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In the paper (Greenblatt in J Funct Anal, https://doi.org/10.1016/j.jfa.2018.05.014 , 2018) the author proved $$L^p$$ Sobolev regularity results for averaging operators over hypersurfaces and connected them to associated Newton polyhedra. In this paper, we use rather different resolution of singularities techniques along with oscillatory integral methods applied to surface measure Fourier transforms to prove $$L^p$$ Sobolev regularity results for a class of averaging operators over surfaces which can be of any codimension.

Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. In this paper, we consider phaseless sampling and reconstruction of real-valued signals in a high-dimensional shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. The determination of a signal in a shift-invariant space, up to a sign, by its magnitude measurements on the whole Euclidean space has been shown in the literature to be equivalent to its nonseparability. In this paper, we introduce an undirected graph associated with the signal in a shift-invariant space and use connectivity of the graph to characterize nonseparability of the signal. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that nonseparable signals in the shift-invariant space can be reconstructed in a stable way from their phaseless samples taken on that set. In this paper, we also propose a reconstruction algorithm which provides an approximation to the original signal when its noisy phaseless samples are available only. Finally, numerical simulations are performed to demonstrate the robustness of the proposed algorithm to reconstruct box spline signals from their noisy phaseless samples.

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 previous papers in this series. We prove that these spaces are perfect sequence spaces. As a consequence we describe the tensor structure of sequential mappings on the spaces of smooth type functions and characterise their adjoint mappings. As an application we prove the universality of the spaces of smooth type functions on compact manifolds without boundary., 23 pages

In this paper, we study the mean value property for both the harmonic functions and the functions in the domain of the Laplacian on the tetrahedral Sierpinski gasket. This paper is a continuation of the work of Strichartz and the first author (Qiu and Strichartz, J Fourier Anal Appl 19:943–966, 2013)where the same property on p.c.f. self-similar sets with Dihedral-3 symmetry was considered.

Given a domain $$\varOmega \subset {\mathbb {R}}^d$$ with positive and finite Lebesgue measure and a discrete set $$\varLambda \subset {\mathbb {R}}^d$$ , we say that $$(\varOmega , \varLambda )$$ is a frame spectral pair if the set of exponential functions $${\mathcal {E}}(\varLambda ):=\{e^{2\pi i \lambda \cdot x}: \lambda \in \varLambda \}$$ is a frame for $$L^2(\varOmega )$$ . Special cases of frames include Riesz bases and orthogonal bases. In the finite setting $${\mathbb {Z}}_N^d$$ , $$d, N\ge 1$$ , a frame spectral pair can be similarly defined. In this paper we show how to construct and obtain new classes of frame spectral pairs in $${\mathbb {R}}^d$$ by “adding” a frame spectral pair in $${\mathbb {R}}^{d}$$ to a frame spectral pair in $${\mathbb {Z}}_N^d$$ . Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. We also remark on the link between the spectral property of a domain and sampling theory.

We consider bilinear pseudo-differential operators with symbols in the bilinear H\"ormander class, $BS_{\rho, \rho}^m$, $m \in \mathbb{R}$, $0 \leq \rho < 1$. The aim of this paper is to discuss low regularity conditions for symbols to assure the boundedness from $L^2 \times L^2$ to $h^1$ and from $L^2 \times bmo$ to $L^2$., Comment: 40 pages

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

In this paper, we consider block thresholding wavelet estimators of spatial regression functions on stationary Gaussian random fields observed over a rectangular domain indexed with $${{\mathbb {Z}}}^2$$, whose covariance function is assumed to satisfy some weak condition. We investigate their asymptotic rates of convergence under the mean integrated squared error when spatial regression functions belong to a large range of Besov function classes $$B^{s}_{p,q}({{\mathbb {R}}}^2)$$. To do this, we derived a result showing the discrepancy between empirical wavelet coefficients and true wavelet coefficients is within certain small rate across above Besov function classes. Based on that, we are able to determine the rates of convergence of our estimators and the supremum norm error over above function classes. The obtained rates of convergence correspond to those established in the standard univariate nonparametric regression with short-range dependence. Therefore, those rates could be considered as sharp as possible. A mild simulation study is carried out to examine the finite sample performance of the proposed estimates.

