Showing papers in "Journal of Fourier Analysis and Applications in 2014"

Stable optimizationless recovery from phaseless linear measurements

Abstract: We address the problem of recovering an n-vector from m linear measurements lacking sign or phase information. We show that lifting and semidefinite relaxation suffice by themselves for stable recovery in the setting of m=O(nlogn) random sensing vectors, with high probability. The recovery method is optimizationless in the sense that trace minimization in the PhaseLift procedure is unnecessary. That is, PhaseLift reduces to a feasibility problem. The optimizationless perspective allows for a Douglas-Rachford numerical algorithm that is unavailable for PhaseLift. This method exhibits linear convergence with a favorable convergence rate and without any parameter tuning.

Hörmander Class of Pseudo-Differential Operators on Compact Lie Groups and Global Hypoellipticity

Abstract: In this paper we give several global characterisations of the Hormander class $$\Psi ^m(G)$$ of pseudo-differential operators on compact Lie groups in terms of the representation theory of the group. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given, in particular of operators that are locally not invertible nor hypoelliptic but globally are. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.

The Mystery of Gabor Frames

Abstract: This is a survey about the theory of Gabor frames. We review the structural results about Gabor frames over a lattice and then discuss the few known results about the fine structure of Gabor frames. We add a new result about the relation between properties of the window and properties of the frame set and conclude with a vision of how a more complete theory of the fine structure might look like.

The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

Abstract: We investigate the weighted bounds for multilinear maximal functions and Calderon–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1

On Sharp Aperture-Weighted Estimates for Square Functions

Abstract: Let Sα,ψ ( f ) be the square function defined by means of the cone in R n+1 + of aperture α, and a standard kernel ψ .L et(w)A p denote the A p characteristic of the weight w. We show that for any 1 < p < ∞ and α ≥ 1,

Phaseless Signal Recovery in Infinite Dimensional Spaces Using Structured Modulations

Abstract: This paper considers the recovery of continuous signals in infinite dimensional spaces from the magnitude of their frequency samples. It proposes a sampling scheme which involves a combination of oversampling and modulations with complex exponentials. Sufficient conditions are given such that almost every signal with compact support can be reconstructed up to a unimodular constant using only its magnitude samples in the frequency domain. Finally it is shown that an average sampling rate of four times the Nyquist rate is enough to reconstruct almost every time-limited signal.

Characterization of Shift-Invariant Spaces on a Class of Nilpotent Lie Groups with Applications

Abstract: Given a simply connected nilpotent Lie group having unitary irreducible representations that are square-integrable modulo the center, we use operator-valued periodization to give a range-function type characterization of shift-invariant spaces of function on the group We then give characterizations of frame and Riesz families for shift-invariant spaces

Step Refinable Functions and Orthogonal MRA on Vilenkin Groups

Abstract: We find necessary and sufficient conditions on refinable step function under which this function generates an orthogonal MRA in the $L_{2}(\mathfrak{G})$ -spaces on Vilenkin group $\mathfrak{G}$ . We consider a class of refinable step functions for which the mask m 0(χ) is constant on cosets $\mathfrak{G}_{-1}^{\bot}\chi$ and its modulus |m 0(χ)| has two values only: 0 and 1. We prove that any refinable step function φ from this class that generates an orthogonal MRA on Vilenkin group $\mathfrak{G}$ has Fourier transform with condition $\operatorname{supp}\hat{\varphi}(\chi)\subset\mathfrak{G}_{p-2}^{\bot}$ . We show the sharpness of this result, too.

Gaussian Cubature Arising from Hybrid Characters of Simple Lie Groups

Abstract: Lie groups with two different root lengths allow two ‘mixed sign’ homomorphisms on their corresponding Weyl groups, which in turn give rise to two families of hybrid Weyl group orbit functions and characters. In this paper we extend the ideas leading to the Gaussian cubature formulas for families of polynomials arising from the characters of irreducible representations of any simple Lie group, to new cubature formulas based on the corresponding hybrid characters. These formulas are new forms of Gaussian cubature in the short root length case and new forms of Radau cubature in the long root case. The nodes for the cubature arise quite naturally from the (computationally efficient) elements of finite order of the Lie group.

