51. 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 

52. 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 

53. 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 

54. 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 

55. CMB 2005 (vol 48 pp. 260)
56. CMB 2004 (vol 47 pp. 475)
 Wade, W. R.

Uniqueness of Almost Everywhere Convergent Vilenkin Series
D. J. Grubb [3] has shown that uniqueness holds, under a
mild growth condition, for Vilenkin series which converge almost
everywhere to zero. We show that, under even less restrictive
growth conditions, one can replace the limit function 0 by an
arbitrary $f\in L^q$, when $q>1$.
Categories:43A75, 42C10 

57. CMB 2004 (vol 47 pp. 3)
58. CMB 2003 (vol 46 pp. 191)
 Kim, YongCheol

Weak Type Estimates of the Maximal Quasiradial BochnerRiesz Operator On Certain Hardy Spaces
Let $\{A_t\}_{t>0}$ be the dilation group in $\mathbb{R}^n$ generated
by the infinitesimal generator $M$ where $A_t=\exp(M\log t)$, and let
$\varrho\in C^{\infty}(\mathbb{R}^n\setminus\{0\})$ be a
$A_t$homogeneous distance function defined on $\mathbb{R}^n$. For
$f\in \mathfrak{S}(\mathbb{R}^n)$, we define the maximal quasiradial
BochnerRiesz operator $\mathfrak{M}^{\delta}_{\varrho}$ of index
$\delta>0$ by
$$
\mathfrak{M}^{\delta}_{\varrho} f(x)=\sup_{t>0}\mathcal{F}^{1}
[(1\varrho/t)_+^{\delta}\hat f ](x).
$$
If $A_t=t I$ and $\{\xi\in \mathbb{R}^n\mid \varrho(\xi)=1\}$ is a
smooth convex hypersurface of finite type, then we prove in an
extremely easy way that $\mathfrak{M}^{\delta}_{\varrho}$ is well
defined on $H^p(\mathbb{R}^n)$ when $\delta=n(1/p1/2)1/2$ and
$0 n(1/p1/2)1/2$ and $0
Categories:42B15, 42B25 

59. CMB 2002 (vol 45 pp. 25)
 Bloom, Steven; Kerman, Ron

Extrapolation of $L^p$ Data from a Modular Inequality
If an operator $T$ satisfies a modular inequality on a rearrangement
invariant space $L^\rho (\Omega,\mu)$, and if $p$ is strictly between
the indices of the space, then the Lebesgue inequality $\int Tf^p
\leq C \int f^p$ holds. This extrapolation result is a partial
converse to the usual interpolation results. A modular inequality for
Orlicz spaces takes the form $\int \Phi (Tf) \leq \int \Phi (C
f)$, and here, one can extrapolate to the (finite) indices $i(\Phi)$
and $I(\Phi)$ as well.
Category:42B25 

60. CMB 2002 (vol 45 pp. 46)
61. CMB 2001 (vol 44 pp. 121)
62. CMB 2000 (vol 43 pp. 355)
 Kelly, Brian P.

A DimensionFree WeakType Estimate for Operators on UMDValued Functions
Let $\T$ denote the unit circle in the complex plane, and let $X$ be a
Banach space that satisfies\break Burkholder's UMD condition. Fix a natural
number, $N \in \N$. Let $\od$ denote the reverse lexicographical order
on $\Z^N$. For each $f \in L^1 (\T^N,X)$, there exists a strongly
measurable function $\wt{f}$ such that formally, for all $\bfn \in
\Z^N$, $\Dual{{\wt{f}}} (\bfn) = i \sgn_\od (\bfn)
\Dual{f} (\bfn)$. In this paper, we present a summation method for
this conjugate function directly analogous to the martingale methods
developed by Asmar and MontgomerySmith for scalarvalued functions.
Using a stochastic integral representation and an application of
Garling's characterization of UMD spaces, we prove that the associated
maximal operator satisfies a weaktype $(1,1)$ inequality with a
constant independent of the dimension~$N$.
Category:42A61 

63. CMB 2000 (vol 43 pp. 330)
 Hare, Kathryn E.

Maximal Operators and Cantor Sets
We consider maximal operators in the plane, defined by Cantor sets of
directions, and show such operators are not bounded on $L^2$ if the
Cantor set has positive Hausdorff dimension.
Keywords:maximal functions, Cantor set, lacunary set Categories:42B25, 43A46 

64. CMB 2000 (vol 43 pp. 63)
 Iosevich, Alex; Lu, Guozhen

Sharpness Results and Knapp's Homogeneity Argument
We prove that the $L^2$ restriction theorem, and $L^p \to L^{p'}$,
$\frac{1}{p}+\frac{1}{p'}=1$, boundedness of the surface averages
imply certain geometric restrictions on the underlying
hypersurface. We deduce that these bounds imply that a certain
number of principal curvatures do not vanish.
Category:42B99 

