Discrete Harmonic Analysis Research Articles

Pointwise convergence of multiple ergodic averages – PCMEA

Pointwise convergence of multiple ergodic averages – PCMEA Given an arbitrary measure preserving system we show that the multilinear ergodic averages sampled along an arbitrary number of sequences coming from a Hardy field converge pointwise almost everywhere. We aim to prove this for as wide a class of Hardy field functions as possible. To do so, we establish a long variational inequality along lacunary sequences which implies a maximal inequality, norm convergence, and pointwise convergence. By a transference argument it suffices prove this long variational inequality in the case that the measure preserving system is the integers. This reduction allows us to use tools from discrete harmonic analysis, additive combinatorics, and analytic number theory. We then give applications in areas such as upcrossings, equidistribution, and combinatorics.

Multiscale Coupling and the Maximum of {mathcal {P}}(phi )_2 Models on the Torus

We establish a coupling between the {mathcal {P}}(phi )_2 measure and the Gaussian free field on the two-dimensional unit torus at all spatial scales, quantified by probabilistic regularity estimates on the difference field. Our result includes the well-studied phi ^4_2 measure. The proof uses an exact correspondence between the Polchinski renormalisation group approach, which is used to define the coupling, and the Boué–Dupuis stochastic control representation for {mathcal {P}}(phi )_2. More precisely, we show that the difference field is obtained from a specific minimiser of the variational problem. This allows to transfer regularity estimates for the small-scales of minimisers, obtained using discrete harmonic analysis tools, to the difference field.As an application of the coupling, we prove that the maximum of the {mathcal {P}}(phi )_2 field on the discretised torus with mesh-size epsilon > 0 converges in distribution to a randomly shifted Gumbel distribution as epsilon rightarrow 0.

Discrete harmonic analysis associated with Jacobi expansions III: The Littlewood–Paley–Stein [formula omitted]-functions and the Laplace type multipliers

The research about harmonic analysis associated with Jacobi expansions carried out in Arenas et al. (2020) and Arenas et al. (2022) is continued in this paper. Given the operator J(α,β)=J(α,β)−I, where J(α,β) is the three-term recurrence relation for the normalized Jacobi polynomials and I is the identity operator, we define the corresponding Littlewood–Paley–Stein gk(α,β)-functions associated with it and we prove an equivalence of norms with weights for them. As a consequence, we deduce a result for Laplace type multipliers.

THE IONESCU–WAINGER MULTIPLIER THEOREM AND THE ADELES

The Ionescu–Wainger multiplier theorem establishes good L p bounds for Fourier multiplier operators localized to major arcs; it has become an indispensible tool in discrete harmonic analysis. We give a simplified proof of this theorem with more explicit constants (removing logarithmic losses that were present in previous versions of the theorem), and give a more general variant involving adelic Fourier multipliers. We also establish a closely related adelic sampling theorem that shows that ℓ p ( Z d ) norms of functions with Fourier transform supported on major arcs are comparable to the L p ( A Z d ) norm of their adelic counterparts.

Discrete Harmonic Analysis Associated with Jacobi Expansions II: the Riesz Transform

The present work is the continuation of our study (Arenas et al. J. Math. Anal. Appl. 490(123996), 21, 2020) on discrete harmonic analysis related to Jacobi expansions. The role of a Laplacian is played by the operator J(α,β) defined by the three-term recurrence relation for the normalised Jacobi polynomials. The main interest is to establish weighted inequalities for the Riesz transform associated with J(α,β). We make use of an appropriate discrete vector-valued local Calderon-Zygmund theory.

Discrete harmonic analysis associated with Jacobi expansions I: The heat semigroup

In this paper we commence the study of discrete harmonic analysis associated with Jacobi orthogonal polynomials of order (α,β). Particularly, we analyse the heat semigroup Wt(α,β), t≥0, related to the operator J(α,β)−I, where J(α,β) is the three-term recurrence relation for the normalised Jacobi polynomials and I is the identity operator. First, we prove the positivity of the operator Wt(α,β) under some suitable restrictions on the parameters α and β. In our main result, we investigate the mapping properties of the maximal operator defined by the heat semigroup in weighted ℓp-spaces using discrete vector-valued local Calderón-Zygmund theory. Moreover, we treat the Poisson semigroup by means of an appropriate subordination identity.

