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Sharp $$\ell ^p$$ ℓ p -Improving Estimates for the Discrete Paraboloid J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2021-01-07 Shival Dasu, Ciprian Demeter, Bartosz Langowski
We prove \(\ell ^p\)-improving estimates for the averaging operator along the discrete paraboloid in the sharp range of p in all dimensions \(n\ge 2\).
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On the Search for Tight Frames of Low Coherence J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-11-23 Xuemei Chen, Douglas P. Hardin, Edward B. Saff
We introduce a projective Riesz s-kernel for the unit sphere \(\mathbb {S}^{d-1}\) and investigate properties of N-point energy minimizing configurations for such a kernel. We show that these configurations, for s and N sufficiently large, form frames that are well-separated (have low coherence) and are nearly tight. Our results suggest an algorithm for computing well-separated tight frames which is
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Weighted Alpert Wavelets J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-11-23 Rob Rahm, Eric T. Sawyer, Brett D. Wick
In this paper we construct a wavelet basis in \(L^{2}({\mathbb {R}}^{n};\mu )\) possessing vanishing moments of a fixed order for a general locally finite positive Borel measure \(\mu \). The approach is based on a clever construction of Alpert in the case of the Lebesgue measure that is appropriately modified to handle the general measures considered here. We then use this new wavelet basis to study
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A Unified View of Space–Time Covariance Functions Through Gelfand Pairs J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-11-20 Christian Berg
We give a characterization of positive definite integrable functions on a product of two Gelfand pairs as an integral of positive definite functions on one of the Gelfand pairs with respect to the Plancherel measure on the dual of the other Gelfand pair. In the very special case where the Gelfand pairs are Euclidean groups and the compact subgroups are reduced to the identity, the characterization
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Generalized 2-Microlocal Frontier Prescription J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-11-20 Ursula Molter, Mariel Rosenblatt
The characterization of local regularity is a fundamental issue in signal and image processing, since it contains relevant information about the underlying systems. The 2-microlocal frontier, a monotone concave downward curve in \(\mathbb {R}^2\), provides a useful way to classify pointwise singularity. In this paper we characterize all functions whose 2-microlocal frontier at a given point \(x_0\)
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Frames and Numerical Approximation II: Generalized Sampling J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-11-20 Ben Adcock, Daan Huybrechs
In a previous paper (Adcock and Huybrechs in SIAM Rev 61(3):443–473, 2019) we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormous flexibility compared to using a basis, but ill-conditioning often prevents the numerical computation of best approximations. We showed that, in spite of said ill-conditioning, approximations
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Complex Phase Retrieval from Subgaussian Measurements J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-11-20 Felix Krahmer, Dominik Stöger
Phase retrieval refers to the problem of reconstructing an unknown vector \(x_0 \in {\mathbb {C}}^n\) or \(x_0 \in {\mathbb {R}}^n \) from m measurements of the form \(y_i = \big \vert \langle \xi ^{\left( i\right) }, x_0 \rangle \big \vert ^2 \), where \( \left\{ \xi ^{\left( i\right) } \right\} ^m_{i=1} \subset {\mathbb {C}}^m \) are known measurement vectors. While Gaussian measurements allow for
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A Note on the $$L^p$$ L p Integrability of a Class of Bochner–Riesz Kernels J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-11-18 Reuben Wheeler
For a general compact variety \(\Gamma \) of arbitrary codimension, one can consider the \(L^p\) mapping properties of the Bochner–Riesz multiplier $$\begin{aligned} m_{\Gamma , \alpha }(\zeta ) \ = \ \mathrm{dist}(\zeta , \Gamma )^{\alpha } \phi (\zeta ) \end{aligned}$$ where \(\alpha > 0\) and \(\phi \) is an appropriate smooth cutoff function. Even for the sphere \(\Gamma = {{\mathbb {S}}}^{N-1}\)
