On a Sequence of Positive Linear Operators Associated with a Continuous Selection of Borel Measures

In this paper we introduce and study a new sequence of positive linear operators acting on the space of Lebesgue-integrable functions on the unit interval. These operators are defined by means of continuous selections of Borel measures and generalize the Kantorovich operators. We investigate their approximation properties by presenting several estimates of the rate of convergence by means of suitable moduli of smoothness. Some shape preserving properties are also shown. Dedicated to the memory of Professor Aldo Cossu     

Moduli of smoothness and approximation on the unit sphere and the unit ball

A new modulus of smoothness based on the Euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness, including direct and inverse theorem for the best approximation by polynomials and its equivalence to a K-functional, defined via partial derivatives in Euler angles. The set of results on the moduli on the sphere serves as a basis for defining new moduli of smoothness and their corresponding K-functionals on the unit ball, which are used to characterize the best approximation by polynomials on the ball.     

Wavelet decompositions of L 2-functionals

Based on distribution-theoretical definitions of L ₂ and Sobolev spaces given by Werner in [P. Werner (1970). A distribution-theoretical approach to certain Lebesgue and Sobolev spaces. J. Math. Anal. Appl., 29, 19–78.] real interpolation, Besov type spaces and approximation spaces with respect to multiresolution approximations are considered. The key for the investigation are generalized moduli of smoothness introduced by Haf in [H. Haf (1992). On the approximation of functionals in Sobolev spaces by singular integrals. Applicable Analysis, 45, 295–308.]. Those moduli of smoothness allow to connect the concept of L ₂-functionals with more recent developments in multiscale analysis, see e.g. [W. Dahmen (1995). Multiscale analysis, approximation, and interpolation spaces. In: C.K. Chui and L.L. Schumaker (Eds.), Approximation Theory VIII, Vol. 2: Wavelets and Multilevel Approximation, pp. 47–88.]. In particular, we derive wavelet characterizations for the Sobolev spaces introduced by Werner and establish stable wavelet decompositions of L ₂-functionals. Generalizations to more general spaces of functionals and applications are also mentioned.     

Approximation by boolean sums of positive linear operators III: Estimates for some numerical approximation schemes

In the present note we intröduce and investigate certain sequences of discrete positive linear operators and Boolean sum modifications of them. The mappings considered are obtained by discretizing a class of transformed convolution-type operators using Gaussian quadrature of appropriate order. For our operators and their modifications we prove pointwise Jackson-type theorems involving the first and second order moduli of smoothness, thus providing new and elegant proofs of earlier results by Timan, Telyakowskii, Gopengauz and DeVore. Due to their discrete structure, optimal order of approximation and ease of computation, the operators appear to be useful for numerical approximation. In an intermediate step we solve an old problem in Approximation Theory; its importance was only recently emphasized in a paper of Butzer.     

Restricted Nonlinear Approximation

We introduce a new form of nonlinear approximation called restricted approximation . It is a generalization of n -term wavelet approximation in which a weight function is used to control the terms in the wavelet expansion of the approximant. This form of approximation occurs in statistical estimation and in the characterization of interpolation spaces for certain pairs of L _p and Besov spaces. We characterize, both in terms of their wavelet coefficients and also in terms of their smoothness, the functions which are approximated with a specified rate by restricted approximation. We also show the relation of this form of approximation with certain types of thresholding of wavelet coefficients. March 31, 1998. Date accepted: January 28, 1999.     

Jackson inequality for Banach spaces on the sphere

The best rate of approximation of functions on the sphere by spherical polynomials is majorized by recently introduced moduli of smoothness. The treatment applies to a wide class of Banach spaces of functions.      

Optimized Tensor-Product Approximation Spaces

This paper is concerned with the construction of optimized grids and approximation spaces for elliptic differential and integral equations. The main result is the analysis of the approximation of the embedding of the intersection of classes of functions with bounded mixed derivatives in standard Sobolev spaces. Based on the framework of tensor-product biorthogonal wavelet bases and stable subspace splittings, the problem is reduced to diagonal mappings between Hilbert sequence spaces. We construct operator adapted finite element subspaces with a lower dimension than the standard full-grid spaces. These new approximation spaces preserve the approximation order of the standard full-grid spaces, provided that certain additional regularity assumptions are fulfilled. The form of the approximation spaces is governed by the ratios of the smoothness exponents of the considered classes of functions. We show in which cases the so-called curse of dimensionality can be broken. The theory covers elliptic boundary value problems as well as boundary integral equations. September 17, 1998. Date revised: March 5, 1999. Date accepted: September 20, 1999.     

Convergence in Variation and Rate of Approximation for Nonlinear Integral Operators of Convolution Type

In this paper we obtain estimates, convergence results and rate of approximation for functions belonging to BV–spaces (spaces of functions with bounded variation) by means of nonlinear convolution integral operators. We treat both the periodic and the non-periodic case using, respectively, the classical Jordan variation and the multidimensional variation in the sense of Tonelli.     

