1. CJM 2015 (vol 68 pp. 109)
 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
$w_{\alpha,\beta}(x)
:=(1+x)^\alpha(1x)^\beta$ 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
Jacksontype estimate involving the weighted DitzianTotik $k$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 Jacksontype 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 shiftinvariant 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, shiftinvariant 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 (coonesided) 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 
