26. CMB 2011 (vol 55 pp. 646)
 Zhou, Jiang; Ma, Bolin

Marcinkiewicz Commutators with Lipschitz Functions in Nonhomogeneous Spaces
Under the assumption that $\mu$ is a nondoubling
measure, we study certain commutators generated by the
Lipschitz function and the Marcinkiewicz integral whose kernel
satisfies a HÃ¶rmandertype condition. We establish the boundedness
of these commutators on the Lebesgue spaces, Lipschitz spaces, and
Hardy spaces. Our results are extensions of known theorems in the
doubling case.
Keywords:non doubling measure, Marcinkiewicz integral, commutator, ${\rm Lip}_{\beta}(\mu)$, $H^1(\mu)$ Categories:42B25, 47B47, 42B20, 47A30 

27. CMB 2011 (vol 56 pp. 194)
 Stefánsson, Úlfar F.

On the Smallest and Largest Zeros of MÃ¼ntzLegendre Polynomials
MÃ¼ntzLegendre
polynomials $L_n(\Lambda;x)$ associated with a
sequence $\Lambda=\{\lambda_k\}$ are obtained by orthogonalizing the
system $(x^{\lambda_0}, x^{\lambda_1}, x^{\lambda_2}, \dots)$ in
$L_2[0,1]$ with respect to the Legendre weight. If the $\lambda_k$'s
are distinct, it is well known that $L_n(\Lambda;x)$ has exactly $n$
zeros $l_{n,n}\lt l_{n1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ on $(0,1)$.
First we prove the following global bound for the smallest zero,
$$
\exp\biggl(4\sum_{j=0}^n \frac{1}{2\lambda_j+1}\biggr) \lt l_{n,n}.
$$
An important consequence is that if the associated MÃ¼ntz space is
nondense in $L_2[0,1]$, then
$$
\inf_{n}x_{n,n}\geq
\exp\biggl({4\sum_{j=0}^{\infty} \frac{1}{2\lambda_j+1}}\biggr)\gt 0,
$$
so
the elements $L_n(\Lambda;x)$ have no zeros close to 0.
Furthermore, we determine the asymptotic behavior of the largest zeros; for $k$ fixed,
$$
\lim_{n\rightarrow\infty} \vert \log l_{k,n}\vert \sum_{j=0}^n
(2\lambda_j+1)= \Bigl(\frac{j_k}{2}\Bigr)^2,
$$
where $j_k$ denotes the $k$th zero of the Bessel function $J_0$.
Keywords:MÃ¼ntz polynomials, MÃ¼ntzLegendre polynomials Categories:42C05, 42C99, 41A60, 30B50 

28. CMB 2011 (vol 55 pp. 555)
 Michalowski, Nicholas; Rule, David J.; Staubach, Wolfgang

Weighted $L^p$ Boundedness of Pseudodifferential Operators and Applications
In this paper we prove weighted norm inequalities with weights in
the $A_p$ classes, for pseudodifferential operators with symbols in
the class ${S^{n(\rho 1)}_{\rho, \delta}}$ that fall outside the
scope of CalderÃ³nZygmund theory. This is accomplished by
controlling the sharp function of the pseudodifferential operator by
HardyLittlewood type maximal functions. Our weighted norm
inequalities also yield $L^{p}$ boundedness of commutators of
functions of bounded mean oscillation with a wide class of operators
in $\mathrm{OP}S^{m}_{\rho, \delta}$.
Keywords:weighted norm inequality, pseudodifferential operator, commutator estimates Categories:42B20, 42B25, 35S05, 47G30 

29. CMB 2011 (vol 55 pp. 708)
30. 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 

31. CMB 2011 (vol 55 pp. 303)
 Han, Yongsheng; Lee, MingYi; Lin, ChinCheng

Atomic Decomposition and Boundedness of Operators on Weighted Hardy Spaces
In this article, we establish a new atomic decomposition for $f\in L^2_w\cap H^p_w$,
where the decomposition converges in $L^2_w$norm rather than in the distribution sense.
As applications of this decomposition, assuming that $T$ is a linear
operator bounded on $L^2_w$ and $0
Keywords:$A_p$ weights, atomic decomposition, CalderÃ³n reproducing formula, weighted Hardy spaces Categories:42B25, 42B30 

