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Search: MSC category 42 ( Fourier analysis )

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1. CMB Online first

Totik, Vilmos
Universality under Szegő's condition
This paper presents a theorem on universality on orthogonal polynomials/random matrices under a weak local condition on the weight function $w$. With a new inequality for polynomials and with the use of fast decreasing polynomials, it is shown that an approach of D. S. Lubinsky is applicable. The proof works at all points which are Lebesgue-points both for the weight function $w$ and for $\log w$.

Keywords:universality, random matrices, Christoffel functions, asymptotics, potential theory
Categories:42C05, 60B20, 30C85, 31A15

2. CMB Online first

Zaatra, Mohamed
Generating some symmetric semi-classical orthogonal polynomials
We show that if $v$ is a regular semi-classical form (linear functional), then the symmetric form $u$ defined by the relation $x^{2}\sigma u = -\lambda v$, where $(\sigma f)(x)=f(x^{2})$ and the odd moments of $u$ are $0$, is also regular and semi-classical form for every complex $\lambda $ except for a discrete set of numbers depending on $v$. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with $u$ and the class of the form $u$ knowing that of $v$. We conclude with an illustrative example.

Keywords:orthogonal polynomials, quadratic decomposition, semi-classical forms, structure relation
Categories:33C45, 42C05

3. CMB Online first

He, Ziyi; Yang, Dachun; Yuan, Wen
Littlewood-Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls
In this paper, the authors characterize second-order Sobolev spaces $W^{2,p}({\mathbb R}^n)$, with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and $n\in\{1,2,3\}$, via the Lusin area function and the Littlewood-Paley $g_\lambda^\ast$-function in terms of ball means.

Keywords:Sobolev space, ball means, Lusin-area function, $g_\lambda^*$-function
Categories:46E35, 42B25, 42B20, 42B35

4. CMB Online first

Han, Yanchang
Embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces
In this article we prove the embedding theorem for inhomogeneous Besov and Triebel-Lizorkin spaces on RD-spaces. The crucial idea is to use the geometric density condition on the measure.

Keywords:spaces of homogeneous type, test function space, distributions, Calderón reproducing formula, Besov and Triebel-Lizorkin spaces, embedding
Categories:42B25, 46F05, 46E35

5. CMB Online first

Hsu, Ming-Hsiu; Lee, Ming-Yi
VMO space associated with parabolic sections and its application
In this paper we define $VMO_\mathcal{P}$ space associated with a family $\mathcal{P}$ of parabolic sections and show that the dual of $VMO_\mathcal{P}$ is the Hardy space $H^1_\mathcal{P}$. As an application, we prove that almost everywhere convergence of a bounded sequence in $H^1_\mathcal{P}$ implies weak* convergence.

Keywords:Monge-Ampere equation, parabolic section, Hardy space, BMO, VMO
Category:42B30

6. CMB Online first

Liu, Feng; Wu, Huoxiong
On the Regularity of the Multisublinear Maximal Functions
This paper is concerned with the study of the regularity for the multisublinear maximal operator. It is proved that the multisublinear maximal operator is bounded on first-order Sobolev spaces. Moreover, two key point-wise inequalities for the partial derivatives of the multisublinear maximal functions are established. As an application, the quasi-continuity on the multisublinear maximal function is also obtained.

Keywords:regularity, multisublinear maximal operator, Sobolev spaces, partial deviative, quasicontinuity
Categories:42B25, 46E35