In this paper the generalized localization principle for the spherical partial sums of the multiple Fourier series in the $$L_2$$-class is proved, that is, if $$f\in L_2({\mathbb {T}}^N)$$ and $$f=0$$ on an open set $$\Omega \subset {\mathbb {T}}^N$$, then it is shown that the spherical partial sums of this function converge to zero almost-everywhere on $$\Omega $$. It has been previously known that the generalized localization is not valid in $$L_p({\mathbb {T}}^N)$$ when $$1\le p

This paper provides analysis for convergence of the singular value thresholding algorithm for solving matrix completion and affine rank minimization problems arising from compressive sensing, signal processing, machine learning, and related topics. A necessary and sufficient condition for the convergence of the algorithm with respect to the Bregman distance is given in terms of the step size sequence $$\{\delta _k\}_{k\in \mathbb {N}}$$ as $$\sum _{k=1}^{\infty }\delta _k=\infty $$. Concrete convergence rates in terms of Bregman distances and Frobenius norms of matrices are presented. Our novel analysis is carried out by giving an identity for the Bregman distance as the excess gradient descent objective function values and an error decomposition after viewing the algorithm as a mirror descent algorithm with a non-differentiable mirror map.

In this paper, a p-adic analogue of the wave equation with Lipschitz source is considered. Since it is hard to arrive the solution of the problem, we propose a regularized method to solve the probl ...

In this paper, we prove that the maximal inequality $$\begin{aligned} \big \Vert \sup _{|t|\tfrac{1}{2}$$ with $$\Omega =\{(x,y)\in \mathbb {R}^2\mid x>0\}$$ and $$\Delta _D=\partial _x^2+(1+x)\partial _y^2$$ . As a direct application, we obtain the pointwise convergence for the free Schrodinger equation $$i\partial _tu+\Delta _D u=0$$ with initial data $$u(0)=f$$ inside strictly convex domain.

Polygons can be seen as closed parameterized curves. Their parameterizations can be chosen as continuous, piecewise linear, periodic functions. Such functions possess a convergent Fourier series. Often polygons are classified with Fourier descriptors defined via Fourier coefficients of the parameterization. This fact motivates the discussion of the approximation error of Fourier partial sums of piecewise linear functions. More generally, the paper investigates convergence rates for periodic splines using elementary techniques of calculus. For example, such splines are used as curve parameterizations for active contours. Error bounds are shown to be best possible. An interesting effect is that the convergence rate at knots is different for odd and even degrees of piecewise polynomials. The slower rate for polynomials of odd degree can be used to detect dominant corners of contours.

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 the Bessel Riesz transforms, non-tangential and radial maximal functions defined via Poisson and heat semigroups, based on the atomic decomposition, the extension of Merryfield's result which connects the product non-tangential maximal function and area function, and on the grand maximal function technique which connects the product non-tangential and radial maximal function. We then obtain directly the decomposition of the product BMO space associated with Bessel operators. These results are a first extension for product Hardy and BMO associated to a differential operator other than the Laplacian and are a major step beyond the Chang--Fefferman setting., Comment: 46 pages

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 yield the sharp bounds for the associated frames, in a broad setting that generalizes several existing constructions. We also examine how Gabor-type frames behave for signals defined on Cayley graphs by exploiting the representation theory of the underlying group. We explore how natural classes of translations can be constructed for Cayley graphs, and how the choice of an eigenbasis can significantly impact the properties of the resulting translation operators and frames on the graph.

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 covers the integrability of the d-dimensional Fourier transform of the radial functions.

