26. CJM 2010 (vol 63 pp. 181)
 Ismail, Mourad E. H.; Obermaier, Josef

Characterizations of Continuous and Discrete $q$Ultraspherical Polynomials
We characterize the continuous $q$ultraspherical polynomials in
terms of the special form of the coefficients in the expansion
$\mathcal{D}_q P_n(x)$ in the basis $\{P_n(x)\}$, $\mathcal{D}_q$
being the AskeyWilson divided difference operator. The polynomials
are assumed to be symmetric, and the connection coefficients
are multiples of the reciprocal of the square of the $L^2$ norm of
the polynomials. A similar characterization is given for the discrete
$q$ultraspherical polynomials. A new proof of the evaluation of
the connection coefficients for big $q$Jacobi polynomials is given.
Keywords:continuous $q$ultraspherical polynomials, big $q$Jacobi polynomials, discrete $q$ultra\spherical polynomials, AskeyWilson operator, $q$difference operator, recursion coefficients Categories:33D45, 42C05 

27. 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 

28. CJM 2010 (vol 62 pp. 1182)
 Yue, Hong

A Fractal Function Related to the JohnNirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$
A borderline case function $f$ for $ Q_{\alpha}({\mathbb R^n})$ spaces
is defined as a Haar wavelet decomposition, with the coefficients
depending on a fixed parameter $\beta>0$. On its support $I_0=[0,
1]^n$, $f(x)$ can be expressed by the binary expansions of the
coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb
R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for
$\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a
JohnNirenberg inequality for $ Q_{\alpha}({\mathbb R^n})$. When
$\beta\neq 1$, $f$ is a selfaffine function. It is continuous almost
everywhere and discontinuous at all dyadic points inside $I_0$. In
addition, it is not monotone along any coordinate direction in any
small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from
$I_0$ to $[\frac{1}{12^{\beta}}, \frac{1}{12^{\beta}}]$, and the
graph of $f$ has a noninteger fractal dimension $n+1\beta$.
Keywords:Haar wavelets, Q spaces, JohnNirenberg inequality, Greedy expansion, selfaffine, fractal, Box dimension Categories:42B35, 42C10, 30D50, 28A80 

29. CJM 2010 (vol 62 pp. 827)
 Ouyang, Caiheng; Xu, Quanhua

BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces
This paper studies the relationship between vectorvalued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1z)^{q1}\\nabla f(z)\^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\f(z)f(z_0)\^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$uniformly convex, where $$P_{z_0}(z)=\frac{1z_0^2}{1\bar{z_0}z^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$uniformly smooth norm.
Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces Categories:46E40, 42B25, 46B20 

30. CJM 2009 (vol 61 pp. 807)
31. CJM 2009 (vol 61 pp. 141)
 Green, Ben; Konyagin, Sergei

On the Littlewood Problem Modulo a Prime
Let $p$ be a prime, and let $f \from \mathbb{Z}/p\mathbb{Z} \rightarrow
\mathbb{R}$ be a function with $\E f = 0$ and $\Vert \widehat{f}
\Vert_1 \leq 1$. Then
$\min_{x \in \Zp} f(x) = O(\log p)^{1/3 + \epsilon}$.
One should think of $f$ as being ``approximately continuous''; our
result is then an ``approximate intermediate value theorem''.
As an immediate consequence we show that if $A \subseteq \Zp$ is a
set of cardinality $\lfloor p/2\rfloor$, then
$\sum_r \widehat{1_A}(r) \gg (\log p)^{1/3  \epsilon}$. This
gives a result on a ``mod $p$'' analogue of Littlewood's wellknown
problem concerning the smallest possible $L^1$norm of the Fourier
transform of a set of $n$ integers.
Another application is to answer a question of Gowers. If $A
\subseteq \Zp$ is a set of size $\lfloor p/2 \rfloor$, then there is
some $x \in \Zp$ such that
\[  A \cap (A + x)  p/4  = o(p).\]
Categories:42A99, 11B99 

32. CJM 2008 (vol 60 pp. 1283)
 Ho, KwokPun

Remarks on LittlewoodPaley Analysis
LittlewoodPaley analysis is generalized in
this article. We show that the compactness of the Fourier support
imposed on the analyzing function can be removed. We also prove
that the LittlewoodPaley decomposition of tempered distributions
converges under a topology stronger than the weakstar topology,
namely, the inductive limit topology. Finally, we construct a
multiparameter LittlewoodPaley analysis and obtain the
corresponding ``renormalization'' for the convergence of this
multiparameter LittlewoodPaley analysis.
Keywords:LittlewoodPaley analysis, distributions Category:42B25 

