Sampling theorem of Hermite type and aliasing error on the Sobolev class of functions

Denote by B _2σ,p (1 < p < ∞) the bandlimited class p-integrable functions whose Fourier transform is supported in the interval [−σ, σ]. It is shown that a function in B _2σ,p can be reconstructed in L _p(ℝ) by its sampling sequences {f (κπ / σ)} _κ∈ℤ and {f’ (κπ / σ)} _κ∈ℤ using the Hermite cardinal interpolation. Moreover, it will be shown that if f belongs to L _p ^r (ℝ), 1 < p < ∞, then the exact order of its aliasing error can be determined. Project supported by the Scientific Research Common Program of Beijing Municipal Commission of Education under grant number KM 200410009010 and by the Natural Science Foundation of China under grant number 10071006     

Testfunctions for quantitative Korovkin theorems in Lp

Quantitative error estimates for the L_p approximation by positive linear operators can be given by using the so-called averaged modulus of smoothness or τ-modulus of first and second order. The approximation error for the three test function e_i, e_i(x)=xⁱ, i=0,1,2 is hereby of special importance. In this paper it is shown that it is possible to give quantitative L_p error estimates where the monomials are replaced by other Tchebychev systems that have certain additional properties.     

Representation ofL p-norms and isometric embedding inL p-spaces

For fixed 1≦p<∞ theL _p-semi-norms onR ⁿ are identified with positive linear functionals on the closed linear subspace ofC(R ⁿ ) spanned by the functions |<ξ, ·>| ^p , ξ∈R ⁿ . For every positive linear functional σ, on that space, the function Φ_σ:R ⁿ →R given by Φ_σ is anL _p-semi-norm and the mapping σ→Φ_σ is 1-1 and onto. The closed linear span of |<ξ, ·>| ^p , ξ∈R ⁿ is the space of all even continuous functions that are homogeneous of degreep, ifp is not an even integer and is the space of all homogeneous polynomials of degreep whenp is an even integer. This representation is used to prove that there is no finite list of norm inequalities that characterizes linear isometric embeddability, in anyL _p unlessp=2. Supported by the National Science Foundation MCS-79-06634 at U.C. Berkeley.     

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.     

Theorems for generalized Favard-Kantorovich and Favard-Durrmeyer operators in exponential function spaces

We consider the Kantorovich and the Durrmeyer type modifications of the generalized Favard operators and we prove some direct approximation theorems for functions f such that w _σ f ∈ L _p (R), where 1 ≤ p ≤ ∞ and w _σ (x) = exp(−σx ²), σ > 0.     

On the bestm-term trigonometric and orthogonal trigonometric approximations of functions from the classesL Ψ β,ρ

We obtain estimates exact in order for the best trigonometric and orthogonal trigonometric approximations of the classesL _Ψ^β,ρ of functions of one variable in the spaceL _q in the case 2<p <q < ∞.     

Dominated estimates of convex combinations of commuting isometries

The principal result of this paper is that the convex combination of two positive, invertible, commuting isometries ofL _p(X,F, μ) 1<p<+∞, one of which is periodic, admits a dominated estimate with constantp/p−1. In establishing this, the following analogue of Linderholm’s theorem is obtained: Let σ and ε be two commuting non-singular point transformations of a Lebesgue Space with τ periodic. Then given ε>O, there exists a periodic non-singular point transformation σ′ such that σ′ commutes with τ and μ(x:σ′x≠σx}<ε. Byan approximation argument, the principal result is applied to the convex combination of two isometries ofL _p (0, 1) induced by point transformations of the form τx=x ^k,k>0 to show that such convex combinations admit a dominated estimate with constantp/p−1. Research supported in part by NSF Grant No. GP-7475. A portion of the contents of this paper is based on the author’s doctoral dissertation written under the direction of Professor R. V. Chacon of the University of Minnesota.     

Cardinal Interpolation with Gaussian Kernels

In this paper, interpolation by scaled multi-integer translates of Gaussian kernels is studied. The main result establishes L _p Sobolev error estimates and shows that the error is controlled by the L _p multiplier norm of a Fourier multiplier closely related to the cardinal interpolant, and comparable to the Hilbert transform. Consequently, its multiplier norm is bounded independent of the grid spacing when 1<p<∞, and involves a logarithmic term when p=1 or ∞.     