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 transform coefficients resulting from the pointwise product of the mask and the sample. The goal is to recover the original signal from such measurements. The algorithmic framework we study herein relies on first inverting a linear system of equations to recover a fraction of the entries in $$x_0 x_0^*$$ and then using non-linear techniques to recover the magnitudes and phases of the entries of $$x_0$$ . Thus, this paper’s contributions are three-fold. First, focusing on the linear part, it expands the theory studying which measurement schemes (i.e., masks, shifts of the sample) yield invertible linear systems, including an analysis of the conditioning of the resulting systems. Second, it analyzes a class of improved magnitude recovery algorithms and, third, it proposes and analyzes algorithms for phase recovery in the ptychographic setting where large shifts—up to $$50\%$$ the size of the mask—are permitted.

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 as focus on the case where they form tight frames for the underlying space. We then give some constructions of regular two-distance sets, in particular, two-distance frames, both tight and non-tight cases. We also supply an example of a non-tight maximal two-distance frame. Connections among two-distance sets, equiangular lines, and quasi-symmetric designs are also discussed. For instance, we give a sufficient condition for constructing sets of equiangular lines from regular two-distance sets, especially from quasi-symmetric designs satisfying certain conditions.

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 measure $${\bar{\mu }}$$ (Dutkay and Jorgensen in Rev. Mat. Iberoamericana 22(1):131–180, 2006). Unitary dilation and translation operators can be defined on $$L^2({\mathcal {R}}, {\bar{\mu }})$$ which satisfy the Baumslag–Solitar group relation, and give rise to a filtration of the Hilbert space $$L^2({\mathcal {R}}, {\bar{\mu }})$$ called a multi-resolution analysis (Dutkay and Jorgensen 2006). The low pass filter function corresponding to this construction can be used to produce a measure, m, on a compact topological group called the 3-solenoid, denoted $${\mathcal {S}}_3$$ (Dutkay in Trans Am Math Soc 358(12):5271–5291, 2006). The Hilbert space $$L^2({\mathcal {S}}_3, m)$$ also admits a unitary representation of the Baumslag–Solitar group, and there exists a generalized Fourier transform between $$L^2({\mathcal {R}}, {\bar{\mu }})$$ and $$L^2({\mathcal {S}}_3,m)$$ (Dutkay 2006). In this paper, we build off of Jorgensen and Dutkay’s work to show that the unitary operators on $$L^2({\mathcal {S}}_3,m)$$ mentioned above are related to each other via a family of partial isometries, which satisfy properties resembling the Cuntz relations.

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}

The focus of this paper is on the mathematical construction of a transform that is invariant to a finite rotation group and is stable to small perturbations. A key step in our theory lies in the construction of directionally sensitive functions that are partially generated by rotations. We call such a family a rotational uniform covering frame and by studying rotations of the frame, we derive the desired operator, which we call the rotational Fourier scattering transform. We prove that the transformation is rotationally invariant to a finite rotation group, is bounded above and below, is non-expansive, and contracts small translations and additive diffeomorphisms. To address the numerical aspects of this theory, we also construct digital versions of the frame and show how to faithfully truncate the transform. We also discuss connections between this new family of directional representations with previously constructed ones.

In this paper, we study the convergence of adaptive Fourier sums for real-valued $$2\pi $$ -periodic functions. For this purpose, we approximate the sequence of classical Fourier coefficients by a short exponential sum with a pre-defined number of $$N+1$$ terms. The obtained approximation can be interpreted as an adaptive N-th Fourier sum with respect to the orthogonal Takenaka-Malmquist basis. Using the theoretical results on rational approximation in Hardy spaces and on the decay of singular values of special infinite Hankel matrices, we show that adaptive Fourier sums can converge essentially faster than classical Fourier sums for a large class of functions. Further, we derive an algorithm to compute almost optimal adaptive Fourier sums. Our numerical results show that the significantly better convergence behavior of adaptive Fourier sums for optimally chosen basis elements can also be achieved in practice. For comparison, we also provide a greedy algorithm to determine an adaptive Fourier sum. This algorithm requires less computational effort but yields essentially slower convergence.

