We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus

There are two parts for this paper. In the first part we extend some results in a recent paper by Du, Guth, Li and Zhang to a more general class of phase functions. The main methods are Bourgain–Demeter’s \(l^2\) decoupling theorem and induction on scales. In the second part we prove some positive results for the maximal extension operator for hypersurfaces with positive principal curvatures. The main

For a normalized root system R in \({\mathbb {R}}^N\) and a multiplicity function \(k\ge 0\) let \({\mathbf {N}}=N+\sum _{\alpha \in R} k(\alpha )\). We denote by \(dw({\mathbf {x}})=\varPi _{\alpha \in R}|\langle {\mathbf {x}},\alpha \rangle |^{k(\alpha )}\,d{\mathbf {x}}\) the associated measure in \({\mathbb {R}}^N\). Let \(L=-\varDelta +V\), \(V\ge 0\), be the Dunkl–Schrödinger operator on \({\mathbb

In this paper we characterize all subspaces of analytic functions in finitely generated shift-invariant spaces with compactly supported generators and provide explicit descriptions of their elements. We illustrate the differences between our characterizations and Strang-Fix or zero conditions on several examples. Consequently, we depict the analytic functions generated by scalar or vector subdivision

We introduce new anisotropic wavelet-type transforms generated by two components: a wavelet measure (or a wavelet function) and a kernel function that naturally generalizes the Gauss and Poisson kernels. The analogues of Calderon’s reproducing formula are established in the framework of the \(L_{p}(\mathbb {R}^{n+1})\)-theory. These wavelet-type transforms have close connection with a significant generalization

We present new estimates in the setting of weighted Lorentz spaces for important operators in Harmonic Analysis such as sparse operators, Bochner–Riesz at the critical index, Hörmander multipliers and rough singular integrals among others.

This article was originally published Online First without Open Access. After publication in volume 26, issue 4, the author decided to opt for Open Choice and to make the article an Open Access publication. Therefore, the copyright of the article has been changed to ©The Author(s) 2021 and the article is forthwith distributed under the terms of the Creative Commons Attribution 4.0 International License

We introduce a set of novel multiscale basis transforms for signals on graphs that utilize their “dual” domains by incorporating the “natural” distances between graph Laplacian eigenvectors, rather than simply using the eigenvalue ordering. These basis dictionaries can be seen as generalizations of the classical Shannon wavelet packet dictionary to arbitrary graphs, and do not rely on the frequency

The operator that first truncates to a neighborhood of the origin in the spatial domain then truncates to a neighborhood of the origin in the spectral domain is investigated in the case of Boolean cubes. This operator is self adjoint on the space of functions spanned by the Laplacian eigenvectors corresponding to small eigenvalues. The eigenspaces of this iterated projection operator are studied through

We consider a disjoint cover (partition) of an undirected weighted finite or infinite graph G by J connected subgraphs (clusters) \(\{S_{j}\}_{j\in J}\) and select functions \(\psi _{j}\) on each of the clusters. For a given signal f on G the set of its weighted average values samples is defined via inner products \(\{\langle f, \psi _{j}\rangle \}_{j\in J}\). The main results of the paper are based

We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket (\(SG\)), which is a fractal set that can be viewed as a limit of a sequence of finite graphs. These orthogonal polynomials arise through the Gram–Schmidt orthogonalisation process applied on the set of monomials on \(SG\) using several notions of a Sobolev inner products. After establishing some recurrence relations for

We introduce a construction of multiscale tight frames on general domains. The frame elements are obtained by spectral filtering of the integral operator associated with a reproducing kernel. Our construction extends classical wavelets as well as generalized wavelets on both continuous and discrete non-Euclidean structures such as Riemannian manifolds and weighted graphs. Moreover, it allows to study

Given a graph \(G=(V,E)\), suppose we are interested in selecting a sequence of vertices \((x_j)_{j=1}^n\) such that \(\left\{ x_1, \dots , x_k\right\} \) is ‘well-distributed’ uniformly in k. We describe a greedy algorithm motivated by potential theory and corresponding developments in the continuous setting. The algorithm performs nicely on graphs and may be of use for sampling problems. We can interpret

Our paper Carypis contains an error in the proof of Proposition. More precisely the estimate claimed in Equation is erroneously motivated.

