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
$w_{\alpha,\beta}(x)
:=(1+x)^\alpha(1x)^\beta$ 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 2014 (vol 67 pp. 527)
 Brugallé, Erwan; Shaw, Kristin

Obstructions to Approximating Tropical Curves in Surfaces Via Intersection Theory
We provide some new local obstructions to
approximating
tropical curves in
smooth tropical surfaces. These obstructions are based on
a
relation between tropical and complex intersection theories which is
also established here. We give
two applications of the methods developed in this paper.
First we classify all locally irreducible approximable 3valent fan tropical
curves in a
fan tropical plane.
Secondly, we prove that a generic nonsingular
tropical surface
in tropical projective 3space contains finitely
many approximable tropical lines
if
it is of degree 3, and contains no approximable tropical lines if
it is of degree 4 or more.
Keywords:tropical geometry, amoebas, approximation of tropical varieties, intersection theory Categories:14T05, 14M25 

3. CJM 2011 (vol 63 pp. 1161)
 Neuwirth, Stefan; Ricard, Éric

Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group
We inspect the relationship between relative Fourier
multipliers on noncommutative LebesgueOrlicz spaces of a discrete
group $\varGamma$ and relative ToeplitzSchur multipliers on
SchattenvonNeumannOrlicz classes. Four applications are given:
lacunary sets, unconditional Schauder bases for the subspace of a
Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the
norm of the Hilbert transform and the Riesz projection on
SchattenvonNeumann classes with exponent a power of 2, and the norm of
Toeplitz Schur multipliers on SchattenvonNeumann classes with
exponent less than 1.
Keywords:Fourier multiplier, Toeplitz Schur multiplier, lacunary set, unconditional approximation property, Hilbert transform, Riesz projection Categories:47B49, 43A22, 43A46, 46B28 

4. CJM 2010 (vol 62 pp. 1419)
 Yang, Dachun; Yang, Dongyong

BMOEstimates for Maximal Operators via Approximations of the Identity with NonDoubling Measures
Let $\mu$ be a nonnegative Radon measure
on $\mathbb{R}^d$ that satisfies the growth condition that there exist
constants $C_0>0$ and $n\in(0,d]$ such that for all $x\in\mathbb{R}^d$ and
$r>0$, ${\mu(B(x,\,r))\le C_0r^n}$, where $B(x,r)$ is the open ball
centered at $x$ and having radius $r$. In this paper, the authors prove
that if $f$ belongs to the $\textrm {BMO}$type space $\textrm{RBMO}(\mu)$ of Tolsa, then
the homogeneous maximal function $\dot{\mathcal{M}}_S(f)$ (when $\mathbb{R}^d$ is not an
initial cube) and the inhomogeneous maximal function
$\mathcal{M}_S(f)$ (when $\mathbb{R}^d$ is an initial cube)
associated with a given approximation of the identity $S$ of Tolsa are
either infinite everywhere or finite almost everywhere,
and in the latter case, $\dot{\mathcal{M}}_S$ and $\mathcal{M}_S$ are bounded from
$\textrm{RBMO}(\mu)$ to the $\textrm {BLO}$type
space $\textrm{RBLO}(\mu)$. The authors also prove that the inhomogeneous
maximal operator $\mathcal{M}_S$ is bounded from the local
$\textrm {BMO}$type space $\textrm{rbmo}(\mu)$
to the local $\textrm {BLO}$type space $\textrm{rblo}(\mu)$.
Keywords:Nondoubling measure, maximal operator, approximation of the identity, RBMO(mu), RBLO(mu), rbmo(mu), rblo(mu) Categories:42B25, 42B30, 47A30, 43A99 

5. CJM 2010 (vol 62 pp. 737)
 Ditzian, Z.; Prymak, A.

Approximation by Dilated Averages and KFunctionals
For a positive finite measure $d\mu(\mathbf{u})$ on $\mathbb{R}^d$
normalized to satisfy $\int_{\mathbb{R}^d}d\mu(\mathbf{u})=1$, the dilated average of
$f( \mathbf{x})$ is given by \[ A_tf(\mathbf{x})=\int_{\mathbb{R}^d}f(\mathbf{x}t\mathbf{u})d\mu(\mathbf{u}). \] It
will be shown that under some mild assumptions on $d\mu(\mathbf{u})$ one has
the equivalence \[ \A_tff\_B\approx \inf \{
(\fg\_B+t^2 \P(D)g\_B): P(D)g\in B\}\quad\text{for }t>0, \]
where $\varphi(t)\approx \psi(t)$ means
$c^{1}\le\varphi(t)/\psi(t)\le c$, $B$ is a Banach space of functions
for which translations are continuous isometries and $P(D)$ is an
elliptic differential operator induced by $\mu$. Many applications are
given, notable among which is the averaging operator with $d\mu(\mathbf{u})=
\frac{1}{m(S)}\chi_S(\mathbf{u})d\mathbf{u}$, where $S$ is a bounded convex set
in $\mathbb{R}^d$ with an interior point, $m(S)$ is the Lebesgue measure of
$S$, and $\chi_S(\mathbf{u})$ is the characteristic function of $S$. The rate
of approximation by averages on the boundary of a convex set under
more restrictive conditions is also shown to be equivalent to an
appropriate $K$functional.
Keywords:rate of approximation, Kfunctionals, strong converse inequality Categories:41A27, 41A35, 41A63 