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Global $${\mathcal {M}}-$$ M - Hypoellipticity, Global $${\mathcal {M}}-$$ M - Solvability and Perturbations by Lower Order Ultradifferential Pseudodifferential Operators J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-11-18 Igor Ambo Ferra, Gerson Petronilho, Bruno de Lessa Victor
We introduce a new class of ultradifferentiable pseudodifferential operators on the torus whose calculus allows us to show that global hypoellipticity, in ultradifferentiable classes, with a finite loss of derivatives of a system of pseudodifferential operators, is stable under perturbations by lower order pseudodifferential operators whose order depends on the loss of derivatives. The key point in
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Bounds on Moments of Weighted Sums of Finite Riesz Products J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-11-16 Aline Bonami, Rafał Latała, Piotr Nayar, Tomasz Tkocz
Let \(n_j\) be a lacunary sequence of integers, such that \(n_{j+1}/n_j\ge r\). We are interested in linear combinations of the sequence of finite Riesz products \(\prod _{j=1}^N(1+\cos (n_j t))\). We prove that, whenever the Riesz products are normalized in \(L^p\) norm (\(p\ge 1\)) and when r is large enough, the \(L^p\) norm of such a linear combination is equivalent to the \(\ell ^p\) norm of the
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A Duality Principle for Groups II: Multi-frames Meet Super-Frames J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-11-09 R. Balan, D. Dutkay, D. Han, D. Larson, F. Luef
The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties
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On Spectral Eigenvalue Problem of a Class of Self-similar Spectral Measures with Consecutive Digits J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-10-20 Cong Wang, Zhi-Yi Wu
Let \(\mu _{p,q}\) be a self-similar spectral measure with consecutive digits generated by an iterated function system \(\{f_i(x)=\frac{x}{p}+\frac{i}{q}\}_{i=0}^{q-1}\), where \(2\le q\in {{\mathbb {Z}}}\) and q|p. It is known that for each \(w=w_1w_2\cdots \in \{-1,1\}^\infty :=\{i_1i_2\cdots :~\text {all}~i_k\in \{-1,1\}\}\), the set $$\begin{aligned} \Lambda _w=\bigg \{\sum _{j=1}^{n}a_j w_j p^{j-1}:a_j\in
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Point-Wise Wavelet Estimation in the Convolution Structure Density Model J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-10-20 Youming Liu, Cong Wu
By using a kernel method, Lepski and Willer establish adaptive and optimal \(L^p\) risk estimations in the convolution structure density model in 2017 and 2019. They assume their density functions to be in a Nikol’skii space. Motivated by their work, we first use a linear wavelet estimator to obtain a point-wise optimal estimation in the same model. We allow our densities to be in a local and anisotropic
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An Analytic Method of Phase Retrieval for X-Ray Phase Contrast Imaging J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-10-20 Victor Palamodov
New lensless diffractive X-ray technic for micro-scale imaging of biological tissue is based on quantitative information on the phase. This method yields improved contrast compared to purely absorption-based tomography but involves a phase retrieval problem since of physical limitation of detectors. An analytic method is proposed in the paper for reconstruction of the ray projection of complex refraction
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On the Limit as $$s\rightarrow 0^+$$ s → 0 + of Fractional Orlicz–Sobolev Spaces J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-10-20 Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková
An extended version of the Maz’ya–Shaposhnikova theorem on the limit as \(s\rightarrow 0^+\) of the Gagliardo–Slobodeckij fractional seminorm is established in the Orlicz space setting. Our result holds in fractional Orlicz–Sobolev spaces associated with Young functions satisfying the \(\Delta _2\)-condition, and, as shown by counterexamples, it may fail if this condition is dropped.