Norm Inequalities in Generalized Morrey Spaces

Abstract: We prove that Calderon-Zygmund operators, Marcinkiewicz operators, maximal operators associated to Bochner-Riesz operators, operators with rough kernel as well as commutators associated to these operators which are known to be bounded on weighted Morrey spaces under appropriate conditions, are bounded on a wide family of function spaces.

On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling

Abstract: The characteristic functional is the infinite-dimensional generalization of the Fourier transform for measures on function spaces. It characterizes the statistical law of the associated stochastic process in the same way as a characteristic function specifies the probability distribution of its corresponding random variable. Our goal in this work is to lay the foundations of the innovation model, a (possibly) non-Gaussian probabilistic model for sparse signals. This is achieved by using the characteristic functional to specify sparse stochastic processes that are defined as linear transformations of general continuous-domain white Levy noises (also called innovation processes). We prove the existence of a broad class of sparse processes by using the Minlos–Bochner theorem. This requires a careful study of the regularity properties, especially the $$L^p$$ -boundedness, of the characteristic functional of the innovations. We are especially interested in the functionals that are only defined for $$p<1$$ since they appear to be associated with the sparser kind of processes. Finally, we apply our main theorem of existence to two specific subclasses of processes with specific invariance properties.

A consistent and stable approach to generalized sampling

Abstract: We consider the problem of generalized sampling, in which one seeks to obtain reconstructions in arbitrary finite dimensional spaces from a finite number of samples taken with respect to an arbitrary orthonormal basis. Typical approaches to this problem consider solutions obtained via the consistent reconstruction technique or as solutions of an overcomplete linear systems. However, the consistent reconstruction technique is known to be non-convergent and ill-conditioned in important cases, such as the recovery of wavelet coefficients from Fourier samples, and whilst the latter approach presents solutions which are convergent and well-conditioned when the system is sufficiently overcomplete, the solution becomes inconsistent with the original measurements. In this paper, we consider generalized sampling via a non-linear minimization problem and prove that the minimizers present solutions which are convergent, stable and consistent with the original measurements. We also provide analysis in the case of recovering wavelets coefficients from Fourier samples. We show that for compactly supported wavelets of sufficient smoothness, there is a linear relationship between the number of wavelet coefficients which can be accurately recovered and the number of Fourier samples available.

Hagedorn Wavepackets in Time-Frequency and Phase Space

Abstract: The Hermite functions are an orthonormalbasis of the space of square integrable functions with favourable approximation properties. Allowing for a flexible localization in position and momentum, the Hagedorn wavepackets generalize the Hermite functions also to several dimensions. Using Hagedorn’s raising and lowering operators, we derive explicit formulas and recurrence relations for the Wigner and FBI transform of the wavepackets and show their relation to the Laguerre polyomials.

Ulyanov-type Inequalities Between Lorentz–Zygmund Spaces

Abstract: We establish inequalities of Ulyanov-type for moduli of smoothness relating the source Lorentz–Zygmund space $$\, L^{p,r}(\log L)^{\alpha -\gamma },\, \gamma >0,$$ and the target space $$\, L^{p^*,s}(\log L)^\alpha $$ over $$\, {\mathbb R}^n$$ if $$\, 1

Smoothing of Commutators for a Hörmander Class of Bilinear Pseudodifferential Operators

Abstract: Commutators of bilinear pseudodifferential operators with symbols in the Hormander class $BS_{1, 0}^{1}$ and multiplication by Lipschitz functions are shown to be bilinear Calderon-Zygmund operators. A connection with a notion of compactness in the bilinear setting for the iteration of the commutators is also made.

Square Function Characterization of Weak Hardy Spaces

Abstract: We obtain a new square function characterization of the weak Hardy space $$H^{p,\infty }$$ for all $$p\in (0,\infty )$$ . This space consists of all tempered distributions whose smooth maximal function lies in weak $$L^p$$ . Our proof is based on interpolation between $$H^p$$ spaces. The main difficulty we overcome is the lack of a good dense subspace of $$H^{p,\infty }$$ which forces us to work with general $$H^{p,\infty }$$ distributions.

Rate of Innovation for (Non-)Periodic Signals and Optimal Lower Stability Bound for Filtering

Abstract: One of fundamental problems in sampling theory is to reconstruct (non-)periodic signals from their filtered signals in a stable way. In this paper, we obtain a universal upper bound to the rate of innovation for signals in a closed linear space, which can be stably reconstructed, via the optimal lower stability bound for filtering on that linear space.