32. CMB 2011 (vol 55 pp. 689)
 Berndt, Ryan

A Pointwise Estimate for the Fourier Transform and Maxima of a Function
We show a pointwise estimate for the Fourier
transform on the line involving the number of times the function
changes monotonicity. The contrapositive of the theorem may be used to
find a lower bound to the number of local maxima of a function. We
also show two applications of the theorem. The first is the two weight
problem for the Fourier transform, and the second is estimating the
number of roots of the derivative of a function.
Keywords:Fourier transform, maxima, two weight problem, roots, norm estimates, DirichletJordan theorem Categories:42A38, 65T99 

33. CMB 2010 (vol 54 pp. 113)
 Hytönen, Tuomas P.

On the Norm of the BeurlingAhlfors Operator in Several Dimensions
The generalized BeurlingAhlfors operator $S$ on
$L^p(\mathbb{R}^n;\Lambda)$, where $\Lambda:=\Lambda(\mathbb{R}^n)$ is the
exterior algebra with its natural Hilbert space norm, satisfies the
estimate
$$\S\_{\mathcal{L}(L^p(\mathbb{R}^n;\Lambda))}\leq(n/2+1)(p^*1),\quad
p^*:=\max\{p,p'\}$$
This improves on earlier results in all dimensions $n\geq 3$. The
proof is based on the heat extension and relies at the bottom on
Burkholder's sharp inequality for martingale transforms.
Categories:42B20, 60G46 

34. CMB 2010 (vol 54 pp. 159)
 Sababheh, Mohammad

Hardy Inequalities on the Real Line
We prove that some inequalities, which are considered to be
generalizations of Hardy's inequality on the circle,
can be modified and proved to be true for functions integrable on the real line.
In fact we would like to show that some constructions that were
used to prove the Littlewood conjecture can be used similarly to
produce real Hardytype inequalities.
This discussion will lead to many questions concerning the
relationship between Hardytype inequalities on the circle and
those on the real line.
Keywords:Hardy's inequality, inequalities including the Fourier transform and Hardy spaces Categories:42A05, 42A99 

35. CMB 2010 (vol 54 pp. 100)
 Fan, Dashan; Wu, Huoxiong

On the Generalized Marcinkiewicz Integral Operators with Rough Kernels
A class of generalized Marcinkiewicz
integral operators is introduced, and, under rather weak conditions
on the integral kernels, the boundedness of such operators on $L^p$
and TriebelLizorkin spaces is established.
Keywords: Marcinkiewicz integral, LittlewoodPaley theory, TriebelLizorkin space, rough kernel, product domain Categories:42B20, , , , , 42B25, 42B30, 42B99 

36. CMB 2010 (vol 54 pp. 172)
 Shayya, Bassam

Measures with Fourier Transforms in $L^2$ of a Halfspace
We prove that if the Fourier transform of a compactly supported
measure is in $L^2$ of a halfspace, then the measure is
absolutely continuous to Lebesgue measure. We then show how this
result can be used to translate information about the
dimensionality of a measure and the decay of its Fourier
transform into geometric information about its support.
Categories:42B10, 28A75 

37. CMB 2010 (vol 53 pp. 491)
 Jizheng, Huang; Liu, Heping

The Weak Type (1,1) Estimates of Maximal Functions on the Laguerre Hypergroup
In this paper, we discuss various maximal functions on the Laguerre hypergroup $\mathbf{K}$ including the heat maximal function, the Poisson maximal function, and the HardyLittlewood maximal function which is consistent with the structure of hypergroup of $\mathbf{K}$. We shall establish the weak type $(1,1)$ estimates for these maximal functions. The $L^p$ estimates for $p>1$ follow from the interpolation. Some applications are included.
Keywords:Laguerre hypergroup, maximal function, heat kernel, Poisson kernel Categories:42B25, 43A62 

38. CMB 2009 (vol 53 pp. 133)
39. CMB 2009 (vol 53 pp. 263)
 Feuto, Justin; Fofana, Ibrahim; Koua, Konin

Weighted Norm Inequalities for a Maximal Operator in Some Subspace of Amalgams
We give weighted norm inequalities for the maximal fractional operator $ \mathcal M_{q,\beta }$ of HardyÂLittlewood and the fractional integral $I_{\gamma}$. These inequalities are established between $(L^{q},L^{p}) ^{\alpha }(X,d,\mu )$ spaces (which are superspaces of Lebesgue spaces $L^{\alpha}(X,d,\mu)$ and subspaces of amalgams $(L^{q},L^{p})(X,d,\mu)$) and in the setting of space of homogeneous type $(X,d,\mu)$. The conditions on the weights are stated in terms of Orlicz norm.
Keywords:fractional maximal operator, fractional integral, space of homogeneous type Categories:42B35, 42B20, 42B25 

