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.     

Boundedness of Riesz transforms for elliptic operators on abstract Wiener spaces

Let (E,H,μ) be an abstract Wiener space and let D_V:=VD, where D denotes the Malliavin derivative and V is a closed and densely defined operator from H into another Hilbert space . Given a bounded operator B on , coercive on the range , we consider the operators A:=V^*BV in H and in , as well as the realisations of the operators and in L^p(E,μ) and respectively, where 1<p<∞. Our main result asserts that the following four assertions are equivalent:

(1) with for ;

(2) admits a bounded H^∞-functional calculus on ;

(3) with for ;

(4) admits a bounded H^∞-functional calculus on .

Moreover, if these conditions are satisfied, then . The equivalence (1)–(4) is a non-symmetric generalisation of the classical Meyer inequalities of Malliavin calculus (where , V=I, ). A one-sided version of (1)–(4), giving L^p-boundedness of the Riesz transform in terms of a square function estimate, is also obtained. As an application let −A generate an analytic C₀-contraction semigroup on a Hilbert space H and let −L be the L^p-realisation of the generator of its second quantisation. Our results imply that two-sided bounds for the Riesz transform of L are equivalent with the Kato square root property for A. The boundedness of the Riesz transform is used to obtain an L^p-domain characterisation for the operator L.

Kolmogorov Entropy for Classes of Convex Functions

Kolmogorov ε-entropy of a compact set in a metric space measures its metric massivity and thus replaces its dimension which is usually infinite. The notion quantifies the compactness property of sets in metric spaces, and it is widely applied in pure and applied mathematics. The ε-entropy of a compact set is the most economic quantity of information that permits a recovery of elements of this set with accuracy ε. In the present article we study the problem of asymptotic behavior of the ε-entropy for uniformly bounded classes of convex functions in L _p -metric proposed by A.I.   Shnirelman. The asymptotic of the Kolmogorov ε-entropy for the compact metric space of convex and uniformly bounded functions equipped with L _p -metric is ε ^−1/2, ε→0₊.      

Walsh-Type Wavelet Packet Expansions

We consider a family of basic nonstationary wavelet packets generated using the Haar filters except for a finite number of scales where we allow the use of arbitrary filters. Such a system, which we call a system of Walsh-type wavelet packets, can be considered as a smooth generalization of the Walsh functions. We show that the basic Walsh-type wavelet packets share a number of metric properties with the Walsh system. We prove that the system constitutes a Schauder basis for L^p( ), 1<p<∞, and we construct an explicit function in L¹( ) for which the expansion fails. Then we prove that expansions of L^p( )-functions, 1<p<∞, in the Walsh-type wavelet packets converge pointwise a.e. Finally, we prove that the analogous results are true for periodic Walsh-type wavelet packets in L^p[0,1).     

Characterizations of spaces in the unit ball of

Let B denote the unit ball of . For 0<p<∞, the holomorphic function spaces Q_p and Q_p,0 on the unit ball of are defined as

and

In this paper, we give some derivative-free, mixture and oscillation characterizations for Q_p and Q_p,0 spaces in the unit ball of .     

Distributions of class Lα

L_α (0 α 1) is a class of infinitely divisible distributions defined by restricting the measure in the Levy-Khinchin formula to a special form. When α = 1, L_α is just the classical class L. Several properties for L_α classes, which are similar to the most important properties for the class L, are established. Also, a conjecture of Wolfe about unimodality of some L_α distributions is disproved by giving a counterexample.     

Besov spaces andK-functionals inL p, 0<p<1*

We investigate Besov spaces and their connection with trigonometric polynomial approximation inL _p[−π,π], algebraic polynomial approximation inL _p[−1,1], algebraic polynomial approximation inL _p(S), and entire function of exponential type approximation inL _p(R), and characterizeK-functionals for certain pairs of function spaces including (L _p[−π,π],B _s ^a(L _p[−π,π])), (L _p(R),_s ^a(L_p(R))), , and , where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r. This project is supported by the National Science Foundation of China.     

On the Asymptotics of Trimmed Best k-Nets

Trimmed best k-nets were introduced in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413) as a robustified L_∞-based quantization procedure. This paper focuses on the asymptotics of this procedure. Also, some possible applications are briefly sketched to motivate the interest of this technique. Consistency and weak limit law are obtained in the multivariate setting. Consistency holds for absolutely continuous distributions without the (artificial) requirement of a trimming level varying with the sample size as in J. A. Cuesta-Albertos, A. Gordaliza and C. Matrán (1998, Statist. Probab. Lett.36, 401–413). The weak convergence will be stated toward a non-normal limit law at a O_P(n^−1/3) rate of convergence. An algorithm for computing trimmed best k-nets is proposed. Also a procedure is given in order to choose an appropriate number of centers, k, for a given data set.     

LMI-Based Robust <Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> Control of Uncertain Neutral Systems with State and Input Delays

Robust stabilization and robust H_∞ control problems for a class of uncertain neutral system with state and input delays are considered. By choosing a Lyapunov-Krasovskii functional, an alternative delay-dependent sufficient condition for the existence of a controller is derived in terms of linear matrix inequalities. Furthermore, a convex optimization problem can be formulated to obtain an H_∞ controller which minimizes an upper H_∞ norm bound of the closed-loop state-input delayed system.     

Connections between stability and asymptotic stability of nonlinear switched systems

Here are considered nonlinear switched systems in which the switching occurs among a class of subsystems that are characterized by input–output properties stated in terms of L_p spaces of signals. The relationships between the L_p stability of each subsystem and the internal stability of the switched system are studied. In particular, conditions on the dwell time of the switching signals that guarantee the asymptotic stability of the overall system are provided. The connections among these conditions and the L_p input–output properties of the subsystems are investigated.     