Book Review: Discrete harmonic analysis

Book Review: Discrete harmonic analysis

Discrete Harmonic Analysis Associated with Ultraspherical Expansions

In this paper we study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted lp-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by the difference operator $$ {\Delta}_{\lambda} f(n):=a_{n}^{\lambda} f(n+1)-2f(n)+a_{n-1}^{\lambda} f(n-1),\quad n\in \mathbb{N}, \lambda >0, $$ where $a_{n}^{\lambda } :=\{(2\lambda +n)(n+1)/[(n+\lambda )(n+1+\lambda )]\}^{1/2}$ , $n\in \mathbb {N}$ , and $a_{-1}^{\lambda }:=0$ . We also prove weighted lp-boundedness properties of transplantation operators associated with the system $\{\varphi _{n}^{\lambda } \}_{n\in \mathbb {N}}$ of ultraspherical functions, a family of eigenfunctions of Δλ. In order to show our results we previously establish a vector-valued local Calderon-Zygmund theorem in our discrete setting.

Restriction estimates of varepsilon -removal type for k-th powers and paraboloids

We obtain restriction estimates of varepsilon -removal type for the set of k-th powers of integers, and for discrete d-dimensional surfaces of the form {(n1,⋯,nd,n1k+⋯+ndk):|n1|,⋯,|nd|⩽N},\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\{ (n_1,\\dots ,n_d,n_1^k + \\cdots + n_d^k) \\,:\\, |n_1|,\\dots ,|n_d| \\leqslant N \\}, \\end{aligned}$$\\end{document}which we term ‘k-paraboloids’. For these surfaces, we obtain a satisfying range of exponents for large values of d, k. We also obtain estimates of varepsilon -removal type in the full supercritical range for k-th powers and for k-paraboloids of dimension d < k(k-2). We rely on a variety of techniques in discrete harmonic analysis originating in Bourgain’s works on the restriction theory of the squares and the discrete parabola.

The one weight inequality of integral operator induced by Hardy kernel

The integral operator induced by the reproducing kernel of H2(D) plays an important role in operator theory on the function spaces. In this paper, by the techniques of discrete harmonic analysis and dyadic methods, the one weight inequality of integral operator induced by Hardy kernel is established. We introduce R 2,2-weight and characterize it by a Carleson box maximal operator M 1/2. Meanwhile, we discuss the p - q boundedness of M 1/2.

Spectral Decomposition of Cyclic Operators in Discrete Harmonic Analysis

For the operators of the discrete Fourier transform, the discrete Vilenkin–Christenson transform, and all linear transpositions of the discrete Walsh transform, we obtain their spectral decompositions and calculate the dimensions of eigenspaces. For complex operators, namely, the discrete Fourier transform and the Vilenkin–Christenson transform, we obtain real projectors on eigenspaces. For the discrete Walsh transform, we consider in detail the Paley and Walsh orderings and a new ordering in which the matrices of operators are symmetric. For operators of linear transpositions of the discrete Walsh transforms with nonsymmetric matrices, we obtain a spectral decomposition with complex projectors on eigenspaces. We also present the Parseval frame for eigenspaces of the discrete Walsh transform.

Discrete harmonic analysis on a Weyl alcove

We introduce a representation of the double affine Hecke algebra at the critical level q=1 in terms of difference-reflection operators and use it to construct an explicit integrable discrete Laplacian on the Weyl alcove corresponding to an element in the center. The Laplacian in question is to be viewed as an integrable discretization of the conventional Laplace operator on Euclidean space perturbed by a delta-potential supported on the reflection hyperplanes of the affine Weyl group. The Bethe Ansatz method is employed to show that our discrete Laplacian and its commuting integrals are diagonalized by a finite-dimensional basis of periodic Macdonald spherical functions.