65. CMB 2000 (vol 43 pp. 17)
 Bak, JongGuk

Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves
Suppose that $\g \in C^2\bigl([0,\infty)\bigr)$ is a realvalued function
such that $\g(0)=\g'(0)=0$, and $\g''(t)\approx t^{m2}$, for some integer
$m\geq 2$. Let $\Gamma (t)=\bigl(t,\g(t)\bigr)$, $t>0$, be a curve in the
plane, and let $d \lambda =dt$ be a measure on this curve. For a
function $f$
on $\bR^2$, let
$$
Tf(x)=(\lambda *f)(x)=\int_0^{\infty} f\bigl(x\Gamma(t)\bigr)\,dt,
\quad x\in\bR^2 .
$$
An elementary proof is given for the optimal $L^p$$L^q$ mapping
properties of $T$.
Categories:42A85, 42B15 

66. CMB 1999 (vol 42 pp. 463)
 Hofmann, Steve; Li, Xinwei; Yang, Dachun

A Generalized Characterization of Commutators of Parabolic Singular Integrals
Let $x=(x_1, \dots, x_n)\in\rz$ and $\dz_\lz x=(\lz^{\az_1}x_1,
\dots,\lz^{\az_n}x_n)$, where $\lz>0$ and $1\le \az_1\le\cdots
\le\az_n$. Denote $\az=\az_1+\cdots+\az_n$. We characterize those
functions $A(x)$ for which the parabolic Calder\'on commutator
$$
T_{A}f(x)\equiv \pv \int_{\mathbb{R}^n}
K(xy)[A(x)A(y)]f(y)\,dy
$$
is bounded on $L^2(\mathbb{R}^n)$, where $K(\dz_\lz x)=\lz^{\az1}K(x)$,
$K$ is smooth away from the origin and satisfies a certain cancellation
property.
Keywords:parabolic singular integral, commutator, parabolic $\BMO$ sobolev space, homogeneous space, T1theorem, symbol Category:42B20 

67. CMB 1999 (vol 42 pp. 344)
68. CMB 1999 (vol 42 pp. 198)
69. CMB 1999 (vol 42 pp. 37)
70. CMB 1998 (vol 41 pp. 398)
 Dziubański, Jacek; Hernández, Eugenio

Bandlimited wavelets with subexponential decay
It is well known that the compactly supported wavelets cannot belong to
the class $C^\infty({\bf R})\cap L^2({\bf R})$. This is also true for
wavelets with exponential decay. We show that one can construct
wavelets in the class $C^\infty({\bf R})\cap L^2({\bf R})$ that are
``almost'' of exponential decay and, moreover, they are
bandlimited. We do this by showing that we can adapt the
construction of the Lemari\'eMeyer wavelets \cite{LM} that
is found in \cite{BSW} so that we obtain bandlimited,
$C^\infty$wavelets on $\bf R$ that have subexponential decay,
that is, for every $0<\varepsilon<1$, there exits $C_\varepsilon>0$
such that $\psi(x)\leq C_\varepsilon e^{x^{1\varepsilon}}$,
$x\in\bf R$. Moreover, all of its derivatives have also
subexponential decay. The proof is constructive and uses the
Gevrey classes of functions.
Keywords:Wavelet, Gevrey classes, subexponential decay Category:42C15 

71. CMB 1998 (vol 41 pp. 478)
72. CMB 1998 (vol 41 pp. 404)
 AlHasan, Abdelnaser J.; Fan, Dashan

$L^p$boundedness of a singular integral operator
Let $b(t)$ be an $L^\infty$ function on $\bR$, $\Omega (\,y')$ be
an $H^1$ function on the unit sphere satisfying the mean zero
property (1) and $Q_m(t)$ be a real polynomial on $\bR$ of degree
$m$ satisfying $Q_m(0)=0$. We prove that the singular integral
operator
$$
T_{Q_m,b} (\,f) (x)=p.v. \int_\bR^n b(y) \Omega(\,y) y^{n} f
\left( xQ_m (y) y' \right) \,dy
$$
is bounded in $L^p (\bR^n)$ for $1
Keywords:singular integral, rough kernel, Hardy space Category:42B20 

73. CMB 1998 (vol 41 pp. 306)
74. CMB 1998 (vol 41 pp. 49)
 Harrison, K. J.; Ward, J. A.; Eaton, LJ.

Stability of weighted darma filters
We study the stability of linear filters associated with certain types of
linear difference equations with variable coefficients. We show that
stability is determined by the locations of the poles of a rational transfer
function relative to the spectrum of an associated weighted shift operator.
The known theory for filters associated with constantcoefficient difference
equations is a special case.
Keywords:Difference equations, adaptive $\DARMA$ filters, weighted shifts,, stability and boundedness, automatic continuity Categories:47A62, 47B37, 93D25, 42A85, 47N70 

75. CMB 1997 (vol 40 pp. 433)