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Norm Inflation for Nonlinear Schrödinger Equations in Fourier–Lebesgue and Modulation Spaces of Negative Regularity J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-10-20 Divyang G. Bhimani, Rémi Carles
We consider nonlinear Schrödinger equations in Fourier–Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less
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Trudinger–Moser Type Inequalities with Vanishing Weights in the Unit Ball J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-10-02 Van Hoang Nguyen
Let \(\mathbf{B} \) denote the unit ball in \({\mathbb {R}}^n\) with \(n\ge 2\). In this paper, we present the balance conditions on the nonlinearity function F and the weight function h such that the weighted Trudinger–Moser type inequalities $$\begin{aligned} \sup _{u \in W^{1,n}_{0}(\mathbf{B }),\, u \text { is radial}, \Vert \nabla u\Vert _{L^n(\mathbf{B })} \le 1} \int _{\mathbf{B }} F(u) h(|x|)
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Uncertainty Principles for Fourier Multipliers J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-09-30 Michael V. Northington
The admittable Sobolev regularity is quantified for a function, w, which has a zero in the d-dimensional torus and whose reciprocal \(u=1/w\) is a (p, q)-multiplier. Several aspects of this problem are addressed, including zero-sets of positive Hausdorff dimension, matrix valued Fourier multipliers, and non-symmetric versions of Sobolev regularity. Additionally, we make a connection between Fourier
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Irregularity of Distribution in Wasserstein Distance J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-09-29 Cole Graham
We study the non-uniformity of probability measures on the interval and circle. On the interval, we identify the Wasserstein-p distance with the classical \(L^p\)-discrepancy. We thereby derive sharp estimates in Wasserstein distances for the irregularity of distribution of sequences on the interval and circle. Furthermore, we prove an \(L^p\)-adapted Erdős–Turán inequality, and use it to extend a
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Herz Spaces Meet Morrey Type Spaces and Complementary Morrey Type Spaces J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-09-14 Humberto Rafeiro; Stefan Samko
We introduce local and global generalized Herz spaces. As one of the main results we show that Morrey type spaces and complementary Morrey type spaces are included into the scale of these Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized Herz spaces with application to Morrey type spaces and their complementary spaces, based on the mentioned inclusion.
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On Symmetric Compactly Supported Wavelets with Vanishing Moments Associated to $$E_d^{(2)}(\mathbb {Z})$$ E d ( 2 ) ( Z ) Dilations J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-09-09 M. L. Arenas; Angel San Antolín
Let A be an expansive linear map on \({{\mathbb {R}}}^d\) preserving the integer lattice and with \(| \det A|=2\). We prove that if there exists a self-affine tile set associated to A, there exists a compactly supported wavelet with any desired number of vanishing moments and some symmetry. We put emphasis on construction of wavelets associated to a linear map A on \({{\mathbb {R}}}^2\) and to the
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On Sets Containing an Affine Copy of Bounded Decreasing Sequences J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-09-09 Tongou Yang
How small can a set be while containing many configurations? Following up on earlier work of Erdős and Kakutani (Colloq Math 4:195–196, 1957), Máthé (Fund Math 213(3):213–219, 2011) and Molter and Yavicoli (Math Proc Camb Soc 168:57–73, 2018), we address the question in two directions. On one hand, if a subset of the real numbers contains an affine copy of all bounded decreasing sequences, then we
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Blaschke Decompositions on Weighted Hardy Spaces J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-08-19 Stephen D. Farnham
Recently, several authors have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either the unwinding series or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of the boundary value of an analytic function \(F: \partial {\mathbb {D}}\rightarrow {\mathbb {C}}\) and by performing an iterative method to obtain a sequence
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Solving Nonlinear p -Adic Pseudo-differential Equations: Combining the Wavelet Basis with the Schauder Fixed Point Theorem J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-08-14 Ehsan Pourhadi; Andrei Yu. Khrennikov; Klaudia Oleschko; María de Jesús Correa Lopez
Recently theory of p-adic wavelets started to be actively used to study of the Cauchy problem for nonlinear pseudo-differential equations for functions depending on the real time and p-adic spatial variable. These mathematical studies were motivated by applications to problems of geophysics (fluids flows through capillary networks in porous disordered media) and the turbulence theory. In this article
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The Boundedness of the (Sub)bilinear Maximal Function Along “Non-flat” Smooth Curves J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-08-03 Alejandra Gaitan; Victor Lie
Let \(\mathcal {NF}\) be the class of smooth non-flat curves near the origin and near infinity introduced in Lie (Am J Math 137(2):313–363, 2015) and let \(\gamma \in \mathcal {NF}\). We show—via a unifying approach relative to the corresponding bilinear Hilbert transform \(H_{\Gamma }\)—that the (sub)bilinear maximal function along curves \(\Gamma =(t,-\gamma (t))\) defined as$$\begin{aligned} M_\Gamma
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A Note on the HRT Conjecture and a New Uncertainty Principle for the Short-Time Fourier Transform J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-07-30 Fabio Nicola; S. Ivan Trapasso
In this note we provide a negative answer to a question raised by Kreisel concerning a condition on the short-time Fourier transform that would imply the HRT conjecture. In particular we provide a new type of uncertainty principle for the short-time Fourier transform which forbids the arrangement of an arbitrary “bump with fat tail” profile.