Given a measure μ of polynomial growth, we refine a deep result by David and Mattila to construct an atomic martingale filtration of supp (μ) which provides the right framework for a dyadic form of nondoubling harmonic analysis. Despite this filtration being highly irregular, its atoms are comparable to balls in the given metric—which in turn are all doubling—and satisfy a weaker but crucial form of regularity. Our dyadic formulation is effective to address three basic questions:(i)A dyadic form of Tolsa’s RBMO space which contains it.(ii)Lerner’s domination and A-type bounds for nondoubling measures.(iii)A noncommutative form of nonhomogeneous Calderón–Zygmund theory. Our martingale RBMO space preserves the crucial properties of Tolsa’s original definition and reveals its interpolation behavior with the L scale in the category of Banach spaces, unknown so far. On the other hand, due to some known obstructions for Haar shifts and related concepts over nondoubling measures, our pointwise domination theorem via sparsity naturally deviates from its doubling analogue. In a different direction, matrix-valued harmonic analysis over noncommutative L spaces has recently produced profound applications. Our analogue for nondoubling measures was expected for quite some time. Finally, we also find a dyadic form of the Calderón–Zygmund decomposition which unifies those by Tolsa and López-Sánchez/Martell/Parcet., The authors thank Xavier Tolsa for bringing to our attention David and Mattila’s construction and for several fruitful discussions on the content of this paper. Both authors were partially supported by CSIC Project PIE 201650E030 and also by ICMAT Severo Ochoa Grant SEV-2015-0554, and the first named author was supported in part by ERC Grant 32501.

A discrete set $$\Lambda \subseteq {\mathbb {R}}^d$$ is called a spectrum for the probability measure $$\mu $$ if the family of functions $$\{e^{2 \pi i \langle \lambda ,\, x\rangle }: \lambda \in \Lambda \}$$ forms an orthonormal basis for the Hilbert space $$L^2(\mu ).$$ In this paper, we will give a characterization of the spectra of self-affine measures generated by compatible pairs in $${\mathbb {R}}^d.$$ As an application, we show, for the Cantor measure $$\mu _{b,~q}$$ on $${\mathbb {R}}$$ with consecutive digit set and any integer $$p\in {\mathbb {Z}}$$ with $$\gcd (p,\,q)=1,$$ that the set $$\begin{aligned} \{\Lambda \subseteq {\mathbb {R}}: \Lambda \ \hbox {and} \ p\Lambda \ \text {are both spectra for }\mu _{b,~q}\text { and }0 \in \Lambda \} \end{aligned}$$ has the cardinality of the continuum.

The fractional Fourier transform $$F_{\theta }(w)$$ with an angle $$\theta $$ of a function f(t) is a generalization of the standard Fourier transform and reduces to it when $$\theta =\pi /2. $$ It has many applications in signal processing and optics because of its close relations with a number of time-frequency representations. It is known that the Wigner distribution of the fractional Fourier transform $$F_{\theta }(w)$$ may be obtained from the Wigner distribution of f by a two-dimensional rotation with the angle $$\theta $$ in the $$t-w$$ plane The fractional Fourier transform has been extended to higher dimensions by taking the tensor product of one-dimensional transforms; hence, resulting in a transform in several but separable variables. It has been shown that the Wigner distribution of the two-dimensional fractional Fourier transform $$F_{\theta ,\phi }(v,w)$$ may be obtained from the Wigner distribution of f(x, y) by a simple four-dimensional rotation with the angle $$\theta $$ in the $$x-y$$ plane and the angle $$\phi $$ in the $$v-w$$ plane. The aim of this paper is two-fold: (1) To introduce a new definition of the two-dimensional fractional Fourier transform that is not a tensor product of two copies of one-dimensional transforms. The new transform, which is more general than the one that exists in the literature, uses a relatively new family of Hermite functions, known as Hermite functions of two complex variables. (2) To give an explicit matrix representation of a four-dimensional rotation that verifies that the Wigner distribution of the new two-dimensional fractional Fourier transform $$F_{\theta ,\phi }(v,w)$$ may be obtained from the Wigner distribution of f(x, y) by a four-dimensional rotation. The matrix representation is more general than the one for the tensor product case and it corresponds to a four-dimensional rotation with two planes of rotations, one with the angle $$(\theta +\phi )/2$$ and the other with the angle $$(\theta -\phi )/2$$ .