We present recent advances in harmonic analysis on infinite graphs. Our approach combines combinatorial tools with new results from the theory of unbounded Hermitian operators in Hilbert space, geometry, boundary constructions, and spectral invariants. We focus on particular classes of infinite graphs, including such weighted graphs which arise in electrical network models, as well as new diagrammatic

Indicator functions mentioned in the title are constructed on an arbitrary nondiscrete locally compact Abelian group of finite dimension. Moreover, they can be obtained by small perturbation from any indicator function fixed beforehand. In the case of a noncompact group, the term “Fourier sums” should be understood as “partial Fourier integrals”. A certain weighted version of the result is also provided

The goal of this article is to survey various results concerning stochastic completeness of graphs. In particular, we present a variety of formulations of stochastic completeness and discuss how a discrepancy between uniqueness class and volume growth criteria in the continuous and discrete settings was ultimately resolved via the use of intrinsic metrics. Along the way, we discuss some equivalent

The angular synchronization problem of estimating a set of unknown angles from their known noisy pairwise differences arises in various applications. It can be reformulated as an optimization problem on graphs involving the graph Laplacian matrix. We consider a general, weighted version of this problem, where the impact of the noise differs between different pairs of entries and some of the differences

We define a Muckenhoup-type condition on weighted Morrey spaces using the Köthe dual of the space. We show that the condition is necessary and sufficient for the boundedness of the maximal operator defined with balls centered at the origin on weighted Morrey spaces. A modified condition characterizes the weighted inequalities for the Calderón operator. We also show that the Muckenhoup-type condition

We prove the continuous dependence of the solution maps for the Euler equations in the (critical) Triebel–Lizorkin spaces, which was not shown in the previous works [6, 7, 9]. The proof relies on the classical Bona–Smith method as [12], where similar result was obtained in critical Besov spaces \(B^1_{\infty ,1}\).

We give applications of the theory of superoscillations to various questions, namely extension of positive definite functions, interpolation of polynomials and also of R-functions; we also discuss possible applications to signal theory and prediction theory of stationary stochastic processes. In all cases, we give a constructive procedure, by way of a limiting process, to get the required results.

In this paper, we work in the setting of Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and the harmonic function theory in this setting introduced by Muckenhoupt–Stein, especially the generalised Cauchy–Riemann equations and the conjugate harmonic functions. We provide the equivalent characterizations of product Hardy spaces associated with Bessel operators in terms of

In the past decade, significant progress has been made to generalize classical tools from Fourier analysis to analyze and process signals defined on networks. In this paper, we propose a new framework for constructing Gabor-type frames for signals on graphs. Our approach uses general and flexible families of linear operators acting as translations. Compared to previous work in the literature, our methods

We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Green’s function and when it gives a non-constant harmonic function which is square integrable.

We show that a real-valued function f in the shift-invariant space generated by a totally positive function of Gaussian type is uniquely determined, up to a sign, by its absolute values \(\{|f(\lambda )|: \lambda \in \Lambda \}\) on any set \(\Lambda \subseteq {\mathbb {R}}\) with lower Beurling density \(D^{-}(\Lambda )>2\). We consider a totally positive function of Gaussian type, i.e., a function

The purpose of the present paper is to discuss the integrability of the Fourier transform of \(L^\infty \)-functions subject to a very weak decay condition. This will include the negative power of the logarithm. For a start, the case when \(d=1\) is investigated by the use of the second mean value theorem. Based on the discussion of this case, a passage to the weighted Hankel transform is done, which

The present paper is devoted to clustering geometric graphs. While the standard spectral clustering is often not effective for geometric graphs, we present an effective generalization, which we call higher-order spectral clustering. It resembles in concept the classical spectral clustering method but uses for partitioning the eigenvector associated with a higher-order eigenvalue. We establish the weak

We establish sharp asymptotically optimal strategies for the problem of online prediction with history dependent experts. The prediction problem is played (in part) over a discrete graph called the d dimensional de Bruijn graph, where d is the number of days of history used by the experts. Previous work Drenska and Kohn (arXiv:2007.12732, 2020) established \(O(\varepsilon )\) optimal strategies for

This paper is dedicated to the proof of Strichartz estimates on the Heisenberg group \({\mathop {\mathbb H}\nolimits }^d\) for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on \({\mathop {\mathbb H}\nolimits }^d\) is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates

Deep generative modeling has led to new and state of the art approaches for enforcing structural priors in a variety of inverse problems. In contrast to priors given by sparsity, deep models can provide direct low-dimensional parameterizations of the manifold of images or signals belonging to a particular natural class, allowing for recovery algorithms to be posed in a low-dimensional space. This dimensionality

In this paper, we establish a Tauberian theorem of Wiener–Ikehara type. As an application, we give a short proof of Blackwell’s renewal theorem of discrete case. We also establish a Dirichlet series version of Blackwell’s renewal theorem as another application.