Discrete Orthogonality of the Malmquist Takenaka System of the Upper Half Plane and Rational Interpolation

Abstract: In the representation of non periodic signals the use of special rational orthogonal systems is more efficient. One of these bases is the Malmquist–Takenaka system for the upper half plane. We will prove the discrete orthogonality of this system. Based on the discretization we introduce a new rational interpolation operator and we will study the properties of this operator. A finite sampling theorem for a special subset of non periodic analytic signals will be presented.

On Radial Functions and Distributions and Their Fourier Transforms

Abstract: We give formulas relating the Fourier transform of a radial function in \(\mathbb{R}^{n}\) and the Fourier transform of the same function in \(\mathbb{R}^{n+1}\), completing the analysis of Grafakos and Teschl (J. Fourier Anal. Appl. 19:167–179, 2013) where the case of \(\mathbb{R}^{n}\) and \(\mathbb{R}^{n+2}\) was considered.

The Broken Ray Transform on the Square

Abstract: We study a particular broken ray transform on the Euclidean unit square and establish injectivity and stability for $$C_{0}^{2}$$ perturbations of a vanishing absorption parameter $$\sigma \equiv 0$$ Given an open subset $$E$$ of the boundary, we measure the attenuation of all broken rays starting and ending at $$E$$ with the standard optical reflection rule applied to $$\partial \Omega {\setminus } E$$ Using the analytic microlocal approach of Frigyik et al for the X-ray transform on generic families of curves, we show injectivity via a path unfolding argument under suitable conditions on the available broken rays Then we show that with a suitable decomposition of the measurement operator via smooth cutoff functions, the associated normal operator is a classical pseudo differential operator of order $$-1$$ , which leads to the desired result

Growth and Integrability of Fourier Transforms on Euclidean Space

Abstract: A fundamental theme in classical Fourier analysis relates smoothness properties of functions to the growth and/or integrability of their Fourier transform. By using a suitable class of $$L^{p}$$ -multipliers, a rather general inequality controlling the size of Fourier transforms for large and small argument is obtained. As consequences, quantitative Riemann–Lebesgue estimates are obtained and an integrability result for the Fourier transform is developed extending ideas used by Titchmarsh in the one dimensional setting.

Refinable Function-Based Construction of Weak (Quasi-)Affine Bi-Frames

Abstract: Refinable function-based affine frames and affine bi-frames have been extensively studied in the literature. All these works are based on some restrictions on refinable functions. This paper addresses what are expected from two general refinable functions. We introduce the notion of weak (quasi-) affine bi-frame; present a refinable function-based construction of weak (quasi-) affine bi-frames; and obtain a fast algorithm associated with weak affine bi-frames. An example is also given to show that our construction is optimal in some sense.

Difference Sets and Positive Exponential Sums I. General Properties

Abstract: We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set A the maximal size of a set whose difference set avoids A will be related to positive exponential sums using frequencies from A.

Revisiting the Concentration Problem of Vector Fields within a Spherical Cap: A Commuting Differential Operator Solution

Abstract: We propose a novel basis of vector functions, the mixed vector spherical harmonics that are closely related to the functions of Sheppard and Torok and help us reduce the concentration problem of tangential vector fields within a spherical cap to an equivalent scalar problem. Exploiting an analogy with previous results published by Grunbaum, Longhi and Perlstadt, we construct a differential operator that commutes with the concentration operator of this scalar problem and propose a stable and convenient method to obtain its eigenfunctions. Having obtained the scalar eigenfunctions, the calculation of tangential vector Slepian functions is straightforward.

Discrete Hardy Spaces

Abstract: We study the boundary behavior of discrete monogenic functions, i.e. null-solutions of a discrete Dirac operator, in the upper and lower half space. Calculating the Fourier symbol of the boundary operator we construct the corresponding discrete Hilbert transforms, the projection operators arising from them, and discuss the notion of discrete Hardy spaces. Hereby, we focus on the 3D-case with the generalization to the n-dimensional case being straightforward.