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On a Characterization of Riesz Bases via Biorthogonal Sequences J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-07-30 Diana T. Stoeva
It is well known that a sequence in a Hilbert space is a Riesz basis if and only if it is a complete Bessel sequence with biorthogonal sequence which is also a complete Bessel sequence. Here we prove that the completeness of one (any one) of the biorthogonal sequences can be removed from the characterization.
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Fourier Series Windowed by a Bump Function J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-07-27 Paul Bergold; Caroline Lasser
We study the Fourier transform windowed by a bump function. We transfer Jackson’s classical results on the convergence of the Fourier series of a periodic function to windowed series of a not necessarily periodic function. Numerical experiments illustrate the obtained theoretical results.
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Continuous Schauder Frames for Banach Spaces J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-07-27 Joseph Eisner; Daniel Freeman
We introduce the notion of a continuous Schauder frame for a Banach space. This is both a generalization of continuous frames for Hilbert spaces and a generalization of unconditional Schauder frames for Banach spaces. Furthermore, we generalize the properties shrinking and boundedly complete to the continuous Schauder frame setting, and prove that many of the fundamental James theorems still hold in
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Fourier Analysis of Periodic Radon Transforms J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-07-24 Jesse Railo
We study reconstruction of an unknown function from its d-plane Radon transform on the flat torus \({\mathbb {T}}^n = {\mathbb {R}}^n /{\mathbb {Z}}^n\) when \(1 \le d \le n-1\). We prove new reconstruction formulas and stability results with respect to weighted Bessel potential norms. We solve the associated Tikhonov minimization problem on \(H^s\) Sobolev spaces using the properties of the adjoint
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Linear Independence of Time–Frequency Translates in $$L^p$$ L p Spaces J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-07-24 Jorge Antezana; Joaquim Bruna; Enrique Pujals
We study the Heil–Ramanathan–Topiwala conjecture in \(L^p\) spaces by reformulating it as a fixed point problem. This reformulation shows that a function with linearly dependent time–frequency translates has a very rigid structure, which is encoded in a family of linear operators. This is used to give an elementary proof that if \(f\in L^p({\mathbb {R}})\), \(p\in [1,2]\), and \(\Lambda \subseteq {\mathbb
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A Generalized Gelfand Pair Attached to a 3-Step Nilpotent Lie Group J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-07-24 Andrea L. Gallo; Linda V. Saal
Let N be a nilpotent Lie group and K a compact subgroup of the automorphism group Aut(N) of N. It is well-known that if \((K < imes N,K)\) is a Gelfand pair then N is at most 2-step nilpotent Lie group. The notion of Gelfand pair was generalized when K is a non-compact group. In this work, we give an example of a 3-step nilpotent Lie group and a non-compact subgroup K of Aut(N) such that \((K < imes
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Real Interpolation of Hardy-Type Spaces and BMO-Regularity J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-07-22 Dmitry V. Rutsky
Let \(\Omega \) be a \(\sigma \)-finite measurable space. Suppose that (X, Y) is a couple of quasi-Banach lattices of measurable functions on \({\mathbb {T}} \times \Omega \) satisfying some additional assumptions. The Hardy-type spaces \(X_A\) consist of functions on \({\mathbb {D}} \times \Omega \) belonging to the Smirnov class \(\mathrm {N}^+\) in the first variable such that their boundary values
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Transport Inequalities on Euclidean Spaces for Non-Euclidean Metrics J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-07-06 Sergey G. Bobkov; Michel Ledoux
We explore upper bounds on Kantorovich transport distances between probability measures on the Euclidean spaces in terms of their Fourier-Stieltjes transforms, with focus on non-Euclidean metrics. The results are illustrated on empirical measures in the optimal matching problem on the real line.