Recently, a large deviation multifractal formalism based on histograms of wavelet leader coefficients, compared to some other wavelet-based formalisms, was proved to be efficient for uniform Holder functions. In this paper, we extend this efficiency for non-uniform Holder functions. We first obtain optimal bounds for both wavelet and wavelet leader histograms for all functions in the critical Besov space $$B^{m/t,q}_t({\mathbb {T}})$$ , where $$t,q>0$$ and $${\mathbb {T}}$$ is the unit torus of $$\mathbb {R}^m$$ . We then compute these histograms for quasi-all functions in $$B^{m/t,q}_t({\mathbb {T}})$$ , in the sense of Baire Category. Although, increasing parts of these histograms have increasing visibility, they coincide only if $$00$$ , however wavelet histograms method covers it only if $$0

The paper introduces a variable exponent space X which has in common with $$L^{\infty }([0,1])$$ the property that the space C([0, 1]) of continuous functions on [0, 1] is a closed linear subspace in it. The associate space of X contains both the Kolmogorov and the Marcinkiewicz examples of functions in $$L^{1}$$ with a.e. divergent Fourier series.

In this paper we study trigonometric series with general monotone coefficients, i.e., satisfying $$\begin{aligned} \sum \limits _{k=n}^{2n} |a_k - a_{k+1}| \le C \sum \limits _{k=[{n}/{\gamma }]}^{[\gamma n]} \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$ for some $$C \ge 1$$ and $$\gamma >1$$ . We first prove the Lebesgue-type inequalities for such series $$\begin{aligned} n|a_n|\le C \omega (f,1/n). \end{aligned}$$ Moreover, we obtain necessary and sufficient conditions for the sum of such series to belong to the generalized Lipschitz, Nikolskii, and Zygmund spaces. We also prove similar results for trigonometric series with weak monotone coefficients, i.e., satisfying $$\begin{aligned} |a_n | \le C \sum \limits _{k=[{n}/{\gamma }]}^{\infty } \frac{|a_k|}{k}, \quad n\in \mathbb {N}, \end{aligned}$$ for some $$C \ge 1$$ and $$\gamma >1$$ . Sharpness of the obtained results is given. Finally, we study the asymptotic results of Salem–Hardy type.

The existence of sets supporting a Borel measure such that its Fourier transform tends to zero at infinity can be traced back to the problem of uniqueness of trigonometric series, studied extensively by Cantor. Given $$\alpha \in (0, 1)$$ , Beurling asked if there exists a subset of the real line of Hausdorff dimension $$\alpha $$ supporting a Borel measure whose Fourier transform converges to zero at infinity with rate $$\alpha /2$$ . Salem answered the question in the affirmative and such sets are now called Salem sets or rounded sets. Kahane showed that images of compact sets by fractional Brownian motion are Salem sets and this was recently extended to Gaussian random fields with stationary increments and to multi-parameter Brownian sheets. He asked if the level sets of fractional Brownian motion are also Salem sets and the problem has remained open since. This paper answers Kahane’s question in the affirmative. The argument is based on the study of oscillatory integrals with non-smooth amplitudes and new properties of the generalised Euler spiral which have independent interest.

The main purpose of this paper is to prove global well-posedness in some scale-invariant weighted Besov spaces and Strichartz spaces for the nonlinear wave equation with competing nonlinearities which have time depending potentials decaying at infinity. We discuss also some new frequency localized estimates in non-isotropic Lebesgue spaces.