一类多元Gauss-Weierstrass 算子线性组合的加权Lp-逼近

本文研究一类多元Gauss-Weierstrass算子的线性组合加Jacobi型权逼近的性质,利用加权矩量不等式及加权K-泛函、光滑模等工具,建立了这类算子在Lp(1≤p≤∞)空间的正、逆定理和逼近阶的特征刻划.     

Norm‐controlled inversion in smooth Banach algebras,II

We show that smoothness implies norm‐controlled inversion: the smoothness of an element a in a Banach algebra with a one‐parameter automorphism group is preserved under inversion, and the norm of the inverse is controlled by the smoothness of a and by spectral data. In our context smooth subalgebras are obtained with the classical constructions of approximation theory and resemble spaces of differentiable functions, Besov spaces or Bessel potential spaces. To treat ultra‐smoothness, we resort to Dales‐Davie algebras. Furthermore, based on Baskakov's work, we derive explicit norm control estimates for infinite matrices with polynomial off‐diagonal decay. This is a quantitative version of Jaffard's theorem.     

Approximation by polynomials in Bergman spaces of slice regular functions in the unit ball

In this paper, we show that the set of quaternionic polynomials is dense in the Bergman spaces of slice regular functions in the unit ball, both of the first and of the second kind. Several proofs are presented, including constructive methods based on the Taylor expansion and on the convolution polynomials. In the last case, quantitative estimates in terms of higher‐order moduli of smoothness and of best approximation quantity are obtained.     

A modulus of smoothness based on an algebraic addition

Summary The purpose of this paper is to present a new approach to smoothness of nonperiodic functions. We consider the space of continuous functions on [−1, 1] as well as the weighted L^p-space and introduce a modulus of smoothness that is based on an algebraic addition ⊕ defined on [−1, 1]. The present paper is mainly concerned with general properties and groundwork, whereas a second paper [4] is devoted to more complex properties, in particular to an equivalent K-functional and to the characterization of best algebraic approximation. Moreover the equivalence with the Butzer-Stens modulus will be shown there.     

A fractional version of the peano-sard theorem

Sard's classical generalization of the Peano kernel theorem provides an extremely useful method for expressing and calculating sharp bounds for approximation errors. The error is expressed in terms of a derivative of the underlying function. However, we can apply the theorem only if the approximation is exact on a certain set of polynomials.

In this paper, we extend the Peano-Sard theorem to the case that the approximation is exact for a class of generalized polynomials (with non-integer exponents). As a result, we obtain an expression for the remainder in terms of a fractional derivative of the function under consideration. This expression permits us to give sharp error bounds as in the classical situation. An application of our results to the classical functional (vanishing on polynomials) gives error bounds of a new type involving weighted Sobolev-type spaces. In this way, we may state estimates for functions with weaker smoothness properties than usual.

The standard version of the Peano-Sard theory is contained in our results as a special case.     

On a class of weak Tchebycheff systems

In this paper we study the approximation power, the existence of a normalized B-basis and the structure of a degree-raising process for spaces of the formrequiring suitable assumptions on the functions u and v. The results about degree raising are detailed for special spaces of this form which have been recently introduced in the area of CAGD.     

Functions related to convexity and smoothness of normed spaces

We introduce a few functions related to convexity and smoothness of normed spaces. Those functions turn out to be moduli of convexity or smoothness or play an intermediate role. We calculate the exact formulas for introduced functions in some classical Banach spaces. An application to geometry of normed spaces is also indicated.     

Convergence for a family of neural network operators in Orlicz spaces

In this paper, we develop the theory for a family of neural network (NN) operators of the Kantorovich type, in the general setting of Orlicz spaces. In particular, a modular convergence theorem is established. In this way, we study the above family of operators in many instances of useful spaces by a unique general approach. The above NN operators provide a constructive approximation process, in which the coefficients, the weights, and the thresholds of the networks needed in order to approximate a given function f , are known. At the end of the paper, several examples of Orlicz spaces, and of sigmoidal activation functions for which the present theory can be applied, are studied in details.     

关于连续函数与可微函数的共凸逼近

关于凸逼近,共凸逼近的各种逼近阶的估计在近十多年来已引起了相当的重视和较多的研究,因为保形渐近的思想在实际问题中有着十分重要的意义。相比较而言,共凸逼近的的研究比凸逼近的情况要复杂,因此,不少关于凸逼近已得到解决的问题对于共凸逼近仍没有结果,关于共凸逼近,1984,Atacir,Sermin证明了。     

Best coapproximation in quotient spaces

As a counterpart to best approximation in normed linear spaces, best coapproximation was introduced by Franchetti and Furi. In this paper, we apply the above coapproximation, and obtain some results on the upper semi-continuity of cometric projection maps. Also we shall determine under what conditions coproximinality can be transmitted to and from quotient spaces.     

Mixed modulus of smoothness with Muckenhoupt weights and approximation by angle

Mixed modulus of smoothness in weighted Lebesgue spaces with Muckenhoupt weights are investigated. Using mixed modulus of smoothness we obtain Potapov type direct and inverse estimates of angular trigonometric approximation of functions in these spaces. Also we obtain equivalences between mixed modulus of smoothness and K-functional and realization functional. Fractional order modulus of smoothness is considered as well.