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Parseval Proximal Neural Networks J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-07-06 Marzieh Hasannasab; Johannes Hertrich; Sebastian Neumayer; Gerlind Plonka; Simon Setzer; Gabriele Steidl
The aim of this paper is twofold. First, we show that a certain concatenation of a proximity operator with an affine operator is again a proximity operator on a suitable Hilbert space. Second, we use our findings to establish so-called proximal neural networks (PNNs) and stable tight frame proximal neural networks. Let \(\mathcal {H}\) and \(\mathcal {K}\) be real Hilbert spaces, \(b \in \mathcal {K}\)
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Pointwise Convergence Along Restricted Directions for the Fractional Schrödinger Equation J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-29 Shobu Shiraki
We consider the pointwise convergence problem for the solution of Schrödinger-type equations along directions determined by a given compact subset of the real line. This problem contains Carleson’s problem as the simplest case and was studied in general by Cho et al. We extend their result from the case of the classical Schrödinger equation to a class of equations which includes the fractional Schrödinger
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Phase Retrieval for Wide Band Signals J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-23 Philippe Jaming; Karim Kellay; Rolando Perez
This study investigates the phase retrieval problem for wide-band signals. We solve the following problem: given \(f\in L^2(\mathbb {R})\) with Fourier transform in \(L^2(\mathbb {R},e^{2c|x|}\,\text{ d }x)\), we find all functions \(g\in L^2(\mathbb {R})\) with Fourier transform in \(L^2(\mathbb {R},e^{2c|x|}\,\text{ d }x)\), such that \(|f(x)|=|g(x)|\) for all \(x\in \mathbb {R}\). To do so, we first
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Sequence Dominance in Shift-Invariant Spaces J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-23 Tomislav Berić; Hrvoje Šikić
We show that a Bessel sequence \(B_\psi \) of integer translates of a square integrable function \(\psi \in L^2(\mathbb {R})\) has the Besselian property if and only if its periodization function \(p_\psi \) is bounded from below. We also give characterizations of Besselian and Hilbertian properties of a general sequence \(B_\psi \) of integer translates in terms of the classical notion of sequence
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Weighted Fourier Inequalities in Lebesgue and Lorentz Spaces J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-23 Erlan Nursultanov; Sergey Tikhonov
In this paper, we obtain sufficient conditions for the weighted Fourier-type transforms to be bounded in Lebesgue and Lorentz spaces. Two types of results are discussed. First, we review the method based on rearrangement inequalities and the corresponding Hardy’s inequalities. Second, we present Hörmander-type conditions on weights so that Fourier-type integral operators are bounded in Lebesgue and
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Density Results for Continuous Frames J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-23 Mishko Mitkovski; Aaron Ramirez
We derive necessary conditions for localization of continuous frames in terms of generalized Beurling densities. As an important application we provide necessary density conditions for sampling, interpolation, and uniform minimality in a very large class of reproducing kernel Hilbert spaces.
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Banach-Valued Modulation Invariant Carleson Embeddings and Outer- $$L^p$$ L p Spaces: The Walsh Case J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-22 Alex Amenta; Gennady Uraltsev
We prove modulation invariant embedding bounds from Bochner spaces \(L^p(\mathbb {W};X)\) on the Walsh group to outer-\(L^p\) spaces on the Walsh extended phase plane. The Banach space X is assumed to be UMD and sufficiently close to a Hilbert space in an interpolative sense. Our embedding bounds imply \(L^p\) bounds and sparse domination for the Banach-valued tritile operator, a discrete model of
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Phase-Retrieval in Shift-Invariant Spaces with Gaussian Generator J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-15 Karlheinz Gröchenig
We study the problem of recovering a function of the form \(f(x) = \sum _{k\in \mathbb {Z}} c_k e^{-(x-k)^2}\) from its phaseless samples \(|f(\lambda )|\) on some arbitrary countable set \(\Lambda \subseteq \mathbb {R}\). For real-valued functions this is possible up to a sign for every separated set with Beurling density \(D^-(\Lambda ) >2\). This result is sharp. For complex-valued functions we
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Wavelet-Type Expansion of the Generalized Rosenblatt Process and Its Rate of Convergence J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-12 Antoine Ayache; Yassine Esmili
Pipiras introduced in the early 2000s an almost surely and uniformly convergent (on compact intervals) wavelet-type expansion of the classical Rosenblatt process. Yet, the issue of estimating, almost surely, its uniform rate of convergence remained an open question. The main goal of our present article is to provide an answer to it in the more general framework of the generalized Rosenblatt process
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The Kato Square Root Problem for Divergence Form Operators with Potential J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-12 Julian Bailey
The Kato square root problem for divergence form elliptic operators with potential \(V : {\mathbb {R}}^{n} \rightarrow {\mathbb {C}}\) is the equivalence statement \(\left\| \left( L + V \right) ^{\frac{1}{2}} u \right\| _{2} \simeq \left\| \nabla u \right\| _{2} + \left\| V^{\frac{1}{2}} u \right\| _{2}\), where \(L + V := - \mathrm {div} (A \nabla ) + V\) and the perturbation A is an \(L^{\infty