33. CJM 2008 (vol 60 pp. 685)
 Savu, Anamaria

Closed and Exact Functions in the Context of GinzburgLandau Models
For a general vector field we exhibit two Hilbert spaces, namely
the space of so called \emph{closed functions} and the space of \emph{exact functions}
and we calculate the codimension of the space of exact functions
inside the larger space of closed functions.
In particular we provide a new approach for the known cases:
the Glauber field and the secondorder GinzburgLandau field
and for the case of the fourthorder GinzburgLandau field.
Keywords:Hermite polynomials, Fock space, Fourier coefficients, Fourier transform, group of symmetries Categories:42B05, 81Q50, 42A16 

34. CJM 2008 (vol 60 pp. 334)
 Curry, Eva

LowPass Filters and Scaling Functions for Multivariable Wavelets
We show that a characterization of scaling functions for
multiresolution analyses given by Hern\'{a}ndez and Weiss and that a
characterization of lowpass filters given by Gundy both hold for
multivariable multiresolution analyses.
Keywords:multivariable multiresolution analysis, lowpass filter, scaling function Categories:42C40, 60G35 

35. CJM 2007 (vol 59 pp. 1223)
 Buraczewski, Dariusz; Martinez, Teresa; Torrea, José L.

CalderÃ³nZygmund Operators Associated to Ultraspherical Expansions
We define the higher order Riesz transforms and the LittlewoodPaley
$g$function
associated to the differential operator $L_\l f(\theta)=f''(\theta)2\l\cot\theta
f'(\theta)+\l^2f(\theta)$. We prove that these operators are
Calder\'{o}nZygmund operators in the homogeneous type space
$((0,\pi),(\sin t)^{2\l}\,dt)$. Consequently, $L^p$ weighted,
$H^1L^1$ and $L^\inftyBMO$ inequalities are obtained.
Keywords:ultraspherical polynomials, CalderÃ³nZygmund operators Categories:42C05, 42C15frcs 

36. CJM 2007 (vol 59 pp. 1207)
 Bu, Shangquan; Le, Christian

$H^p$Maximal Regularity and Operator Valued Multipliers on Hardy Spaces
We consider maximal regularity in the $H^p$ sense for the Cauchy
problem $u'(t) + Au(t) = f(t)\ (t\in \R)$, where $A$ is a closed
operator on a Banach space $X$ and $f$ is an $X$valued function
defined on $\R$. We prove that if $X$ is an AUMD Banach space,
then $A$ satisfies $H^p$maximal regularity if and only if $A$ is
Rademacher sectorial of type $<\frac{\pi}{2}$. Moreover we find an
operator $A$ with $H^p$maximal regularity that does not have the
classical $L^p$maximal regularity. We prove a related Mikhlin
type theorem for operator valued Fourier multipliers on Hardy
spaces $H^p(\R;X)$, in the case when $X$ is an AUMD Banach space.
Keywords:$L^p$maximal regularity, $H^p$maximal regularity, Rademacher boundedness Categories:42B30, 47D06 

37. CJM 2007 (vol 59 pp. 276)
 Bernardis, A. L.; MartínReyes, F. J.; Salvador, P. Ortega

Weighted Inequalities for HardySteklov Operators
We characterize the pairs of weights $(v,w)$ for which the
operator $Tf(x)=g(x)\int_{s(x)}^{h(x)}f$ with $s$ and $h$
increasing and continuous functions is of strong type
$(p,q)$ or weak type $(p,q)$ with respect to the pair
$(v,w)$ in the case $0
Keywords:HardySteklov operator, weights, inequalities Categories:26D15, 46E30, 42B25 

38. CJM 2006 (vol 58 pp. 1121)
 Bownik, Marcin; Speegle, Darrin

The Feichtinger Conjecture for Wavelet Frames, Gabor Frames and Frames of Translates
The Feichtinger conjecture is considered for three special families of
frames. It is shown that if a wavelet frame satisfies a certain weak
regularity condition, then it can be written as the finite union of
Riesz basic sequences each of which is a wavelet system. Moreover, the
above is not true for general wavelet frames. It is also shown that a
supadjoint Gabor frame can be written as the finite union of Riesz
basic sequences. Finally, we show how existing techniques can be
applied to determine whether frames of translates can be written as
the finite union of Riesz basic sequences. We end by giving an example
of a frame of translates such that any Riesz basic subsequence must
consist of highly irregular translates.
Keywords:frame, Riesz basic sequence, wavelet, Gabor system, frame of translates, paving conjecture Categories:42B25, 42B35, 42C40 

