1. CMB 2016 (vol 60 pp. 872)
 Xu, Xu; Zhu, Laiyi

Rational Function Operators from Poisson Integrals
In this paper, we construct two classes of rational function
operators by using the Poisson integrals of the function on the
whole real
axis. The convergence rates of the uniform and mean approximation
of such rational function operators on the whole real axis are
studied.
Keywords:rational function operators, Poisson integrals, convergence rate, uniform approximation, mean approximation Categories:41A20, 41A25, 41A35 

2. CMB 2014 (vol 57 pp. 708)
 Brannan, Michael

Strong Asymptotic Freeness for Free Orthogonal Quantum Groups
It is known that the normalized standard generators of the free
orthogonal quantum group $O_N^+$ converge in distribution to a free
semicircular system as $N \to \infty$. In this note, we
substantially improve this convergence result by proving that, in
addition to distributional convergence, the operator norm of any
noncommutative polynomial in the normalized standard generators of
$O_N^+$ converges as $N \to \infty$ to the operator norm of the
corresponding noncommutative polynomial in a standard free
semicircular system. Analogous strong convergence results are obtained
for the generators of free unitary quantum groups. As applications of
these results, we obtain a matrixcoefficient version of our strong
convergence theorem, and we recover a well known $L^2$$L^\infty$ norm
equivalence for noncommutative polynomials in free semicircular
systems.
Keywords:quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decay Categories:46L54, 20G42, 46L65 

3. CMB 2012 (vol 56 pp. 544)
4. CMB 2011 (vol 55 pp. 242)
 Cegrell, Urban

Convergence in Capacity
In this note we study the convergence of sequences of MongeAmpÃ¨re measures $\{(dd^cu_s)^n\}$,
where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.
Keywords:complex MongeAmpÃ¨re operator, convergence in capacity, plurisubharmonic function Categories:32U20, 31C15 

5. CMB 2011 (vol 55 pp. 424)
 Yang, Jianbin; Li, Song

Convergence Rates of Cascade Algorithms with Infinitely Supported Masks
We investigate the solutions of refinement equations of the form
$$
\phi(x)=\sum_{\alpha\in\mathbb
Z^s}a(\alpha)\:\phi(Mx\alpha),
$$ where the function $\phi$
is in $L_p(\mathbb R^s)$$(1\le p\le\infty)$, $a$ is an infinitely
supported sequence on $\mathbb Z^s$ called a refinement mask, and
$M$ is an $s\times s$ integer matrix such that
$\lim_{n\to\infty}M^{n}=0$. Associated with the mask $a$ and $M$ is
a linear operator $Q_{a,M}$ defined on $L_p(\mathbb R^s)$ by
$Q_{a,M} \phi_0:=\sum_{\alpha\in\mathbb
Z^s}a(\alpha)\phi_0(M\cdot\alpha)$. Main results of this paper are
related to the convergence rates of $(Q_{a,M}^n
\phi_0)_{n=1,2,\dots}$ in $L_p(\mathbb R^s)$ with mask $a$ being
infinitely supported. It is proved that under some appropriate
conditions on the initial function $\phi_0$, $Q_{a,M}^n \phi_0$
converges in $L_p(\mathbb R^s)$ with an exponential rate.
Keywords:refinement equations, infinitely supported mask, cascade algorithms, rates of convergence Categories:39B12, 41A25, 42C40 

6. CMB 2011 (vol 55 pp. 15)
7. CMB 2011 (vol 54 pp. 566)
 Zhou, XiangJun; Shi, Lei; Zhou, DingXuan

Nonuniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows
We consider approximation of multivariate functions in Sobolev
spaces by high order Parzen windows in a nonuniform sampling
setting. Sampling points are neither i.i.d. nor regular, but are
noised from regular grids by nonuniform shifts of a probability
density function. Sample function values at sampling points are
drawn according to probability measures with expected values being
values of the approximated function. The approximation orders are
estimated by means of regularity of the approximated function, the
density function, and the order of the Parzen windows, under
suitable choices of the scaling parameter.
Keywords:multivariate approximation, Sobolev spaces, nonuniform randomized sampling, high order Parzen windows, convergence rates Categories:68T05, 62J02 

