Equivalence theorems concerning the convergence of the Bernstein polynomialsB n f are well known for continuous functionsf in the sup-norm. The purpose of this paper is to extend these ...
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Equivalence theorems concerning the convergence of the Bernstein polynomialsB
n
f are well known for continuous functionsf in the sup-norm. The purpose of this paper is to extend these results for functionsf, Riemann integrable on [0, 1], We have therefore to consider the seminorm
$$\left\| f \right\|_\delta : = \int_0^1 {\mathop {\sup }\limits_{y \in U_\delta (x)} |f(y)|dx,U_\delta (x): = \{ y \in [01]:|x - y \leqslant \delta \} ,}$$
depending also on the increment δ>0. In terms of correspondingτ-moduli direct and inverse theorems are established, e.g.,
$$\left\| f \right\|_\delta : = \int_0^1 {\mathop {\sup }\limits_{y \in U_\delta (x)} |f(y)|dx,U_\delta (x): = \{ y \in [01]:|x - y \leqslant \delta \} ,}$$
Moreover, a similar result is obtained for the error functional appearing in the recently introduced definition of Riemann convergence.
The best estimates for the approximation error of functions, defined on a finite interval, by algebraic polynomials and piecewise polynomial functions are obtained in the case when the errors are measured in the norms...
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The best estimates for the approximation error of functions, defined on a finite interval, by algebraic polynomials and piecewise polynomial functions are obtained in the case when the errors are measured in the norms of Sobolev and Besov spaces. We indicate the weighted Besov spaces, whose functions satisfy Jackson-type and Bernstein-type inequalities and, as a consequence, direct and inverse approximation theorems. In a number of cases, exact constants are indicated in the estimates.
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 ...
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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.
direct estimates for the Bernstein operator are presented by the Ditzian-Totik modulus of smoothness omega(phi)(2)(f, delta), whereby the step-weight phi is a function such that phi(2) is concave. The inversedirectio...
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direct estimates for the Bernstein operator are presented by the Ditzian-Totik modulus of smoothness omega(phi)(2)(f, delta), whereby the step-weight phi is a function such that phi(2) is concave. The inversedirection will be established for those step-weights phi for which phi(2) and phi(2)/phi(2), phi(x) = root x(1-x), are concave functions. This combines the classical estimate (phi = 1) and the estimate developed by Ditzian and Totik (phi = phi). In particular, the cases phi = phi(lambda),( )lambda is an element of [0, 1], are included.
In this paper we give the direct approximation theorem, the inverse theorem, and the equivalence theorem for Szasz-Durrmeyer-Bezier operators in the space L-p[0, infinity) (1 <= p <= infinity) with Ditzian-Totik...
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In this paper we give the direct approximation theorem, the inverse theorem, and the equivalence theorem for Szasz-Durrmeyer-Bezier operators in the space L-p[0, infinity) (1 <= p <= infinity) with Ditzian-Totik modulus.
We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights w having finitely many zeros and singularities (i.e., points where w becomes infinite) on an interval and no...
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We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights w having finitely many zeros and singularities (i.e., points where w becomes infinite) on an interval and not too "rapidly changing" away from these zeros and singularities. This class of doubling weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian Totik weights. We approximate in the weighted L-p (quasi) norm parallel to f parallel to(p,w) with 0 < p < infinity, where parallel to f parallel to(p,w) := (integral(1)(-1) vertical bar f (u) vertical bar(P) w(u)du)(1/p). Equivalence type results involving related realization functionals are also discussed. (C) 2015 Elsevier Inc. All rights reserved.
In this paper, we prove some basic results concerning the best approximation of vector-valued functions by algebraic and trigonometric polynomials with coefficients in normed spaces, called generalized polynomials. Th...
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In this paper, we prove some basic results concerning the best approximation of vector-valued functions by algebraic and trigonometric polynomials with coefficients in normed spaces, called generalized polynomials. Thus we obtain direct and inverse theorems for the best approximation by generalized polynomials and results concerning the existence (and uniqueness) of best approximation generalized polynomials.
作者：
Müller, MWUniv Dortmund
Inst Angew Math Lehrstuhl Approximat Theorie VIII D-44221 Dortmund Germany
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 Sablonniere are cons...
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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 Sablonniere 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 Garrunaoperators [6].
We obtain matching direct and inverse theorems for the degree of weighted L p -approximation by polynomials with the Jacobi weights (1 - x)(alpha) (1 + x)(beta) . Combined, the estimates yield a constructive character...
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We obtain matching direct and inverse theorems for the degree of weighted L p -approximation by polynomials with the Jacobi weights (1 - x)(alpha) (1 + x)(beta) . Combined, the estimates yield a constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials. In addition, we prove Whitney type inequalities which are of independent interest. (C) 2018 Elsevier Inc. All rights reserved.
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