Taylor Series Expansion in Discrete Clifford Analysis

Discrete Clifford analysis is a discrete higher-dimensional function theory which corresponds simultaneously to a refinement of discrete harmonic analysis and to a discrete counterpart of Euclidean Clifford analysis. The discrete framework is based on a discrete Dirac operator that combines both forward and backward difference operators and on the splitting of the basis elements \(\mathbf{e}_j = \mathbf{e}_j^+ + \mathbf{e}_j^-\) into forward and backward basis elements \(\mathbf{e}_j^\pm \). For a systematic development of this function theory, an indispensable tool is the Taylor series expansion, which decomposes a discrete (monogenic) function in terms of discrete homogeneous (monogenic) building blocks. The latter are the so-called discrete Fueter polynomials. For a discrete function, the authors assumed a series expansion which is formally equivalent to the Taylor series expansion in Euclidean Clifford analysis; however, attention needed to be paid to the geometrical conditions on the domain of the function, the convergence and the equivalence to the given discrete function. We furthermore applied the theory to discrete delta functions and investigated the connection with Shannon sampling theorem (Bell Sys Tech J 27:379–423, 1948). We found that any discrete function admits a series expansion into discrete homogeneous polynomials and any discrete monogenic function admits a Taylor series expansion in terms of the discrete Fueter polynomials, i.e. discrete homogeneous monogenic polynomials. Although formally the discrete Taylor series expansion of a function resembles the continuous Taylor series expansion, the main difference is that there is no restriction on discrete functions to be represented as infinite series of discrete homogeneous polynomials. Finally, since the continuous expansion of the Taylor series expansion of discrete delta functions is a sinc function, the discrete Taylor series expansion lays a link with Shannon sampling.

Spectral measures and Cuntz algebras

We consider a family of measures μ \mu supported in R d \mathbb {R}^d and generated in the sense of Hutchinson by a finite family of affine transformations. It is known that interesting sub-families of these measures allow for an orthogonal basis in L 2 ( μ ) L^2(\mu ) consisting of complex exponentials, i.e., a Fourier basis corresponding to a discrete subset Γ \Gamma in R d \mathbb {R}^d . Here we offer two computational devices for understanding the interplay between the possibilities for such sets Γ \Gamma (spectrum) and the measures μ \mu themselves. Our computations combine the following three tools: duality, discrete harmonic analysis, and dynamical systems based on representations of the Cuntz C ∗ C^* -algebras O N \mathcal O_N .

Limit theorems of the theory of discrete periodic splines

It is known from the discrete harmonic analysis that the interpolation problem with equidistant interpolation points has a unique solution. If the right-hand sides in the interpolation problem are fixed, the spline depends on two parameters: the spline order and the number of points located between neighboring interpolation points. We find explicit expressions for the limits of interpolation spllines with respect to each parameter separately and show that both repeated limits exist. We also prove that these repeated limits are equal and their value is an interpolation trigonometric polynomial. Bibliography: 10 titles. Illustrations: 2 figures.

Lattice instantons in the large dimension limit

We give a lattice construction of the discretizations of the topologically nontrivial maps S2n−1→S n . For n=1, 2, 4, 8, these are the Hopf maps. The construction, based on Barnes-Wall lattices, Reed-Muller error-correcting codes, and Hadamard matrices, generalizes to n=2 i for i any integer. Manton's result for the cases n=2 and 4 (i.e., the monopole and instanton) are included. We argue that discrete harmonic analysis will be exact in the infinite dimension limit.

Changes in the pulse wave form with flow through vessels with repetitive saccular dilatations and stenosis.

Several patients with fibroplasia of the renal arteries without significant stenosis were found to have surgically reversible renal hypertension. The hypothesis is advanced that receptors in the kidney may respond to altered wave form by causing an increase in blood pressure. Various models of normal and pathological renal arteries were made and tested in a pulse duplicator. A discrete harmonic analysis was made of input and output waves. The results indicate that nodular, non-stenotic models significantly alter the systolic rise in perfusion pressures to the kidney. Therefore, irregularity of vessel lumen without stenosis may be sufficient to alter organ function.

Digital synthesis of correlated stationary noise

In this note we propose a method of generating stationary noise with a prescribed auto-covariance function by digital methods. The need for such a technique often arises in testing the performance of data processing and engineering systems, where inputs corrupted with correlated noise (of a known form) are required. The technique is quite simple and produces strict-sense stationary noise which agrees approximately with R ( τ ), the prescribed auto-covariance function (acf), over an interval [- T 0 , T 0 ]. The method consists of approximating the spectral density by a periodic process with spectral lines, and then synthesizing the periodic noise with random phases and appropriate amplitudes. In order to simplify discussion of the statistical properties of the noise generated, the technique is first presented in terms of exact harmonic analysis. In practice, discrete harmonic analysis as presented in the third section is used.