8. CMB 2010 (vol 53 pp. 503)
 Kurenok, V. P.

The Time Change Method and SDEs with Nonnegative Drift
Using the time change method we show how to construct a solution to the stochastic equation $dX_t=b(X_{t})dZ_t+a(X_t)dt$ with a nonnegative drift $a$ provided there exists a solution to the auxililary equation $dL_t=[a^{1/\alpha}b](L_{t})d\bar Z_t+dt$ where $Z, \bar Z$ are two symmetric stable processes of the same index $\alpha\in(0,2]$. This approach allows us to prove the existence of solutions for both stochastic equations for the values $0<\alpha<1$ and only measurable coefficients $a$ and $b$ satisfying some conditions of boundedness. The existence proof for the auxililary equation uses the method of integral estimates in the sense of Krylov.
Keywords:Onedimensional SDEs, symmetric stable processes, nonnegative drift, time change, integral estimates, weak convergence Categories:60H10, 60J60, 60J65, 60G44 

9. CMB 2009 (vol 52 pp. 627)
 Yu, Dan Sheng; Zhou, Ping; Zhou, Song Ping

On $L^{1}$Convergence of Fourier Series under the MVBV Condition
Let $f\in L_{2\pi }$ be a realvalued even function with its Fourier series $%
\frac{a_{0}}{2}+\sum_{n=1}^{\infty }a_{n}\cos nx,$ and let
$S_{n}(f,x) ,\;n\geq 1,$ be the $n$th partial sum of the Fourier series. It
is well known that if the nonnegative sequence $\{a_{n}\}$ is decreasing and
$\lim_{n\rightarrow \infty }a_{n}=0$, then%
\begin{equation*}
\lim_{n\rightarrow \infty }\Vert fS_{n}(f)\Vert _{L}=0
\text{ if
and only if }\lim_{n\rightarrow \infty }a_{n}\log n=0.
\end{equation*}%
We weaken the monotone condition in this classical result to the socalled
mean value bounded variation (MVBV) condition. The generalization of the
above classical result in realvalued function space is presented as a
special case of the main result in this paper, which gives the $L^{1}$%
convergence of a function $f\in L_{2\pi }$ in complex space. We also give
results on $L^{1}$approximation of a function $f\in L_{2\pi }$ under the
MVBV condition.
Keywords:complex trigonometric series, $L^{1}$ convergence, monotonicity, mean value bounded variation Categories:42A25, 41A50 

10. CMB 2009 (vol 52 pp. 315)
 Yi, Taishan; Zou, Xingfu

Generic QuasiConvergence for Essentially Strongly OrderPreserving Semiflows
By employing the limit set
dichotomy for essentially strongly orderpreserving semiflows and
the assumption that limit sets have infima and suprema in the
state space, we prove a generic quasiconvergence principle
implying the existence of an open and dense set of stable
quasiconvergent points. We also apply this generic quasiconvergence principle
to a model for biochemical feedback in protein
synthesis and obtain some results about the model which are of theoretical
and realistic significance.
Keywords:Essentially strongly orderpreserving semiflow, compactness, quasiconvergence Categories:34C12, 34K25 

11. CMB 2008 (vol 51 pp. 372)
12. CMB 2006 (vol 49 pp. 256)
 Neelon, Tejinder

A BernsteinWalsh Type Inequality and Applications
A BernsteinWalsh type inequality for $C^{\infty }$ functions of several
variables is derived, which then is applied to obtain analogs and
generalizations of the following classical theorems: (1) BochnakSiciak
theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real
analytic on every line is real analytic; (2) ZornLelong theorem: if a
double power series $F(x,y)$\ converges on a set of lines of positive
capacity then $F(x,y)$\ is convergent; (3) AbhyankarMohSathaye theorem:
the transfinite diameter of the convergence set of a divergent series is
zero.
Keywords:BernsteinWalsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power series Categories:32A05, 26E05 