This paper deals with a theoretical mathematical analysis of a Cauchy problem for the time-fractional diffusion-wave equation in the upper half-plane, \(x\in \mathbb {R}\), \(t\in \mathbb {R}^+\), where the Caputo fractional derivative of order \(\alpha \in \left( 0,2\right) \) is considered. An explicit solution to this Cauchy problem is obtained via separation of variables. A first proof of the validity of the obtained results is provided for a certain kind of initial conditions. Throughout this work a new expression of the solution to this problem and its utility for carrying out rigurous proofs are presented. Finally, several new properties of the solution are obtained.

Both wavelet and atomic decomposition techniques are essential tools in the study of function spaces nowadays, but they both have their advantages and disadvantages. The celebrated bridge between both concepts was given by the compactly supported Daubechies wavelets which can be interpreted as atoms. In this paper we deal with the converse direction, that is, we present a fairly general approach how to construct compactly supported wavelets when an atomic decomposition is known already. The main idea is Taylor’s expansion combined with our new, so-called $$\varkappa $$ -convergence assumption in the admitted sequence spaces. We finally exemplify our main result and collect some known and new settings where such a wavelet decomposition is obtained, e.g., in spaces of Besov or Triebel–Lizorkin type with a doubling weight.

New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. The main result of the paper is an asymptotic formula for the cosine Fourier transform. Such relations have previously been known only for the sine Fourier transform. For this, not only a different space is considered but also a new way of proving such theorems is applied. Interrelations of various function spaces are studied in this context. The obtained results are used for obtaining completely new results on the integrability of trigonometric series.

In this paper, bilinear pseudo-differential operators with symbols in the bilinear Hormander symbol class $$BS^{m}_{1,1}$$ on Triebel–Lizorkin spaces are discussed. As a result, we can obtain the Kato–Ponce inequality in local Hardy spaces.

In this paper we consider the following problem of phase retrieval: given a collection of real-valued band-limited functions $$\{\psi _{\lambda }\}_{\lambda \in \Lambda }\subset L^2(\mathbb {R}^d)$$ that constitutes a semi-discrete frame, we ask whether any real-valued function $$f \in L^2(\mathbb {R}^d)$$ can be uniquely recovered from its unsigned convolutions $${\{|f *\psi _\lambda |\}_{\lambda \in \Lambda }}$$ . We find that under some mild assumptions on the semi-discrete frame and if f has exponential decay at $$\infty $$ , it suffices to know $$|f *\psi _\lambda |$$ on suitably fine lattices to uniquely determine f (up to a global sign factor). We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of $$L^2(\mathbb {R}^d)$$ , $$d=1,2$$ , we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.

In this paper we study singular oscillatory integrals with a nonlinear phase function. We prove estimates of \(L^2 \rightarrow L^2\) and \(L^p\rightarrow L^p\) type.

The maximum selection principle allows to give expansions, in an adaptive way, of functions in the Hardy space \(\mathbf H_2\) of the disk in terms of Blaschke products. The expansion is specific to the given function. Blaschke factors and products have counterparts in the unit ball of \(\mathbb C^N\), and this fact allows us to extend in the present paper the maximum selection principle to the case of functions in the Drury–Arveson space of functions analytic in the unit ball of \(\mathbb C^N\). This will give rise to an algorithm which is a variation in this higher dimensional case of the greedy algorithm. We also introduce infinite Blaschke products in this setting and study their convergence.

In this paper, we are going to describe the first and second complex interpolations of closed subspaces of Morrey spaces, based on our previous results in [11]. Our results will be general enough because we are going to deal with abstract linear subspaces satisfying the lattice condition only. We also consider the closure in Morrey spaces on bounded domains of the set of smooth functions with compact support. Here, we do not require the smoothness condition on domains.