Local Hardy spaces and summability of Fourier transforms

New Wiener amalgam spaces are introduced for local Hardy spaces. A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from the amalgam space to W(L_p,ℓ_∞). This implies the almost everywhere convergence of the θ-means for all fW(L₁,ℓ_∞)L₁.     

The L p -curvature images of convex bodies and L p -projection bodies

Associated with the L _p -curvature image defined by Lutwak, some inequalities for extended mixed p-affine surface areas of convex bodies and the support functions of L _p -projection bodies are established. As a natural extension of a result due to Lutwak, an L _p -type affine isoperimetric inequality, whose special cases are L _p -Busemann-Petty centroid inequality and L _p -affine projection inequality, respectively, is established. Some L _p -mixed volume inequalities involving L _p -projection bodies are also established.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-mixed intersection bodies

In this paper the author first introduce a new concept of L _p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L _p intersection body to L _p -mixed intersection body. Inequalities for L _p -dual mixed volumes of L _p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

Lp-Approximable Sequences of Vectors and Limit Distribution of Quadratic Forms of Random Variables

The properties of L₂-approximable sequences established here form a complete toolkit for statistical results concerning weighted sums of random variables, where the weights are nonstochastic sequences approximated in some sense by square-integrable functions and the random variables are “two-wing” averages of martingale differences. The results constitute the first significant advancement in the theory of L₂-approximable sequences since 1976 when Moussatat introduced a narrower notion of L₂-generated sequences. The method relies on a study of certain linear operators in the spaces L_p and l_p. A criterion of L_p-approximability is given. The results are new even when the weight generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.     

On Lp-Generalization of a Theorem of Adamyan, Arov, and Kreın

The paper deals with problems relating to the theory of Hankel operators. Let G be a bounded simple connected domain with the boundary Γ consisting of a closed analytic Jordan curve. Denote by _n,p(G), 1p<∞, the class of all meromorphic functions on G that can be represented in the form h=β/α, where β belongs to the Smirnov class E_p(G), α is a polynomial degree at most n, α0. We obtain estimates of s-numbers of the Hankel operator A_f constructed from fL_p(Γ), 1p<∞, in terms of the best approximation Δ_n,p of f in the space L_p(Γ) by functions belonging to the class _n,p(G).     

Approximation of smooth functions on compact two-point homogeneous spaces

Estimates of Kolmogorov n-widths and linear n-widths , (1q∞) of Sobolev's classes , (r>0, 1p∞) on compact two-point homogeneous spaces (CTPHS) are established. For part of (p,q)[1,∞]×[1,∞], sharp orders of or were obtained by Bordin et al. (J. Funct. Anal. 202(2) (2003) 307). In this paper, we obtain the sharp orders of and for all the remaining (p,q). Our proof is based on positive cubature formulas and Marcinkiewicz–Zygmund-type inequalities on CTPHS.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

Properties of convergence for -Bernstein polynomials

In this paper, we discuss properties of the ω,q-Bernstein polynomials introduced by S. Lewanowicz and P. Woźny in [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT 44 (1) (2004) 63–78], where fC[0,1], ω,q>0, ω≠1,q⁻¹,…,q⁻ⁿ⁺¹. When ω=0, we recover the q-Bernstein polynomials introduced by [G.M. Phillips, Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518]; when q=1, we recover the classical Bernstein polynomials. We compute the second moment of , and demonstrate that if f is convex and ω,q(0,1) or (1,∞), then are monotonically decreasing in n for all x[0,1]. We prove that for ω(0,1), q_n(0,1], the sequence converges to f uniformly on [0,1] for each fC[0,1] if and only if lim_n→∞q_n=1. For fixed ω,q(0,1), we prove that the sequence converges for each fC[0,1] and obtain the estimates for the rate of convergence of by the modulus of continuity of f, and the estimates are sharp in the sense of order for Lipschitz continuous functions.     

Superapproximation and Commutator Properties of Discrete Orthogonal Projections for Continuous Splines

This paper builds upon the L_p-stability results for discrete orthogonal projections on the spaces S_h of continuous splines of order r obtained by R. D. Grigorieff and I. H. Sloan in (1998, Bull. Austral. Math. Soc.58, 307–332). Properties of such projections were proved with a minimum of assumptions on the mesh and on the quadrature rule defining the discrete inner product. The present results, which include superapproximation and commutator properties, are similar to those derived by I. H. Sloan and W. Wendland (1999, J. Approx. Theory97, 254–281) for smoothest splines on uniform meshes. They are expected to have applications (as in I. H. Sloan and W. Wendland, Numer. Math. (1999, 83, 497–533)) to qualocation methods for non-constant-coefficient boundary integral equations, as well as to the wide range of other numerical methods in which quadrature is used to evaluate L₂-inner products. As a first application, we consider the most basic variable-coefficient boundary integral equation, in which the constant-coefficient operator is the identity. The results are also extended to the case of periodic boundary conditions, in order to allow appplication to boundary integral equations on closed curves.     

Point sets with low <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-discrepancy

In this paper we study the L _p -discrepancy of digitally shifted Hammersley point sets. While it is known that the (unshifted) Hammersley point set (which is also known as Roth net) with N points has L _p -discrepancy (p an integer) of order (log N)/N, we show that there always exists a shift such that the digitally shifted Hammersley point set has L _p -discrepancy (p an even integer) of order which is best possible by a result of W. Schmidt. Further we concentrate on the case p = 2. We give very tight lower and upper bounds for the L ₂-discrepancy of digitally shifted Hammersley point sets which show that the value of the L ₂-discrepancy of such a point set mostly depends on the number of zero coordinates of the shift and not so much on the position of these. This work is supported by the Austrian Research Fund (FWF), Project P17022-N12 and Project S8305.