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Some Embeddings of Morrey Spaces with Critical Smoothness J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-12 Dorothee D. Haroske; Susana D. Moura; Leszek Skrzypczak
We study embeddings of Besov–Morrey spaces \({{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\) and of Triebel–Lizorkin–Morrey spaces \({{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\) in the limiting cases when the smoothness s equals \(s_o=d\max (1/u-p/u,0)\) or \(s_{\infty }=d/u\), which is related to the embeddings in \(L_1^{\mathrm {loc}}({{{\mathbb {R}}}^d})\) or in \(L_{\infty }({{{\mathbb
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Regular Two-Distance Sets J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-12 Peter G. Casazza; Tin T. Tran; Janet C. Tremain
This paper makes a deep study of regular two-distance sets. A set of unit vectors X in Euclidean space \({\mathbb {R}}^n\) is said to be regular two-distance set if the inner product of any pair of its vectors is either \(\alpha \) or \(\beta \), and the number of \(\alpha \)’s (and hence \(\beta \)’s) on each row of the Gram matrix of X is the same. We present various properties of these sets as well
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Daubechies’ Time–Frequency Localization Operator on Cantor Type Sets I J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-12 Helge Knutsen
We study Daubechies’ time–frequency localization operator, which is characterized by a window and weight function. We consider a Gaussian window and a spherically symmetric weight as this choice yields explicit formulas for the eigenvalues, with the Hermite functions as the associated eigenfunctions. Inspired by the fractal uncertainty principle in the separate time–frequency representation, we define
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Quantum Harmonic Analysis on Lattices and Gabor Multipliers J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-12 Eirik Skrettingland
We develop a theory of quantum harmonic analysis on lattices in \({\mathbb {R}}^{2d}\). Convolutions of a sequence with an operator and of two operators are defined over a lattice, and using corresponding Fourier transforms of sequences and operators we develop a version of harmonic analysis for these objects. We prove analogues of results from classical harmonic analysis and the quantum harmonic analysis
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Domains Without Dense Steklov Nodal Sets J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-11 Oscar P. Bruno; Jeffrey Galkowski
This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem$$\begin{aligned} -\Delta \phi _{\sigma _j}=0,\quad \hbox { on }\,\,\Omega ,\quad \partial _\nu \phi _{\sigma _j}=\sigma _j \phi _{\sigma _j}\quad \hbox { on }\,\,\partial \Omega \end{aligned}$$in two-dimensional domains \(\Omega \). In particular, this paper presents a dense
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Time-Changed Dirac–Fokker–Planck Equations on the Lattice J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-08 Nelson Faustino
A time-changed discretization for the Dirac equation is proposed. More precisely, we consider a Dirac equation with discrete space and continuous time perturbed by a time-dependent diffusion term \(\sigma ^2Ht^{2H-1}\) that seamlessly describes a latticizing version of the time-changed Fokker–Planck equation carrying the Hurst parameter \(00\) and \(0<\alpha <\frac{1}{2}\)) preserves the main features
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On the Fourier Transforms of Nonlinear Self-similar Measures J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-06-05 Zhanqi Zhang; Yingqing Xiao
In-homogeneous self-similar measures can be viewed as special cases of nonlinear self-similar measures. In this paper, we study the asymptotic behaviour of the Fourier transforms of nonlinear self-similar measures. Some typical examples are exhibited, and we show that the Fourier transforms of those measures are usually localized, i.e., the Fourier transforms decay rapidly at \(\infty \). We also discuss
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Improving Estimates for Discrete Polynomial Averages J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-05-28 Rui Han; Vjekoslav Kovač; Michael T. Lacey; José Madrid; Fan Yang
For a polynomial P mapping the integers into the integers, define an averaging operator \(A_{N} f(x):=\frac{1}{N}\sum _{k=1}^N f(x+P(k))\) acting on functions on the integers. We prove sufficient conditions for the \(\ell ^{p}\)-improving inequality$$\begin{aligned} \Vert A_N f\Vert _{\ell ^q(\mathbb {Z})} \lesssim _{P,p,q} N^{-d(\frac{1}{p}-\frac{1}{q})} \Vert f\Vert _{\ell ^p(\mathbb {Z})}, \qquad
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Sampling and Reconstruction by Means of Weighted Inverses J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-05-27 M. Laura Arias; M. Celeste Gonzalez
In this article, we address the problem of reconstructing an element in a Hilbert space from its samples by means of a weighted least square approximation. We show how this problem is linked with the notions of weighted inverses, weighted projections and an angle condition known as compatibility. In addition, we study perfect reconstruction operators and their relationship with the previous problem
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A Uniqueness Result for Light Ray Transform on Symmetric 2-Tensor Fields J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-05-21 Venkateswaran P. Krishnan; Soumen Senapati; Manmohan Vashisth
We study light ray transform of symmetric 2-tensor fields defined on a bounded time-space domain in \({\mathbb {R}}^{1+n}\) for \(n\ge 3\). We prove a uniqueness result for such light ray transforms. More precisely, we characterize the kernel of the light ray transform vanishing near a fixed direction at each point in the time-space domain.