This paper first shows that the Riemann localisation property holds for the Fourier-Laplace series partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimension. By Riemann localisation on the sphere \(\mathbb {S}^{d}\subset \mathbb {R}^{d+1}\), \(d\ge 2\), we mean that for a suitable subset X of \(\mathbb {L}_{p}(\mathbb {S}^{d})\), \(1\le p\le \infty \), the \(\mathbb {L}_{p}\)-norm of the Fourier local convolution of \(f\in X\) converges to zero as the degree goes to infinity. The Fourier local convolution of f at \(\mathbf {x}\in \mathbb {S}^{d}\) is the Fourier convolution with a modified version of f obtained by replacing values of f by zero on a neighbourhood of \(\mathbf {x}\). The failure of Riemann localisation for \(d>2\) can be overcome by considering a filtered version: we prove that for a sphere of any dimension and sufficiently smooth filter the corresponding local convolution always has the Riemann localisation property. Key tools are asymptotic estimates of the Fourier and filtered kernels.

Sparsity-driven image recovery methods assume that images of interest can be sparsely approximated under some suitable system. As discontinuities of 2D images often show geometrical regularities along image edges with different orientations, an effective sparsifying system should have high orientation selectivity. There have been enduring efforts on constructing discrete frames and tight frames for improving the orientation selectivity of tensor product real-valued wavelet bases/frames. In this paper, we studied the general theory of discrete Gabor frames for finite signals, and constructed a class of discrete 2D Gabor frames with optimal orientation selectivity for sparse image approximation. Besides high orientation selectivity, the proposed multi-scale discrete 2D Gabor frames also allow us to simultaneously exploit sparsity prior of cartoon image regions in spatial domain and the sparsity prior of textural image regions in local frequency domain. Using a composite sparse image model, we showed the advantages of the proposed discrete Gabor frames over the existing wavelet frames in several image recovery experiments.

In this paper, we develop a new method based on the Laplace transform to study the Clifford-Fourier transform. First, the kernel of the Clifford-Fourier transform in the Laplace domain is obtained. When the dimension is even, the inverse Laplace transform may be computed and we obtain the explicit expression for the kernel as a finite sum of Bessel functions. We equally obtain the plane wave decomposition and find new integral representations for the kernel in all dimensions. Finally we define and compute the formal generating function for the even dimensional kernels., 15 pages

The Poisson summation formula (PSF) describes the equivalence between the sampling of an analog signal and the periodization of its frequency spectrum. In engineering textbooks, the PSF is usually stated formally without explicit conditions on the signal for the formula to hold. By contrast, in the mathematics literature, the PSF is commonly stated and proven in the pointwise sense for various types of \(L_1\) signals. This \(L_1\) assumption is, however, too restrictive for many signal-processing tasks that demand the sampling of possibly growing signals. In this paper, we present two generalized versions of the PSF for d-dimensional signals of polynomial growth. In the first generalization, we show that the PSF holds in the space of tempered distributions for every continuous and polynomially growing signal. In the second generalization, the PSF holds in a particular negative-order Sobolev space if we further require that \(d/2+\varepsilon \) derivatives of the signal are bounded by some polynomial in the \(L_2\) sense.

We construct a polyanalytic extension of analytic (Poisson) wavelet frames, leading to a new system of wavelet superframes. Superframes have been introduced in signal analysis as a tool for the multiplexing of signals—encoding several signals as a single one with the purpose of sharing a communication channel. In this paper, a new system of vector-valued wavelets is constructed by selecting as elements of the analyzing vector the first n elements from an explicit orthogonal basis of the space of admissible functions depending on a parameter \(\alpha \). We show that if the resulting discrete affine system indexed by the set \(\Gamma (a,b)=\{a^{m}bk,a^{m}\}_{k,m\in \mathbb {Z} }\) is a wavelet superframe, then the estimate $$\begin{aligned} b\log a

