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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 Askey--Wilson 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, Askey--Wilson operator, $q$-difference operator, recursion coefficientsCategories:33D45, 42C05

27. CJM 2010 (vol 62 pp. 1419)

Yang, Dachun; Yang, Dongyong
 BMO-Estimates for Maximal Operators via Approximations of the Identity with Non-Doubling 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:Non-doubling 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 John-Nirenberg 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 John--Nirenberg inequality for $Q_{\alpha}({\mathbb R^n})$. When $\beta\neq 1$, $f$ is a self-affine 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 $[-\frac{1}{1-2^{-\beta}}, \frac{1}{1-2^{-\beta}}]$, and the graph of $f$ has a non-integer fractal dimension $n+1-\beta$. Keywords:Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimensionCategories: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 vector-valued 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 $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ 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 $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ 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 spacesCategories:46E40, 42B25, 46B20

30. CJM 2009 (vol 61 pp. 807)

Hong, Sunggeum; Kim, Joonil; Yang, Chan Woo
 Maximal Operators Associated with Vector Polynomials of Lacunary Coefficients We prove the $L^p(\mathbb{R}^d)$ ($1 Categories:42B20, 42B25 31. CJM 2009 (vol 61 pp. 141) Green, Ben; Konyagin, Sergei  On the Littlewood Problem Modulo a Prime Let$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$'' analogue of Littlewood's well-known problem concerning the smallest possible$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, Kwok-Pun  Remarks on Littlewood--Paley Analysis Littlewood--Paley 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 Littlewood--Paley decomposition of tempered distributions converges under a topology stronger than the weak-star topology, namely, the inductive limit topology. Finally, we construct a multiparameter Littlewood--Paley analysis and obtain the corresponding renormalization'' for the convergence of this multiparameter Littlewood--Paley analysis. Keywords:Littlewood--Paley analysis, distributionsCategory:42B25 33. CJM 2008 (vol 60 pp. 685) Savu, Anamaria  Closed and Exact Functions in the Context of Ginzburg--Landau 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 second-order Ginzburg--Landau field and for the case of the fourth-order Ginzburg--Landau field. Keywords:Hermite polynomials, Fock space, Fourier coefficients, Fourier transform, group of symmetriesCategories:42B05, 81Q50, 42A16 34. CJM 2008 (vol 60 pp. 334) Curry, Eva  Low-Pass 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 low-pass filters given by Gundy both hold for multivariable multiresolution analyses. Keywords:multivariable multiresolution analysis, low-pass filter, scaling functionCategories:42C40, 60G35 35. CJM 2007 (vol 59 pp. 1223) Buraczewski, Dariusz; Martinez, Teresa; Torrea, José L.  CalderÃ³n--Zygmund Operators Associated to Ultraspherical Expansions We define the higher order Riesz transforms and the Littlewood--Paley$g$-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}n--Zygmund operators in the homogeneous type space$((0,\pi),(\sin t)^{2\l}\,dt)$. Consequently,$L^p$weighted,$H^1-L^1$and$L^\infty-BMO$inequalities are obtained. Keywords:ultraspherical polynomials, CalderÃ³n--Zygmund operatorsCategories: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 boundednessCategories:42B30, 47D06 37. CJM 2007 (vol 59 pp. 276) Bernardis, A. L.; Martín-Reyes, F. J.; Salvador, P. Ortega  Weighted Inequalities for Hardy--Steklov 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:Hardy--Steklov operator, weights, inequalitiesCategories: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 sup-adjoint 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 conjectureCategories:42B25, 42B35, 42C40

39. CJM 2006 (vol 58 pp. 548)

Galanopoulos, P.; Papadimitrakis, M.
 Hausdorff and Quasi-Hausdorff Matrices on Spaces of Analytic Functions We consider Hausdorff and quasi-Hausdorff 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 non-convex 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 one-dimensional and investigate non-convex 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 problemsCategories:42B10, 26D15, 42A82, 42A05

41. CJM 2006 (vol 58 pp. 154)

Prestini, Elena
 Singular Integrals on Product Spaces Related to the Carleson Operator We prove $L^p(\mathbb T^2)$ boundedness, $1 Categories:42B20, 42B08 42. CJM 2004 (vol 56 pp. 655) Tao, Xiangxing; Wang, Henggeng  On the Neumann Problem for the SchrÃ¶dinger Equations with Singular Potentials in Lipschitz Domains We consider the Neumann problem for the Schr\"odinger equations$-\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, $H^p$ spaceCategories: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 $(L^p-L^2)$-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 Bernstein-Szeg\"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 Bernstein-Szeg\"o polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside 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 $$T_{\Omega,\alpha}f(x) = \pv \int_{R^n} b(|y|) \Omega(y') |y|^{-n-\alpha} f(x-y)\,dy,$$ 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 $S^{n-1}$ 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 vector-valued 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 vector-valued 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, frameCategories: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$\varphi(x,y)=|x-y|^{-r}$,$0\le r <2$, the result is sharp. Generalizations to dimensions$d>2\$ are indicated. Categories:42B20, 46B70, 47G10
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