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26. CMB 2011 (vol 55 pp. 708)

Demeter, Ciprian
Improved Range in the Return Times Theorem
We prove that the Return Times Theorem holds true for pairs of $L^p-L^q$ functions, whenever $\frac{1}{p}+\frac{1}{q}<\frac{3}{2}$.

Keywords:Return Times Theorem, maximal multiplier, maximal inequality
Categories:42B25, 37A45

27. 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

28. CMB 2011 (vol 55 pp. 303)

Han, Yongsheng; Lee, Ming-Yi; Lin, Chin-Cheng
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, Calderón reproducing formula, weighted Hardy spaces
Categories:42B25, 42B30

29. 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, Dirichlet-Jordan theorem
Categories:42A38, 65T99

30. CMB 2010 (vol 54 pp. 113)

Hytönen, Tuomas P.
On the Norm of the Beurling-Ahlfors Operator in Several Dimensions
The generalized Beurling-Ahlfors 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

31. 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 Hardy-type inequalities. This discussion will lead to many questions concerning the relationship between Hardy-type 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

32. 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 Triebel--Lizorkin spaces is established.

Keywords: Marcinkiewicz integral, Littlewood--Paley theory, Triebel--Lizorkin space, rough kernel, product domain
Categories:42B20, , , , , 42B25, 42B30, 42B99

33. CMB 2010 (vol 54 pp. 172)

Shayya, Bassam
Measures with Fourier Transforms in $L^2$ of a Half-space
We prove that if the Fourier transform of a compactly supported measure is in $L^2$ of a half-space, 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

34. 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 Hardy--Littlewood 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

35. CMB 2009 (vol 53 pp. 133)

Moritoh, Shinya; Tomoeda, Kyoko
A Further Decay Estimate for the Dziubański-Hernández Wavelets
We give a further decay estimate for the Dziubański-Hernández wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley-Wiener theorem for ultradifferentiable functions.

Keywords:wavelets, ultradifferentiable functions
Categories:42C40, 46E10

36. 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 Hardy–Littlewood 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

37. 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 real-valued 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 f-S_{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 so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued 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

38. CMB 2009 (vol 52 pp. 521)

Chen, Yanping; Ding, Yong
The Parabolic Littlewood--Paley Operator with Hardy Space Kernels
In this paper, we give the $L^p$ boundedness for a class of parabolic Littlewood--Paley $g$-function with its kernel function $\Omega$ is in the Hardy space $H^1(S^{n-1})$.

Keywords:parabolic Littlewood-Paley operator, Hardy space, rough kernel
Categories:42B20, 42B25

39. 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 Christoffel--Darboux formulas are presented for the first time.

Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, Christoffel-Darboux

40. CMB 2008 (vol 51 pp. 487)

Betancor, Jorge J.; Mart\'{\i}nez, Teresa; Rodr\'{\i}guez-Mesa, Lourdes
Laplace Transform Type Multipliers for Hankel Transforms
In this paper we establish that Hankel multipliers of Laplace transform type are bounded from $L^p(w)$ into itself when $1
Keywords:Hankel transform, Laplace transform, multiplier, Calderón--Zygmund

41. CMB 2008 (vol 51 pp. 348)

Casazza, Peter G.; Christensen, Ole
The Reconstruction Property in Banach Spaces and a Perturbation Theorem
Perturbation theory is a fundamental tool in Banach space theory. However, the applications of the classical results are limited by the fact that they force the perturbed sequence to be equivalent to the given sequence. We will develop a more general perturbation theory that does not force equivalence of the sequences.


42. CMB 2007 (vol 50 pp. 85)

Han, Deguang
Classification of Finite Group-Frames and Super-Frames
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, group-frames, frame representations, disjoint frames
Categories:42C15, 46C05, 47B10

43. CMB 2006 (vol 49 pp. 414)

Jiang, Liya; Jia, Houyu; Xu, Han
Commutators Estimates on Triebel--Lizorkin Spaces
In this paper, we consider the behavior of the commutators of convolution operators on the Triebel--Lizorkin spaces $\dot{F}^{s, q} _p$.

Keywords:commutators, Triebel--Lizorkin spaces, paraproduct
Categories:42B, 46F

44. CMB 2006 (vol 49 pp. 438)

Mercer, Idris David
Unimodular Roots of\\ Special Littlewood Polynomials
We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1}$ a Littlewood polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ self-reciprocal if $\alpha(z) = z^{n-1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if $n = 2m+1$ and $a_{m+j} = (-1)^j a_{m-j}$ 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 self-reciprocal Littlewood polynomial must have a zero on the unit circle.

Categories:26C10, 30C15, 42A05

45. CMB 2006 (vol 49 pp. 3)

Al-Salman, 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}n--Zygmund 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

46. 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 $$ |P_n^{(s)}(x)|\leq P_n^{(s)}(1)\bigl(|x|^n +\frac{n-1}{2 s+1}(1-|x|^n)\bigr),$$ 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

47. 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\"ormander--Mihlin 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

48. CMB 2005 (vol 48 pp. 260)

Oberlin, Daniel M.
A Restriction Theorem for a \\$k$-Surface in $\mathbb R ^n$
We establish a sharp Fourier restriction estimate for a measure on a $k$-surface in $\mathbb R ^n$, where $n=k(k+3)/2$.

Keywords:Fourier restriction

49. 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

50. CMB 2004 (vol 47 pp. 3)

Al-Salman, Ahmad; Pan, Yibiao
Singular Integrals With Rough Kernels
In this paper we establish the $L^p$ boundedness of a class of singular integrals with rough kernels associated to polynomial mappings.

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