6. CJM 2007 (vol 59 pp. 3)
 Biller, Harald

Holomorphic Generation of Continuous Inverse Algebras
We study complex commutative Banach algebras
(and, more generally, continuous
inverse algebras) in which the holomorphic functions of a fixed $n$tuple
of elements are dense. In particular, we characterize the compact subsets
of~$\C^n$ which appear as joint spectra of such $n$tuples. The
characterization is compared with several established notions of holomorphic
convexity by means of approximation
conditions.
Keywords:holomorphic functional calculus, commutative continuous inverse algebra, holomorphic convexity, Stein compacta, meromorphic convexity, holomorphic approximation Categories:46H30, 32A38, 32E30, 41A20, 46J15 

7. CJM 2006 (vol 58 pp. 249)
8. CJM 2004 (vol 56 pp. 776)
 Lim, Yongdo

Best Approximation in Riemannian Geodesic Submanifolds of Positive Definite Matrices
We explicitly describe
the best approximation in
geodesic submanifolds of positive definite matrices
obtained from involutive
congruence transformations on the
CartanHadamard manifold ${\mathrm{Sym}}(n,{\Bbb R})^{++}$ of
positive definite matrices.
An explicit calculation for the minimal distance
function from the geodesic submanifold
${\mathrm{Sym}}(p,{\mathbb R})^{++}\times
{\mathrm{Sym}}(q,{\mathbb R})^{++}$ block diagonally embedded in
${\mathrm{Sym}}(n,{\mathbb R})^{++}$ is
given in terms of metric and
spectral geometric means, Cayley transform, and Schur
complements of positive definite matrices when $p\leq 2$ or $q\leq 2.$
Keywords:Matrix approximation, positive, definite matrix, geodesic submanifold, CartanHadamard manifold,, best approximation, minimal distance function, global tubular, neighborhood theorem, Schur complement, metric and spectral, geometric mean, Cayley transform Categories:15A48, 49R50, 15A18, 53C3 

9. CJM 2002 (vol 54 pp. 1305)
10. CJM 2002 (vol 54 pp. 1121)
 Bao, Jiguang

Fully Nonlinear Elliptic Equations on General Domains
By means of the Pucci operator, we construct a function $u_0$, which plays
an essential role in our considerations, and give the existence and regularity
theorems for the bounded viscosity solutions of the generalized Dirichlet
problems of second order fully nonlinear elliptic equations on the general
bounded domains, which may be irregular. The approximation method, the accretive
operator technique and the Caffarelli's perturbation theory are used.
Keywords:Pucci operator, viscosity solution, existence, $C^{2,\psi}$ regularity, Dini condition, fully nonlinear equation, general domain, accretive operator, approximation lemma Categories:35D05, 35D10, 35J60, 35J67 

11. CJM 2002 (vol 54 pp. 945)
 Boivin, André; Gauthier, Paul M.; Paramonov, Petr V.

Approximation on Closed Sets by Analytic or Meromorphic Solutions of Elliptic Equations and Applications
Given a homogeneous elliptic partial differential operator $L$ with constant
complex coefficients and a class of functions (jetdistributions) which
are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and
which belong locally to a Banach space $V$, we consider the problem of
approximating in the norm of $V$ the functions in this class by ``analytic''
and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth,
Arakelyan (including tangential) and Carleman type theorems for a large class
of Banach spaces $V$ and operators $L$. Important applications to boundary
value problems of solutions of homogeneous elliptic partial differential
equations are obtained, including the solution of a generalized Dirichlet
problem.
Keywords:approximation on closed sets, elliptic operator, strongly elliptic operator, $L$meromorphic and $L$analytic functions, localization operator, Banach space of distributions, Dirichlet problem Categories:30D40, 30E10, 31B35, 35Jxx, 35J67, 41A30 

12. CJM 1999 (vol 51 pp. 117)
13. 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 

14. 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 

15. CJM 1997 (vol 49 pp. 963)
 Lin, Huaxin

Homomorphisms from $C(X)$ into $C^*$algebras
Let $A$ be a simple $C^*$algebra
with real rank zero, stable rank one and weakly
unperforated $K_0(A)$ of countable rank. We show that
a monomorphism $\phi\colon C(S^2) \to A$ can be approximated
pointwise by homomorphisms from $C(S^2)$ into $A$ with
finite dimensional range if and only if certain index
vanishes. In particular, we show that every homomorphism
$\phi$ from $C(S^2)$ into a UHFalgebra can be approximated
pointwise by homomorphisms from $C(S^2)$ into the UHFalgebra
with finite dimensional range. As an application, we show
that if $A$ is a simple $C^*$algebra of real rank zero
and is an inductive limit of matrices over $C(S^2)$ then
$A$ is an AFalgebra. Similar results for tori are also
obtained. Classification of ${\bf Hom}\bigl(C(X),A\bigr)$
for lower dimensional spaces is also studied.
Keywords:Homomorphism of $C(S^2)$, approximation, real, rank zero, classification Categories:46L05, 46L80, 46L35 

16. 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 