39. CJM 2006 (vol 58 pp. 548)
 Galanopoulos, P.; Papadimitrakis, M.

Hausdorff and QuasiHausdorff Matrices on Spaces of Analytic Functions
We consider Hausdorff and quasiHausdorff matrices as operators
on classical spaces of analytic functions such as the Hardy and
the Bergman spaces, the Dirichlet space, the Bloch spaces and $\BMOA$. When the generating
sequence of the matrix is the moment sequence of a measure $\mu$,
we find the conditions on $\mu$ which are equivalent to the boundedness
of the matrix on the various spaces.
Categories:47B38, 46E15, 40G05, 42A20 

40. CJM 2006 (vol 58 pp. 401)
 Kolountzakis, Mihail N.; Révész, Szilárd Gy.

On Pointwise Estimates of Positive Definite Functions With Given Support
The following problem has been suggested by Paul Tur\' an. Let
$\Omega$ be a symmetric convex body in the Euclidean space $\mathbb R^d$
or in the torus $\TT^d$. Then, what is the largest possible value
of the integral of positive definite functions that are supported
in $\Omega$ and normalized with the value $1$ at the origin? From
this, Arestov, Berdysheva and Berens arrived at the analogous
pointwise extremal problem for intervals in $\RR$. That is, under
the same conditions and normalizations, the supremum of possible
function values at $z$ is to be found for any given point
$z\in\Omega$. However, it turns out that the problem for the real
line has already been solved by Boas and Kac, who gave several
proofs and also mentioned possible extensions to $\RR^d$ and to
nonconvex domains as well.
Here we present another approach to the problem, giving the
solution in $\RR^d$ and for several cases in~$\TT^d$. Actually, we
elaborate on the fact that the problem is essentially
onedimensional and investigate nonconvex open domains as well.
We show that the extremal problems are equivalent to some more
familiar ones concerning trigonometric polynomials, and thus find
the extremal values for a few cases. An analysis of the
relationship between the problem for $\RR^d$ and that for $\TT^d$
is given, showing that the former case is just the limiting case
of the latter. Thus the hierarchy of difficulty is established, so
that extremal problems for trigonometric polynomials gain renewed
recognition.
Keywords:Fourier transform, positive definite functions and measures, TurÃ¡n's extremal problem, convex symmetric domains, positive trigonometric polynomials, dual extremal problems Categories:42B10, 26D15, 42A82, 42A05 

41. CJM 2006 (vol 58 pp. 154)
42. CJM 2004 (vol 56 pp. 655)
 Tao, Xiangxing; Wang, Henggeng

On the Neumann Problem for the SchrÃ¶dinger Equations with Singular Potentials in Lipschitz Domains
We consider the Neumann problem for the Schr\"odinger equations $\Delta u+Vu=0$,
with singular nonnegative potentials $V$ belonging to the reverse H\"older class
$\B_n$, in a connected Lipschitz domain $\Omega\subset\mathbf{R}^n$. Given
boundary data $g$ in $H^p$ or $L^p$ for $1\epsilon
Keywords:Neumann problem, SchrÃ¶dinger equation, Lipschitz, domain, reverse HÃ¶lder class, $H^p$ space Categories:42B20, 35J10 

43. CJM 2004 (vol 56 pp. 431)
 Rosenblatt, Joseph; Taylor, Michael

Group Actions and Singular Martingales II, The Recognition Problem
We continue our investigation in [RST] of a martingale formed by picking a
measurable set $A$ in a compact group $G$, taking random rotates of $A$, and
considering measures of the resulting intersections, suitably normalized. Here
we concentrate on the inverse problem of recognizing $A$ from a small amount of
data from this martingale. This leads to problems in harmonic analysis on $G$,
including an analysis of integrals of products of Gegenbauer polynomials.
Categories:43A77, 60B15, 60G42, 42C10 

44. CJM 2003 (vol 55 pp. 1134)
 Casarino, Valentina

Norms of Complex Harmonic Projection Operators
In this paper we estimate the $(L^pL^2)$norm of the complex
harmonic projectors $\pi_{\ell\ell'}$, $1\le p\le 2$, uniformly
with respect to the indexes $\ell,\ell'$. We provide sharp
estimates both for the projectors $\pi_{\ell\ell'}$, when
$\ell,\ell'$ belong to a proper angular sector in $\mathbb{N}
\times \mathbb{N}$, and for the projectors $\pi_{\ell 0}$ and
$\pi_{0 \ell}$. The proof is based on an extension of a complex
interpolation argument by C.~Sogge. In the appendix, we prove in a
direct way the uniform boundedness of a particular zonal kernel in
the $L^1$ norm on the unit sphere of $\mathbb{R}^{2n}$.
Categories:43A85, 33C55, 42B15 

