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