\mathcal {R}_{s}-Sectorial Operators and Generalized Triebel-Lizorkin Spaces

Abstract: We introduce a notion of generalized Triebel-Lizorkin spaces associated with sectorial operators in Banach function spaces. Our approach is based on holomorphic functional calculus techniques. Using the concept of \(\mathcal {R}_{s}\)-sectorial operators, which in turn is based on the notion of \(\mathcal {R}_{s}\)-bounded sets of operators introduced by Lutz Weis, we obtain a neat theory including equivalence of various norms and a precise description of real and complex interpolation spaces. Another main result of this article is that an \(\mathcal {R}_{s}\)-sectorial operator always has a bounded H∞-functional calculus in its associated generalized Triebel-Lizorkin spaces.

Optimal Frame Completions with Prescribed Norms for Majorization

Abstract: Given a finite sequence of vectors $$\mathcal F_0$$ in a $$d$$ -dimensional complex Hilbert space $${{\mathcal {H}}}$$ we characterize in a complete and explicit way the optimal completions of $$\mathcal F_0$$ obtained by appending a finite sequence of vectors with prescribed norms, where optimality is measured with respect to majorization (of the eigenvalues of the frame operators of the completed sequences). Indeed, we construct (in terms of a fast algorithm) a vector—that depends on the eigenvalues of the frame operator of the initial sequence $${\mathcal {F}}_0$$ and the sequence of prescribed norms—that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence $${\mathcal {F}}_0$$ we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completions with prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem.

Spectral Measures Associated with the Factorization of the Lebesgue Measure on a Set via Convolution

Abstract: Let $$Q$$ be a fundamental domain of some full-rank lattice in $${\mathbb {R}}^d$$ and let $$\mu $$ and $$ u $$ be two positive Borel measures on $${\mathbb {R}}^d$$ such that the convolution $$\mu * u $$ is a multiple of $$\chi _Q$$ . We consider the problem as to whether or not both measures must be spectral (i.e. each of their respective associated $$L^2$$ space admits an orthogonal basis of exponentials) and we show that this is the case when $$Q = [0,1]^d$$ . This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral measures. In addition, we propose a generalized Fuglede’s Conjecture for spectral measures on $${\mathbb {R}}^1$$ and we show that it implies the classical Fuglede’s Conjecture on $${\mathbb {R}}^1$$ .

On the Dynamics of WKB Wave Functions Whose Phase are Weak KAM Solutions of H–J Equation

Abstract: In the framework of toroidal Pseudodifferential operators on the flat torus \({\mathbb {T}}^n := ({\mathbb {R}} / 2\pi {\mathbb {Z}})^n\) we begin by proving the closure under composition for the class of Weyl operators \(\mathrm {Op}^w_\hbar (b)\) with symbols \(b \in S^m (\mathbb {T}^n \times \mathbb {R}^n)\). Subsequently, we consider \(\mathrm {Op}^w_\hbar (H)\) when \(H=\frac{1}{2} |\eta |^2 + V(x)\) where \(V \in C^\infty ({\mathbb {T}}^n)\) and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrodinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrodinger equation to the solution of the Liouville equation on \(\mathbb {T}^n \times {\mathbb {R}}^n\) written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space \(H^{1} (\mathbb {T}^n; {\mathbb {C}})\) with phase functions in the class of Lipschitz continuous weak KAM solutions (positive and negative type) of the Hamilton–Jacobi equation \(\frac{1}{2} |P+ abla _x v (P,x)|^2 + V(x) = \bar{H}(P)\) for \(P \in \ell {\mathbb {Z}}^n\) with \(\ell >0\), and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of \(P+ abla _x v\).

$$\ell ^p$$-Linear Independence of the System of Integer Translates

Abstract: Various properties of the system \({\mathcal {B}}_{\psi }\) of integer translates of a square integrable function \(\psi \in L^2({\mathbb {R}})\) can be completely described in terms of the periodization function \(p_{\psi }(\xi )=\sum _{k\in {\mathbb {Z}}}|\widehat{\psi }(\xi +k)|^2\) In this paper, we consider the problem of \(\ell ^p\)-linear independence, where \(p>2\) The results we present include the method of construction for one type of counterexamples to several naturally taken conjectures, a new sufficient condition for \(\ell ^p\)-linear independence and a characterization theorem having an additional assumption on \({\mathcal {B}}_{\psi }\) In the latter, we obtain the characterization in terms of the sets of multiplicity of Lebesgue measure zero