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

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.

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.

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.

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

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

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

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

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

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.

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, \(b \in \mathcal {K}\)

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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,

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

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

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

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

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

The Cantor Set supports a Borel probability measure known as the Hutchinson measure which satisfies a well known fixed point relationship (Hutchinson in Indiana University Math J 30(5):713–747, 1981). Previously it has been shown by Jorgensen and Dutkay that the Cantor set can be extended to an inflated Cantor set, \({\mathcal {R}}\), on a subset of the real line, which supports an extended Hutchinson

We study the wave propagation speed problem on metric measure spaces, emphasizing on self-similar sets that are not post-critically finite. We prove that a sub-Gaussian lower heat kernel estimate leads to infinite propagation speed, extending a result of Lee (Infinite propagation speed for wave solutions on some p.c.f. fractals, https://archive.org/details/arxiv-1111.2938) to include bounded and unbounded

We show quantitative (in terms of the radius) \(l^p\)-improving estimates for the discrete spherical averages along the primes. These averaging operators were defined in [1] and are discrete, prime variants of Stein’s spherical averages. The proof uses a precise decomposition of the Fourier multiplier.

In this paper we show that the computational complexity of the iterative thresholding and K-residual-means (ITKrM) algorithm for dictionary learning can be significantly reduced by using dimensionality-reduction techniques based on the Johnson–Lindenstrauss lemma. The dimensionality reduction is efficiently carried out with the fast Fourier transform. We introduce the iterative compressed-thresholding

Let X be a metric space equipped with a measure satisfying the doubling and reverse doubling conditions. In this paper, we develop the theory of new localized Hardy spaces \(H^p_\rho (X)\) for \(\frac{n}{n+1}

This paper is centred on the spectral study of a random Fourier matrix, that is an \(n\times n\) matrix A with (j, k) entries given by \(\exp (2i\pi m Y_j Z_k)\), where \(Y_j\) and \(Z_k\) are two i.i.d sequences of random variables and \(1\le m\le n\) is a real number. When the \(Y_j\) and \(Z_k\) are uniformly distributed on a bounded symmetric interval, this may be seen as a random discretization

Local boundary smoothness of an analytic function f in the unit ball of \({\mathbb {C}}^n\) is compared to the smoothness of its modulus. We prove that in dimensions 2 and higher two different (and natural) conditions imposed on the zeros of f imply two different drops of its smoothness compared to the smoothness of |f|. We also show that some of the drops are the best possible.

Let G be a separable unimodular locally compact group of type I, and let N be a unimodular closed normal subgroup of G of type I, such that G/N is compact. Let for \(1

Suppose that G is a finite Abelian group and write \({\mathcal {W}}(G)\) for the set of cosets of subgroups of G. We show that if \(f:G \rightarrow {\mathbb {Z}}\) satisfies the estimate \(\Vert f\Vert _{A(G)} \le M\) with respect to the Fourier algebra norm, then there is some \(z:{\mathcal {W}}(G) \rightarrow {\mathbb {Z}}\) such that $$\begin{aligned} f=\sum _{W \in {\mathcal {W}}(G)}{z(W)1_W}\quad

Let \(\Delta \) be a closed, cocompact subgroup of \(G \times \widehat{G}\), where G is a second countable, locally compact abelian group. Using localization of Hilbert \(C^*\)-modules, we show that the Heisenberg module \(\mathcal {E}_{\Delta }(G)\) over the twisted group \(C^*\)-algebra \(C^*(\Delta ,c)\) due to Rieffel can be continuously and densely embedded into the Hilbert space \(L^2(G)\). This

The Paley–Littlewood square function

One classical result in greedy approximation theory is that almost-greedy and semi-greedy bases are equivalent in the context of Schauder bases in Banach spaces with finite cotype. This result was proved by Dilworth et al. (Studia Math 159:67–101, 2003) and, recently, in the study of Berná (J Math Anal Appl 470:218–225, 2019), the author proved that the condition of finite cotype can be removed. One

A method for constructing non-uniform filter banks is presented. Starting from a uniform system of translates, generated by a prototype filter, a non-uniform covering of the frequency axis is obtained by composition with a warping function. The warping function is a \({\mathcal {C}}^1\)-diffeomorphism that determines the frequency progression and can be chosen freely, apart from minor technical restrictions

In Analog-to-digital (A/D) conversion, signal decimation has been proven to greatly improve the efficiency of data storage while maintaining high accuracy. When one couples signal decimation with the \(\Sigma \Delta \) quantization scheme, the reconstruction error decays exponentially with respect to the bit-rate. We build on our previous result, which extended signal decimation to finite frames, albeit

In this paper we explore orthogonal systems in \(\mathrm {L}_2(\mathbb {R})\) which give rise to a skew-Hermitian, tridiagonal differentiation matrix. Surprisingly, allowing the differentiation matrix to be complex leads to a particular family of rational orthogonal functions with favourable properties: they form an orthonormal basis for \(\mathrm {L}_2(\mathbb {R})\), have a simple explicit formulae

Consider L groups of point sources or spike trains, with the lth group represented by \(x_l(t)\). For a function \(g:\mathbb {R}\rightarrow \mathbb {R}\), let \(g_l(t) = g(t/\mu _l)\) denote a point spread function with scale \(\mu _l > 0\), and with \(\mu _1< \cdots < \mu _L\). With \(y(t) = \sum _{l=1}^{L} (g_l \star x_l)(t)\), our goal is to recover the source parameters given samples of y, or given

In this paper we show that the local Kato type smoothing estimates are essentially equivalent to the global Kato type smoothing estimates for some class of dispersive equations including the Schrödinger equation. From this we immediately have two results as follows. One is that the known local Kato smoothing estimates are sharp. The sharp regularity ranges of the global Kato smoothing estimates are

It is known that if (p n ) n ∈ℕ is a sequence of orthogonal polynomials in L 2([-1,1],w(x)dx), then the roots are distributed according to an arcsine distribution π -1(1 - x 2)-1 dx for a wide variety of weights w(x). We connect this to a result of the Hilbert transform due to Tricomi: if f(x)(1 - x 2)1/4 ∈ L 2(-1,1) and its Hilbert transform Hf vanishes on (-1,1), then the function f is a multiple

We propose a refinement of the Robertson-Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is stronger than the RSUP. In particular, and unlike the RSUP, which can be saturated by many phase space functions, the refined RSUP can be saturated by pure Gaussian Wigner functions only. Moreover, the new principle is technically as simple as the standard RSUP. In addition

We investigate sharp frame bounds of Gabor frames with chirped Gaussians and rectangular lattices or, equivalently, the case of the standard Gaussian and general lattices. We prove that for even redundancy and standard Gaussian window the hexagonal lattice minimizes the upper frame bound using a result by Montgomery on minimal theta functions.

In a recent paper by T. Strohmer and R. Vershynin ["A Randomized Kaczmarz Algorithm with Exponential Convergence", Journal of Fourier Analysis and Applications, published online on April 25, 2008] a "randomized Kaczmarz algorithm" is proposed for solving systems of linear equations [Formula: see text] . In that algorithm the next equation to be used in an iterative Kaczmarz process is selected with