Convex polynomial and spline approximation inL p,O<p<∞

We prove that a convex functionf ∈ L _p[−1, 1], 0<p<∞, can be approximated by convex polynomials with an error not exceeding Cω ₃ ^ϕ (f,1/n)_p where ω ₃ ^ϕ (f,·) is the Ditzian-Totik modulus of smoothness of order three off. We are thus filling the gap between previously known estimates involving ω ₃ ^ϕ (f,1/n)_p, and the impossibility of having such estimates involving ω₄. We also give similar estimates for the approximation off by convexC ⁰ andC ¹ piecewise quadratics as well as convexC ² piecewise cubic polynomials. Communicated by Dietrich Braess     

Regularizing algorithms for localization of breakpoints of noisy functions

The problem of localizing the singularities (breakpoints) of functions that are noisy in the spaces L _p , 1 < p < ∞, or C is considered. A wide class of smoothing algorithms that determine the number and location of breakpoints is constructed. In addition, for the case when a function is noisy in C, a finite-difference method is constructed. For the proposed methods, convergence theorems are proved and approximation accuracy estimates for the location of breakpoints are obtained. The lower estimates obtained in this paper show the order-optimality of the methods. For all the methods constructed, their capacity of separating close breakpoints is investigated.     

Approximation to the Sobolev classesW q r of functions of several variables by bilinear forms inL p for 2≤q≤p≤∞

We study the approximation of functions of several variables by bilinear forms that are the pairwise products of functions of fewer variables. The order of approximation of Sobolev classesW _q ^r by bilinear forms inL _p for 2≤q≤p≤∞ is found. Translated by N. K. Kulman Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 18–34, July, 1997.     

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.     

Rates of convex approximation in non-hilbert spaces

This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis” subsetS of a function space. Specifically, the focus is on therate at which the lowest achievable error can be reduced as larger subsets ofS are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including, in particular, a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces; in particular, forL _p spaces with 1<p<∞, theO (n ^−1/2) bounds of Barron and Jones are recovered whenp=2. One motivation for the questions studied here arises from the area of “artificial neural networks,” where the problem can be stated in terms of the growth in the number of “neurons” (the elements ofS) needed in order to achieve a desired error rate. The focus on non-Hilbert spaces is due to the desire to understand approximation in the more “robust” (resistant to exemplar noise)L _p, 1 ≤p<2, norms. The techniques used borrow from results regarding moduli of smoothness in functional analysis as well as from the theory of stochastic processes on function spaces.     

The Stability in W s,p (Γ) Spaces of L 2-Projections on Some Convex Sets

ABSTRACT

For a polygonal open bounded subset of ?², of boundary Γ, we study stability estimates for the projection operator from L ¹(Γ) on a convex set K _h of continuous piecewise affine functions satisfying bound constraints. We establish stability estimates in L ^p (Γ) and in W ^s,p (Γ) for 1 ≤ p ≤ ∞ and 0 < s ≤ 1. This kind of result plays a crucial role in error estimates for the numerical approximation of optimal control problems of partial differential equations with bilateral control constraints.     

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.     

Interpolation error estimates in W 1,p for degenerate Q 1 isoparametric elements

Optimal order error estimates in H ¹, for the Q ₁ isoparametric interpolation were obtained in Acosta and Durán (SIAM J Numer Anal37, 18–36, 1999) for a very general class of degenerate convex quadrilateral elements. In this work we show that the same conlusions are valid in W ^1,p for 1≤ p < 3 and we give a counterexample for the case p ≥ 3, showing that the result cannot be generalized for more regular functions. Despite this fact, we show that optimal order error estimates are valid for any p ≥ 1, keeping the interior angles of the element bounded away from 0 and π, independently of the aspect ratio. We also show that the restriction on the maximum angle is sharp for p ≥ 3.     

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.     

Approximative characteristics of the isotropic classes of periodic functions of many variables

Exact-order estimates are obtained for the best orthogonal trigonometric approximations of the Besov (B _p,θ ^r ) and Nukol’skii (H _p ^r ) classes of periodic functions of many variables in the metric of L _q , 1 ≤ p, q ≤ ∞. We also establish the orders of the best approximations of functions from the same classes in the spaces L ₁ and L _∞ by trigonometric polynomials with the corresponding spectrum.     

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.