45. CJM 2003 (vol 55 pp. 1019)
 Handelman, David

More Eventual Positivity for Analytic Functions
Eventual positivity problems for real convergent Maclaurin series lead
to density questions for sets of harmonic functions. These are solved
for large classes of series, and in so doing, asymptotic estimates are
obtained for the values of the series near the radius of convergence
and for the coefficients of convolution powers.
Categories:30B10, 30D15, 30C50, 13A99, 41A58, 42A16 

46. CJM 2003 (vol 55 pp. 576)
 Lukashov, A. L.; Peherstorfer, F.

Automorphic Orthogonal and Extremal Polynomials
It is well known that many polynomials which solve extremal problems
on a single interval as the Chebyshev or the BernsteinSzeg\"o
polynomials can be represented by trigonometric functions and their
inverses. On two intervals one has elliptic instead of trigonometric
functions. In this paper we show that the counterparts of the Chebyshev
and BernsteinSzeg\"o polynomials for several intervals can be represented
with the help of automorphic functions, socalled SchottkyBurnside
functions. Based on this representation and using the SchottkyBurnside
automorphic functions as a tool several extremal properties of such
polynomials as orthogonality properties, extremal properties with
respect to the maximum norm, behaviour of zeros and recurrence
coefficients {\it etc.} are derived.
Categories:42C05, 30F35, 31A15, 41A21, 41A50 

47. CJM 2003 (vol 55 pp. 504)
 Chen, Jiecheng; Fan, Dashan; Ying, Yiming

Certain Operators with Rough Singular Kernels
We study the singular integral operator
$$
T_{\Omega,\alpha}f(x) = \pv \int_{R^n} b(y) \Omega(y')
y^{n\alpha} f(xy)\,dy,
$$
defined on all test functions $f$,where $b$ is a bounded function, $\alpha\geq 0$,
$\Omega(y')$ is an integrable function on the unit sphere $S^{n1}$ satisfying
certain cancellation conditions. We prove that, for $1
Categories:42B20, 42B25, 42B15 

48. CJM 2002 (vol 54 pp. 1165)
 Blasco, Oscar; Arregui, José Luis

Multipliers on Vector Valued Bergman Spaces
Let $X$ be a complex Banach space and let $B_p(X)$ denote the
vectorvalued Bergman space on the unit disc for $1\le p<\infty$. A
sequence $(T_n)_n$ of bounded operators between two Banach spaces $X$
and $Y$ defines a multiplier between $B_p(X)$ and $B_q(Y)$
(resp.\ $B_p(X)$ and $\ell_q(Y)$) if for any function $f(z) =
\sum_{n=0}^\infty x_n z^n$ in $B_p(X)$ we have that $g(z) =
\sum_{n=0}^\infty T_n (x_n) z^n$ belongs to $B_q(Y)$ (resp.\
$\bigl( T_n (x_n) \bigr)_n \in \ell_q(Y)$). Several results on these
multipliers are obtained, some of them depending upon the Fourier or
Rademacher type of the spaces $X$ and $Y$. New properties defined by
the vectorvalued version of certain inequalities for Taylor
coefficients of functions in $B_p(X)$ are introduced.
Categories:42A45, 46E40 

49. CJM 2002 (vol 54 pp. 634)
 Weber, Eric

Frames and Single Wavelets for Unitary Groups
We consider a unitary representation of a discrete countable abelian
group on a separable Hilbert space which is associated to a cyclic
generalized frame multiresolution analysis. We extend Robertson's
theorem to apply to frames generated by the action of the group.
Within this setup we use Stone's theorem and the theory of projection
valued measures to analyze wandering frame collections. This yields a
functional analytic method of constructing a wavelet from a
generalized frame multi\resolution analysis in terms of the frame
scaling vectors. We then explicitly apply our results to the action
of the integers given by translations on $L^2({\mathbb R})$.
Keywords:wavelet, multiresolution analysis, unitary group representation, frame Categories:42C40, 43A25, 42C15, 46N99 

50. CJM 2001 (vol 53 pp. 1031)
 Sampson, G.; Szeptycki, P.

The Complete $(L^p,L^p)$ Mapping Properties of Some Oscillatory Integrals in Several Dimensions
We prove that the operators $\int_{\mathbb{R}_+^2} e^{ix^a \cdot
y^b} \varphi (x,y) f(y)\, dy$ map $L^p(\mathbb{R}^2)$ into itself
for $p \in J =\bigl[\frac{a_l+b_l}{a_l+(\frac{b_l r}{2})},\frac{a_l+b_l}
{a_l(1\frac{r}{2})}\bigr]$ if $a_l,b_l\ge 1$ and $\varphi(x,y)=xy^{r}$,
$0\le r <2$, the result is sharp. Generalizations to dimensions $d>2$
are indicated.
Categories:42B20, 46B70, 47G10 