This paper gives a survey of methods for the construction of space–frequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general context, the notion of frequency is defined using the spectrum of a distinguished differential operator on the manifold, typically the Laplace–Beltrami operator. Our exposition starts with the case of the real line, which serves as motivation and blueprint for the material in the subsequent sections. After the discussion of the real line, our presentation starts out in the most abstract setting proving rather general sampling-type results for appropriately defined Paley–Wiener vectors in Hilbert spaces. These results allow a handy construction of Paley–Wiener frames in \(L_2(\mathbf {M})\), for a Riemann manifold of bounded geometry, essentially by taking a partition of unity in frequency domain. The discretization of the associated integral kernels then gives rise to frames consisting of smooth functions in \(L_2(\mathbf {M})\), with fast decay in space and frequency. These frames are used to introduce new norms in corresponding Besov spaces on \(\mathbf {M}\). For compact Riemannian manifolds the theory extends to \(L_p\) and associated Besov spaces. Moreover, for compact homogeneous manifolds, one obtains the so-called product property for eigenfunctions of certain operators and proves cubature formulae with positive coefficients which allow to construct Parseval frames that characterize Besov spaces in terms of coefficient decay. The general theory is exemplified with the help of various concrete and relevant examples which include the unit sphere and the Poincare half plane.

It is well-known that curvelets provide optimal approximations for so-called cartoon images which are defined as piecewise $$C^2$$ -functions, separated by a $$C^2$$ singularity curve. In this paper, we consider the more general case of piecewise $$C^\beta $$ -functions, separated by a $$C^\beta $$ singularity curve for $$\beta \in (1,2]$$ . We first prove a benchmark result for the possibly achievable best N-term approximation rate for this more general signal model. Then we introduce what we call $$\alpha $$ -curvelets, which are systems that interpolate between wavelet systems on the one hand ( $$\alpha = 1$$ ) and curvelet systems on the other hand ( $$\alpha = \frac{1}{2}$$ ). Our main result states that those frames achieve this optimal rate for $$\alpha = \frac{1}{\beta }$$ , up to $$\log $$ -factors.

In the present paper, we study various Erdős type geometric problems in the setting of the integers modulo q, where \(q=p^l\) is an odd prime power. More precisely, we prove certain results about the distribution of triangles and triangle areas among the points of \(E\subset \mathbb {Z}_q^2\).

In this paper we propose to develop harmonic analysis on the Poincare ball $${{\mathbb {B}}_{t}^{n}}$$ , a model of the $$n$$ -dimensional real hyperbolic space. The Poincare ball $${{\mathbb {B}}_{t}^{n}}$$ is the open ball of the Euclidean $$n$$ -space $$\mathbb {R}^n$$ with radius $$t >0$$ , centered at the origin of $$\mathbb {R}^n$$ and equipped with Mobius addition, thus forming a Mobius gyrogroup where Mobius addition in the ball plays the role of vector addition in $$\mathbb {R}^n.$$ For any $$t>0$$ and an arbitrary parameter $$\sigma \in \mathbb {R}$$ we study the $$(\sigma ,t)$$ -translation, the $$(\sigma ,t)$$ -convolution, the eigenfunctions of the $$(\sigma ,t)$$ -Laplace–Beltrami operator, the $$(\sigma ,t)$$ -Helgason Fourier transform, its inverse transform and the associated Plancherel’s Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $$t \rightarrow +\infty $$ the resulting hyperbolic harmonic analysis on $${{\mathbb {B}}_{t}^{n}}$$ tends to the standard Euclidean harmonic analysis on $$\mathbb {R}^n,$$ thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $${{\mathbb {B}}_{t}^{n}}$$ .

The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator., 34 pages

Ramanujan’s Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several definite integrals and power series, which explains why it is referred to as the “Master Theorem”. In this paper we prove an analogue of Ramanujan’s Master theorem for the hypergeometric Fourier transform associated with root systems. This theorem generalizes to arbitrary positive multiplicity functions the results previously proven by the same authors for the spherical Fourier transform on semisimple Riemannian symmetric spaces.

In this paper we study the Cauchy problem for new classes of parabolic type pseudodifferential equations over the rings of finite adeles and adeles. We show that the adelic topology is metrizable and give an explicit metric. We find explicit representations of the fundamental solutions (the heat kernels). These fundamental solutions are transition functions of Markov processes which are adelic analogues of the Archimedean Brownian motion. We show that the Cauchy problems for these equations are well-posed and find explicit representations of the evolution semigroup and formulas for the solutions of homogeneous and non-homogeneous equations., 36 pages