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On Sampling and Interpolation by Model Sets J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-05-20 Christoph Richard; Christoph Schumacher
We refine a result of Matei and Meyer on stable sampling and stable interpolation for simple model sets. Our setting is model sets in locally compact abelian groups and Fourier analysis of unbounded complex Radon measures as developed by Argabright and de Lamadrid. This leads to a refined version of the underlying model set duality between sampling and interpolation. For rather general model sets,
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Concentration Estimates for Band-Limited Spherical Harmonics Expansions via the Large Sieve Principle J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-05-15 M. Speckbacher; T. Hrycak
We study a concentration problem on the unit sphere \(\mathbb {S}^2\) for band-limited spherical harmonics expansions using large sieve methods. We derive upper bounds for concentration in terms of the maximum Nyquist density. Our proof uses estimates of the spherical harmonics coefficients of certain zonal filters. We also demonstrate an analogue of the classical large sieve inequality for spherical
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The Norm of the Fourier Transform on Compact or Discrete Abelian Groups J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-05-07 Mokshay Madiman; Peng Xu
We calculate the norm of the Fourier operator from \(L^p(X)\) to \(L^q({\hat{X}})\) when X is an infinite locally compact abelian group that is, furthermore, compact or discrete. This subsumes the sharp Hausdorff–Young inequality on such groups. In particular, we identify the region in (p, q)-space where the norm is infinite, generalizing a result of Fournier, and setting up a contrast with the case
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Orthogonal Polynomials and Fourier Orthogonal Series on a Cone J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-04-02 Yuan Xu
Orthogonal polynomials and the Fourier orthogonal series on a cone in \({{\mathbb {R}}}^{d+1}\) are studied. It is shown that orthogonal polynomials with respect to the weight function \((1-t)^{\gamma }(t^2-\Vert x\Vert ^2)^{\mu -\frac{1}{2}}\) on the cone \({{\mathbb {V}}}^{d+1} = \{(x,t): \Vert x\Vert \le t \le 1\}\) are eigenfunctions of a second order differential operator, with eigenvalues depending
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Spectral Multipliers on 2-Step Stratified Groups, I J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-03-19 Mattia Calzi
Given a 2-step stratified group which does not satisfy a slight strengthening of the Moore–Wolf condition, a sub-Laplacian \({\mathcal {L}}\) and a family \({\mathcal {T}}\) of elements of the derived algebra, we study the convolution kernels associated with the operators of the form \(m({\mathcal {L}}, -\,i {\mathcal {T}})\). Under suitable conditions, we prove that: (i) if the convolution kernel
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$$L^p$$Lp -Maximal Regularity for a Class of Degenerate Integro-differential Equations with Infinite Delay in Banach Spaces J. Fourier Anal. Appl. (IF 1.442) Pub Date : 2020-03-16 Rafael Aparicio; Valentin Keyantuo
Using the theory of operator-valued Fourier multipliers, we establish characterizations for well-posedness of a large class of degenerate integro-differential equations of second order in time in Banach spaces. We are concerned with the spaces \(L^p({\mathbb {R}},X), \, 1\leqslant p<\infty \) where X is a given Banach space. When X is a UMD space and \(1