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Search: MSC category 41A25 ( Rate of convergence, degree of approximation )

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1. CJM Online first

Kopotun, Kirill; Leviatan, Dany; Shevchuk, Igor
Constrained approximation with Jacobi weights
In this paper, we prove that, for $\ell=1$ or $2$, the rate of best $\ell$-monotone polynomial approximation in the $L_p$ norm ($1\leq p \leq \infty$) weighted by the Jacobi weight $w_{\alpha,\beta}(x) :=(1+x)^\alpha(1-x)^\beta$ with $\alpha,\beta\gt -1/p$ if $p\lt \infty$, or $\alpha,\beta\geq 0$ if $p=\infty$, is bounded by an appropriate $(\ell+1)$st modulus of smoothness with the same weight, and that this rate cannot be bounded by the $(\ell+2)$nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given.

Keywords:constrained approximation, Jacobi weights, weighted moduli of smoothness, exact estimates, exact orders
Categories:41A29, 41A10, 41A15, 41A17, 41A25

2. CJM 2005 (vol 57 pp. 1224)

Kopotun, K. A.; Leviatan, D.; Shevchuk, I. A.
Convex Polynomial Approximation in the Uniform Norm: Conclusion
Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been left some interesting open questions. In this paper we give final answers to all those open problems. We are able to say, for each $r$th differentiable convex function, whether or not its degree of convex polynomial approximation in the uniform norm may be estimated by a Jackson-type estimate involving the weighted Ditzian-Totik $k$th modulus of smoothness, and how the constants in this estimate behave. It turns out that for some pairs $(k,r)$ we have such estimate with constants depending only on these parameters. For other pairs the estimate is valid, but only with constants that depend on the function being approximated, while there are pairs for which the Jackson-type estimate is, in general, invalid.

Categories:41A10, 41A25, 41A29

3. CJM 1999 (vol 51 pp. 546)

Felten, M.
Strong Converse Inequalities for Averages in Weighted $L^p$ Spaces on $[-1,1]$
Averages in weighted spaces $L^p_\phi[-1,1]$ defined by additions on $[-1,1]$ will be shown to satisfy strong converse inequalities of type A and B with appropriate $K$-functionals. Results for higher levels of smoothness are achieved by combinations of averages. This yields, in particular, strong converse inequalities of type D between $K$-functionals and suitable difference operators.

Keywords:averages, $K$-functionals, weighted spaces, strong converse inequalities
Categories:41A25, 41A63

4. CJM 1997 (vol 49 pp. 944)

Jia, R. Q.; Riemenschneider, S. D.; Zhou, D. X.
Approximation by multiple refinable functions
We consider the shift-invariant space, $\bbbs(\Phi)$, generated by a set $\Phi=\{\phi_1,\ldots,\phi_r\}$ of compactly supported distributions on $\RR$ when the vector of distributions $\phi:=(\phi_1,\ldots,\phi_r)^T$ satisfies a system of refinement equations expressed in matrix form as $$ \phi=\sum_{\alpha\in\ZZ}a(\alpha)\phi(2\,\cdot - \,\alpha) $$ where $a$ is a finitely supported sequence of $r\times r$ matrices of complex numbers. Such {\it multiple refinable functions} occur naturally in the study of multiple wavelets. The purpose of the present paper is to characterize the {\it accuracy} of $\Phi$, the order of the polynomial space contained in $\bbbs(\Phi)$, strictly in terms of the refinement mask $a$. The accuracy determines the $L_p$-approximation order of $\bbbs(\Phi)$ when the functions in $\Phi$ belong to $L_p(\RR)$ (see Jia~[10]). The characterization is achieved in terms of the eigenvalues and eigenvectors of the subdivision operator associated with the mask $a$. In particular, they extend and improve the results of Heil, Strang and Strela~[7], and of Plonka~[16]. In addition, a counterexample is given to the statement of Strang and Strela~[20] that the eigenvalues of the subdivision operator determine the accuracy. The results do not require the linear independence of the shifts of $\phi$.

Keywords:Refinement equations, refinable functions, approximation, order, accuracy, shift-invariant spaces, subdivision
Categories:39B12, 41A25, 65F15

5. CJM 1997 (vol 49 pp. 1034)

Saff, E. B.; Stahl, H.
Ray sequences of best rational approximants for $|x|^\alpha$
The convergence behavior of best uniform rational approximations $r^\ast_{mn}$ with numerator degree~$m$ and denominator degree~$n$ to the function $|x|^\alpha$, $\alpha>0$, on $[-1,1]$ is investigated. It is assumed that the indices $(m,n)$ progress along a ray sequence in the lower triangle of the Walsh table, {\it i.e.} the sequence of indices $\{ (m,n)\}$ satisfies $$ {m\over n}\rightarrow c\in [1, \infty)\quad\hbox{as } m+ n\rightarrow\infty. $$ In addition to the convergence behavior, the asymptotic distribution of poles and zeros of the approximants and the distribution of the extreme points of the error function $|x|^\alpha - r^\ast_{mn} (x)$ on $[-1,1]$ will be studied. The results will be compared with those for paradiagonal sequences $(m=n+2[\alpha/2])$ and for sequences of best polynomial approximants.

Keywords:Walsh table, rational approximation, best approximation,, distribution of poles and zeros.
Categories:41A25, 41A44

6. CJM 1997 (vol 49 pp. 74)

Hu, Y. K.; Kopotun, K. A.; Yu, X. M.
Constrained approximation in Sobolev spaces
Positive, copositive, onesided and intertwining (co-onesided) polynomial and spline approximations of functions $f\in\Wp^k\mll$ are considered. Both uniform and pointwise estimates, which are exact in some sense, are obtained.

Keywords:Constrained approximation, polynomials, splines, degree of, approximation, $L_p$ space, Sobolev space
Categories:41A10, 41A15, 41A25, 41A29

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