We prove \(\ell ^p\)-improving estimates for the averaging operator along the discrete paraboloid in the sharp range of p in all dimensions \(n\ge 2\).

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

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

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

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 \(x_0\)

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

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

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 \(\Gamma = {{\mathbb {S}}}^{N-1}\)

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

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

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

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

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

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

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.

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

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|)

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

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

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.

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

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

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

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

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 dense

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