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our papers (Dasgupta and Ruzhansky in Trans Am Math Soc 368(12):8481–8498, 2016) and (Dasgupta and Ruzhansky in Trans Am Math Soc Ser B 5:81–101, 2018). We prove that these spaces are perfect sequence spaces. As a consequence we describe

We introduce and investigate an adaptation of Fourier series to set-valued functions (multifunctions, SVFs) of bounded variation. In our approach we define an analogue of the partial sums of the Fourier series with the help of the Dirichlet kernel using the newly defined weighted metric integral. We derive error bounds for these approximants. As a consequence, we prove that the sequence of the partial

The wavelet group and wavelet representation associated with shifts coming from a two dimensional crystal symmetry group \(\Gamma \) and dilations by powers of 3, are defined and studied. The main result is an explicit decomposition of this \(3\Gamma \)-wavelet representation into irreducible representations of the wavelet group. Because we prove that the \(3\Gamma \)-wavelet representation is multiplicity

Due to the many applications in Magnetic Resonance Imaging (MRI), Nuclear Magnetic Resonance (NMR), radio interferometry, helium atom scattering etc., the theory of compressed sensing with Fourier transform measurements has reached a mature level. However, for binary measurements via the Walsh transform, the theory has long been merely non-existent, despite the large number of applications such as

We prove a sharp general inequality estimating the distance of two probability measures on a compact Lie group in the Wasserstein metric in terms of their Fourier transforms. We use a generalized form of the Wasserstein metric, related by Kantorovich duality to the family of functions with an arbitrarily prescribed modulus of continuity. The proof is based on smoothing with a suitable kernel, and a

Restricted isometry property (RIP) provides a near isometric map for sparse signals. RIP of structured random matrices has played a key role for dimensionality reduction and recovery from compressive measurements. In a companion paper, we have developed a unified theory for RIP of group structured measurement operators on generalized sparsity models. The implication of the extended result will be further

I will describe joint work with the late Bob Warner. We knew that \(H^1(R)\) gave better results for singular integrals than \(L^1(R)\); our question was: Would the same be true for spectral synthesis?. We extend the Beurling–Pollard argument to give sufficient conditions for spectral synthesis in \(H^1(R)\). We motivate and construct a class of Q-scets which satisfy the boundary and union property

In 2019, Grafakos and Stockdale introduced an \(L^q\) mean Hörmander condition and proved a “limited-range” Calderón–Zygmund theorem. Comparing their theorem with the classical one, it requires weaker assumptions and implies the \(L^p\) boundedness for the “limited-range” instead of \(1< p < \infty \). However, in this paper, we show that the \(L^q\) mean Hörmander condition is actually enough to obtain

In this paper, we study the spectrality and frame-spectrality of exponential systems of the type \(E(\Lambda ,\varphi ) = \{e^{2\pi i \lambda \cdot \varphi (x)}: \lambda \in \Lambda \}\) where the phase function \(\varphi \) is a Borel measurable which is not necessarily linear. A complete characterization of pairs \((\Lambda ,\varphi )\) for which \(E(\Lambda ,\varphi )\) is an orthogonal basis or

We study an approach to solving the phase retrieval problem as it arises in a phase-less imaging modality known as ptychography. In ptychography, small overlapping sections of an unknown sample (or signal, say \(x_0\in \mathbb {C}^{d}\)) are illuminated one at a time, often with a physical mask between the sample and light source. The corresponding measurements are the noisy magnitudes of the Fourier

Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016) and possibly severely ill-conditioned in

Based on hierarchical partitions, we provide the construction of Haar-type tight framelets on any compact set \(K\subseteq \mathbb {R}^d\). In particular, on the unit block \([0,1]^d\), such tight framelets can be built to be with adaptivity and directionality. We show that the adaptive directional Haar tight framelet systems can be used for digraph signal representations. Some examples are provided

We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid \(\varepsilon \)-approximations of arithmetic progressions. Some of these estimates are in terms of Szemerédi bounds. In particular, we answer a question of Fraser, Saito and Yu (IMRN 14:4419–4430, 2019) and considerably improve their bounds. We also show that Hausdorff dimension is equivalent

We prove that a variety of oscillatory and polynomial Carleson operators are uniformly bounded on the family of parameters under considerations. As a particular application of our techniques, we prove uniform bounds for oscillatory Carleson operators near a single scale version of the quadratic Carleson operator.

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.