric Fourier series of functions in the Lebesgue space L _p[0,2]²was studied by M. I. Dyachenko. In this paper, we also consider the problems of the convergence of double Fourier series in Pringsheim"s sense with respect to the trigonometric as well as the Walsh systems of functions in the Lebesgue space L _p[0,1]², p=(p₁,p₂), endowed with a mixed norm, in the particular case when the coefficients of the series in question are monotone with respect to each of the indices. We shall obtain theorems which generalize those of M. I. Dyachenko to the case when p is a vector. We shall also show that our theorems in the case of trigonometric Fourier series are best possible.     

Linear best approximation using a class ofk-majorl p norms

Consideration is given to problems of linear best approximation using a variant of the usuall _p norms referred to ask-majorl _p norms, for the cases when 1<p<. The underlying problem is the minimization of a piecewise smooth function. Best approximations are characterized, and a descent algorithm is developed.     

Low Rank Approximation of a Hankel Matrix by Structured Total Least Norm

The structure preserving rank reduction problem arises in many important applications. The singular value decomposition (SVD), while giving the closest low rank approximation to a given matrix in matrix L ₂ norm and Frobenius norm, may not be appropriate for these applications since it does not preserve the given structure. We present a new method for structure preserving low rank approximation of a matrix, which is based on Structured Total Least Norm (STLN). The STLN is an efficient method for obtaining an approximate solution to an overdetermined linear system AX B, preserving the given linear structure in the perturbation [E F] such that (A + E)X = B + F. The approximate solution can be obtained to minimize the perturbation [E F] in the L _p norm, where p = 1, 2, or . An algorithm is described for Hankel structure preserving low rank approximation using STLN with L _p norm. Computational results are presented, which show performances of the STLN based method for L ₁ and L ₂ norms for reduced rank approximation for Hankel matrices.     

On Approximation with Spline Generated Framelets

We characterize the approximation spaces associated with the best n-term approximation in L_p(R) by elements from a tight wavelet frame associated with a spline scaling function. The approximation spaces are shown to be interpolation spaces between L_p and classical Besov spaces, and the result coincides with the result for nonlinear approximation with an orthonormal wavelet with the same smoothness as the spline scaling function. We also show that, under certain conditions, the Besov smoothness can be measured in terms of the sparsity of expansions in the wavelet frame, just like the nonredundant wavelet case. However, the characterization now holds even for wavelet frame systems that do not have the usually required number of vanishing moments, e.g., for systems built through the Unitary Extension Principle, which can have no more than one vanishing moment. Using these results, we describe a fast algorithm that takes as input any function and provides a near sparsest expansion of it in the framelet system as well as approximants that reach the optimal rate of nonlinear approximation. Together with the existence of a fast algorithm, the absence of the need for vanishing moments may have an important qualitative impact for applications to signal compression, as high vanishing moments usually introduce a Gibbs-type phenomenon (or ringing artifacts)in the approximants.     

Greedy algorithms in Banach spaces

We study efficiency of approximation and convergence of two greedy type algorithms in uniformly smooth Banach spaces. The Weak Chebyshev Greedy Algorithm (WCGA) is defined for an arbitrary dictionary D and provides nonlinear m-term approximation with regard to D. This algorithm is defined inductively with the mth step consisting of two basic substeps: (1) selection of an mth element _m ^c from D, and (2) constructing an m-term approximant G _m ^c . We include the name of Chebyshev in the name of this algorithm because at the substep (2) the approximant G _m ^c is chosen as the best approximant from Span( ₁ ^c ,..., _m ^c ). The term Weak Greedy Algorithm indicates that at each substep (1) we choose _m ^c as an element of D that satisfies some condition which is t _m -times weaker than the condition for _m ^c to be optimal (t _m =1). We got error estimates for Banach spaces with modulus of smoothness (u)u ^q , 1<q2. We proved that for any f from the closure of the convex hull of D the error of m-term approximation by WCGA is of order (1+t ₁ ^p ++t _m ^p )^–1/p , 1/p+1/q=1. Similar results are obtained for Weak Relaxed Greedy Algorithm (WRGA) and its modification. In this case an approximant G ^r _m is a convex linear combination of 0,₁ ^r ,..., ^r _m . We also proved some convergence results for WCGA and WRGA.     

Local convergence in a generalized Fermat-Weber problem

In the Fermat-Weber problem, the location of a source point in ^N is sought which minimizes the sum of weighted Euclidean distances to a set of destinations. A classical iterative algorithm known as the Weiszfeld procedure is used to find the optimal location. Kuhn proves global convergence except for a denumerable set of starting points, while Katz provides local convergence results for this algorithm. In this paper, we consider a generalized version of the Fermat-Weber problem, where distances are measured by anl _p norm and the parameterp takes on a value in the closed interval [1, 2]. This permits the choice of a continuum of distance measures from rectangular (p=1) to Euclidean (p=2). An extended version of the Weiszfeld procedure is presented and local convergence results obtained for the generalized problem. Linear asymptotic convergence rates are typically observed. However, in special cases where the optimal solution occurs at a singular point of the iteration functions, this rate can vary from sublinear to quadratic. It is also shown that for sufficiently large values ofp exceeding 2, convergence of the Weiszfeld algorithm will not occur in general.     

Best constants of harmonic approximation on classes associated with the Laplace operator

We compute the best constants of approximation by entire functions of spherical type and by trigonometric polynomials of spherical degree on classes of functions f satisfying the condition Δ^kf_Lp1, where p=1 or 2 and Δ is the Laplace operator.