Search results

Search: MSC category 42 ( Fourier analysis )

 Expand all        Collapse all Results 1 - 25 of 54

1. CMB Online first

Christensen, Ole; Kim, Hong Oh; Kim, Rae Young
 On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle The unitary extension principle (UEP) by Ron and Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators, and prove that this condition is also sufficient to ensure that an extension with just two generators is possible. Keywords:Bessel sequences, frames, extension of wavelet Bessel system to tight frame, wavelet systems, unitary extension principleCategories:42C15, 42C40

2. CMB Online first

Currey, B.; Mayeli, A.
 The Orthonormal Dilation Property for Abstract Parseval Wavelet Frames In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as $\pi(\Gamma)\psi$, where $\pi$ is a unitary representation of a wavelet group and $\Gamma$ is the abstract pseudo-lattice $\Gamma$. We prove a condition in order that a Parseval frame $\pi(\Gamma)\psi$ can be dilated to an orthonormal basis of the form $\tau(\Gamma)\Psi$ where $\tau$ is a super-representation of $\pi$. For a subclass of groups that includes the case where the translation subgroup is Heisenberg, we show that this condition always holds, and we cite familiar examples as applications. Keywords:frame, dilation, wavelet, Baumslag-Solitar group, shearletCategories:43A65, 42C40, 42C15

3. CMB Online first

Fu, Xiaoye; Gabardo, Jean-Pierre
 Dimension Functions of Self-Affine Scaling Sets In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, and $d_1,d_2\in\mathbb{R}^n$. We show that the dimension function of $K$ must be constant if either $n=1$ or $2$ or one of the digits is $0$, and that it is bounded by $2\lvert K\rvert$ for any $n$. Keywords:scaling set, self-affine tile, orthonormal multiwavelet, dimension functionCategory:42C40

4. CMB Online first

Oberlin, Richard
 Estimates for Compositions of Maximal Operators with Singular Integrals We prove weak-type $(1,1)$ estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $\Delta^*\Psi$ where $\Delta^*$ is Bourgain's maximal multiplier operator and $\Psi$ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the $L^q$ operator norm when $1 \lt q \lt 2.$ We also consider associated variation-norm estimates. Keywords:maximal operator calderon-zygmundCategory:42A45

5. CMB 2011 (vol 56 pp. 326)

Erdoğan, M. Burak; Oberlin, Daniel M.
 Restricting Fourier Transforms of Measures to Curves in $\mathbb R^2$ We establish estimates for restrictions to certain curves in $\mathbb R^2$ of the Fourier transforms of some fractal measures. Keywords:Fourier transforms of fractal measures, Fourier restrictionCategories:42B10, 28A12

6. CMB 2011 (vol 56 pp. 3)

Aïssiou, Tayeb
 Semiclassical Limits of Eigenfunctions on Flat $n$-Dimensional Tori We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an $n$-dimensional flat torus $\mathbb T^{n}$, and the Fourier transform of squares of the eigenfunctions $|\varphi_\lambda|^2$ of the Laplacian have uniform $l^n$ bounds that do not depend on the eigenvalue $\lambda$. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on $\mathbb T^{n+2}$. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an $n$-dimensional sphere $S^n(\lambda)$ of radius $\sqrt{\lambda}$, and we use it in the proof. Keywords:semiclassical limits, eigenfunctions of Laplacian on a torus, quantum limitsCategories:58G25, 81Q50, 35P20, 42B05

7. CMB 2011 (vol 55 pp. 646)

Zhou, Jiang; Ma, Bolin
 Marcinkiewicz Commutators with Lipschitz Functions in Non-homogeneous 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¶rmander-type 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

8. CMB 2011 (vol 56 pp. 194)

Stefánsson, Úlfar F.
 On the Smallest and Largest Zeros of MÃ¼ntz-Legendre Polynomials MÃ¼ntz-Legendre 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 $l_{n,n}\lt l_{n-1,n}\lt \cdots \lt l_{2,n}\lt l_{1,n}$ 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 non-dense 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, MÃ¼ntz-Legendre polynomialsCategories:42C05, 42C99, 41A60, 30B50

9. 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 CalderÃ³n-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood 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 estimatesCategories:42B20, 42B25, 35S05, 47G30

10. 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 inequalityCategories:42B25, 37A45

11. 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 convergenceCategories:39B12, 41A25, 42C40 12. 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, CalderÃ³n reproducing formula, weighted Hardy spacesCategories:42B25, 42B30 13. 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 theoremCategories:42A38, 65T99 14. 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 15. 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 spacesCategories:42A05, 42A99 16. 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 domainCategories:42B20, , , , , 42B25, 42B30, 42B99 17. 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 18. 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 kernelCategories:42B25, 43A62 19. 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 }$of HardyÂLittlewood and the fractional integral$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)$. The conditions on the weights are stated in terms of Orlicz norm. Keywords:fractional maximal operator, fractional integral, space of homogeneous typeCategories:42B35, 42B20, 42B25 20. CMB 2009 (vol 53 pp. 133) Moritoh, Shinya; Tomoeda, Kyoko  A Further Decay Estimate for the DziubaÅski-HernÃ¡ndez Wavelets We give a further decay estimate for the DziubaÅski-HernÃ¡ndez wavelets that are band-limited and have subexponential decay. This is done by constructing an appropriate bell function and using the Paley-Wiener theorem for ultradifferentiable functions. Keywords:wavelets, ultradifferentiable functionsCategories:42C40, 46E10 21. CMB 2009 (vol 52 pp. 627) Yu, Dan Sheng; Zhou, Ping; Zhou, Song Ping  On$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}$convergence, monotonicity, mean value bounded variationCategories:42A25, 41A50 22. CMB 2009 (vol 52 pp. 521) Chen, Yanping; Ding, Yong  The Parabolic Littlewood--Paley Operator with Hardy Space Kernels In this paper, we give the$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})$. Keywords:parabolic Littlewood-Paley operator, Hardy space, rough kernelCategories:42B20, 42B25 23. CMB 2009 (vol 52 pp. 95) Miranian, L.  Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory In the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel--Darboux formulas are presented for the first time. Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, Christoffel-DarbouxCategory:42C99 24. CMB 2008 (vol 51 pp. 487) Betancor, Jorge J.; Mart\'{\i}nez, Teresa; Rodr\'{\i}guez-Mesa, Lourdes  Laplace Transform Type Multipliers for Hankel Transforms In this paper we establish that Hankel multipliers of Laplace transform type are bounded from$L^p(w)$into itself when$1 Keywords:Hankel transform, Laplace transform, multiplier, CalderÃ³n--ZygmundCategory:42

25. 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. Category:42C15
 Page 1 2 3 Next