40. 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 

41. CMB 2009 (vol 52 pp. 521)
42. CMB 2009 (vol 52 pp. 95)
 Miranian, L.

Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory
In the work presented below the classical subject of orthogonal
polynomials on the unit
circle is discussed in the matrix setting. An explicit matrix
representation of the matrix valued orthogonal polynomials in terms of
the moments of the measure is presented. Classical recurrence
relations are revisited using the matrix representation of the
polynomials. The matrix expressions for the kernel polynomials and the
ChristoffelDarboux formulas are presented for the first time.
Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, ChristoffelDarboux Category:42C99 

43. CMB 2008 (vol 51 pp. 487)
44. CMB 2008 (vol 51 pp. 348)
45. CMB 2007 (vol 50 pp. 85)
 Han, Deguang

Classification of Finite GroupFrames and SuperFrames
Given a finite group $G$, we examine the classification of all
frame representations of $G$ and the classification of all
$G$frames, \emph{i.e.,} frames induced by group representations of $G$.
We show that the exact number of equivalence classes of $G$frames
and the exact number of frame representations can be explicitly
calculated. We also discuss how to calculate the largest number
$L$ such that there exists an $L$tuple of strongly disjoint
$G$frames.
Keywords:frames, groupframes, frame representations, disjoint frames Categories:42C15, 46C05, 47B10 

46. CMB 2006 (vol 49 pp. 414)
47. CMB 2006 (vol 49 pp. 438)
 Mercer, Idris David

Unimodular Roots of\\ Special Littlewood Polynomials
We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n1} z^{n1}$ a Littlewood
polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ selfreciprocal
if $\alpha(z) = z^{n1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if
$n = 2m+1$ and $a_{m+j} = (1)^j a_{mj}$ for all $j$. It has been observed
that Littlewood polynomials with particularly high minimum modulus on
the unit
circle in $\bC$ tend to be skewsymmetric. In this paper, we prove that a
skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle,
as well as providing a new proof of the known result that a selfreciprocal
Littlewood polynomial must have a zero on the unit circle.
Categories:26C10, 30C15, 42A05 

48. CMB 2006 (vol 49 pp. 3)
 AlSalman, Ahmad

On a Class of Singular Integral Operators With Rough Kernels
In this paper, we study the $L^p$ mapping properties of a class of singular
integral operators with rough kernels belonging to certain block spaces. We
prove that our operators are bounded on $L^p$ provided that their kernels
satisfy a size condition much weaker than that for the classical
Calder\'{o}nZygmund singular integral operators. Moreover, we present an
example showing that our size condition is optimal. As a consequence of our
results, we substantially improve a previously known result on certain maximal
functions.
Keywords:Singular integrals, Rough kernels, Square functions,, Maximal functions, Block spaces Categories:42B20, 42B15, 42B25 

49. CMB 2005 (vol 48 pp. 382)
 De Carli, Laura

Uniform Estimates of Ultraspherical Polynomials of Large Order
In this paper we prove the sharp inequality
$$ P_n^{(s)}(x)\leq
P_n^{(s)}(1)\bigl(x^n +\frac{n1}{2 s+1}(1x^n)\bigr),$$
where
$P_n^{(s)}(x)$ is the classical ultraspherical polynomial of
degree $n$ and order $s\ge n\frac{1+\sqrt 5}{4}$. This inequality
can be refined in $[0,z_n^s]$ and $[z_n^s,1]$, where $z_n^s$
denotes the largest zero of $P_n^{(s)}(x)$.
Categories:42C05, 33C47 

50. CMB 2005 (vol 48 pp. 370)
 Daly, J. E.; Fridli, S.

Trigonometric Multipliers on $H_{2\pi}$
In this paper we consider multipliers on the real Hardy space
$H_{2\pi}$. It is known that the Marcinkiewicz and the
H\"ormanderMihlin conditions are sufficient for the corresponding
trigonometric multiplier to be bounded on $L_{2\pi}^p$, $1
Keywords:Multipliers, Hardy space Categories:42A45, 42A50, 42A85 