7. CMB 2014 (vol 58 pp. 432)

Yang, Dachun; Yang, Sibei
Second-order Riesz Transforms and Maximal Inequalities Associated with Magnetic Schrödinger Operators
Let $A:=-(\nabla-i\vec{a})\cdot(\nabla-i\vec{a})+V$ be a magnetic Schrödinger operator on $\mathbb{R}^n$, where $\vec{a}:=(a_1,\dots, a_n)\in L^2_{\mathrm{loc}}(\mathbb{R}^n,\mathbb{R}^n)$ and $0\le V\in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ satisfy some reverse Hölder conditions. Let $\varphi\colon \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ for any given $x\in\mathbb{R}^n$ is an Orlicz function, $\varphi(\cdot,t)\in {\mathbb A}_{\infty}(\mathbb{R}^n)$ for all $t\in (0,\infty)$ (the class of uniformly Muckenhoupt weights) and its uniformly critical upper type index $I(\varphi)\in(0,1]$. In this article, the authors prove that second-order Riesz transforms $VA^{-1}$ and $(\nabla-i\vec{a})^2A^{-1}$ are bounded from the Musielak-Orlicz-Hardy space $H_{\varphi,\,A}(\mathbb{R}^n)$, associated with $A$, to the Musielak-Orlicz space $L^{\varphi}(\mathbb{R}^n)$. Moreover, the authors establish the boundedness of $VA^{-1}$ on $H_{\varphi, A}(\mathbb{R}^n)$. As applications, some maximal inequalities associated with $A$ in the scale of $H_{\varphi, A}(\mathbb{R}^n)$ are obtained.

Keywords:Musielak-Orlicz-Hardy space, magnetic Schrödinger operator, atom, second-order Riesz transform, maximal inequality
Categories:42B30, 42B35, 42B25, 35J10, 42B37, 46E30

8. CMB 2014 (vol 58 pp. 19)

Chen, Jiecheng; Hu, Guoen
Compact Commutators of Rough Singular Integral Operators
Let $b\in \mathrm{BMO}(\mathbb{R}^n)$ and $T_{\Omega}$ be the singular integral operator with kernel $\frac{\Omega(x)}{|x|^n}$, where $\Omega$ is homogeneous of degree zero, integrable and has mean value zero on the unit sphere $S^{n-1}$. In this paper, by Fourier transform estimates and approximation to the operator $T_{\Omega}$ by integral operators with smooth kernels, it is proved that if $b\in \mathrm{CMO}(\mathbb{R}^n)$ and $\Omega$ satisfies a certain minimal size condition, then the commutator generated by $b$ and $T_{\Omega}$ is a compact operator on $L^p(\mathbb{R}^n)$ for appropriate index $p$. The associated maximal operator is also considered.

Keywords:commutator,singular integral operator, compact operator, maximal operator
Category:42B20

9. CMB 2014 (vol 58 pp. 144)

Olevskii, Victor
Localization and Completeness in $L_2({\mathbb R})$
We give a necessary and sufficient condition for a sequence to be a localization set for a determining average sampler.

Keywords:localization, completeness, average sampling
Categories:42C30, 94A20

10. CMB 2014 (vol 57 pp. 834)

Koh, Doowon
Restriction Operators Acting on Radial Functions on Vector Spaces Over Finite Fields
We study $L^p-L^r$ restriction estimates for algebraic varieties $V$ in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties $V$ lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties $V$ are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

Keywords:finite fields, radial functions, restriction operators
Categories:42B05, 43A32, 43A15

11. CMB 2013 (vol 57 pp. 463)

Bownik, Marcin; Jasper, John
Constructive Proof of Carpenter's Theorem
We give a constructive proof of Carpenter's Theorem due to Kadison. Unlike the original proof our approach also yields the real case of this theorem.

Keywords:diagonals of projections, the Schur-Horn theorem, the Pythagorean theorem, the Carpenter theorem, spectral theory
Categories:42C15, 47B15, 46C05

12. CMB 2013 (vol 57 pp. 254)

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 principle
Categories:42C15, 42C40

13. CMB 2013 (vol 56 pp. 729)

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, shearlet
Categories:43A65, 42C40, 42C15

14. CMB 2013 (vol 56 pp. 745)

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 function
Category:42C40

15. CMB 2012 (vol 56 pp. 801)

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-zygmund
Category:42A45

16. 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 restriction
Categories:42B10, 28A12

17. 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 limits
Categories:58G25, 81Q50, 35P20, 42B05

18. 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 Hörmander-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

19. CMB 2011 (vol 56 pp. 194)

Stefánsson, Úlfar F.
On the Smallest and Largest Zeros of Müntz-Legendre Polynomials
Müntz-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 Müntz 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:Müntz polynomials, Müntz-Legendre polynomials
Categories:42C05, 42C99, 41A60, 30B50

20. 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 Calderón-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 estimates
Categories:42B20, 42B25, 35S05, 47G30

21. 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

22. 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

23. 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

24. 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

25. 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
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