Marcinkiewicz-Zygmund Inequality Research Articles

An abstract approach to Marcinkiewicz-Zygmund inequalities for approximation and quadrature in modulation spaces

We study the approximation and quadrature error of points that satisfy Marcinkiewicz-Zygmund inequalities. First, we investigate the use of Marcinkiewicz-Zygmund inequalities in an abstract Hilbert space for an abstract approximation and quadrature rule. The setting is then specified to Sobolev spaces induced by Freud weights e − 2 σ | x | α e^{-2\sigma |x|^\alpha } with α > 1 \alpha >1 and σ > 0 \sigma >0 , and we derive specific bounds for the approximation and quadrature error. For the Gaussian weight e − 2 π x 2 e^{-2\pi x^2} , we verify that the Sobolev spaces essentially coincide with a specific class of modulation spaces that are well known in (time-frequency) analysis.

Constructive subsampling of finite frames with applications in optimal function recovery

In this paper we present new constructive methods, random and deterministic, for the efficient subsampling of finite frames in Cm. Based on a suitable random subsampling strategy, we are able to extract from any given frame with bounds 0<A≤B<∞ (and condition B/A) a similarly conditioned reweighted subframe consisting of merely O(mlog⁡m) elements. Further, utilizing a deterministic subsampling method based on principles developed by Batson, Spielman, and Srivastava to control the spectrum of sums of Hermitian rank-1 matrices, we are able to reduce the number of elements to O(m) (with a constant close to one). By controlling the weights via a preconditioning step, we can, in addition, preserve the lower frame bound in the unweighted case. This permits the derivation of new quasi-optimal unweighted (left) Marcinkiewicz-Zygmund inequalities for L2(D,ν) with constructible node sets of size O(m) for m-dimensional subspaces of bounded functions. Those can be applied e.g. for (plain) least-squares sampling reconstruction of functions, where we obtain new quasi-optimal results avoiding the Kadison-Singer theorem. Numerical experiments indicate the applicability of our results.

Approximation and Localized Polynomial Frame on Double Hyperbolic and Conic Domains

We study approximation and localized polynomial frames on a bounded double hyperbolic or conic surface and the domain bounded by such a surface and hyperplanes. The main work follows the framework developed recently in Xu (J Funct Anal 281(12):109257, 2021) for homogeneous spaces that are assumed to contain highly localized kernels constructed via a family of orthogonal polynomials. The existence of such kernels will be established with the help of closed-form formulas for the reproducing kernels. The main results provide construction of semi-discrete localized tight frame in weighted $$L^2$$ norm and characterization of best approximation by polynomials on our domains. Several intermediate results, including the Marcinkiewicz–Zygmund inequalities, positive cubature rules, Christoffel functions, and Bernstein inequalities, are shown to hold for doubling weights defined via the intrinsic distance on the domain.

On the quadrature exactness in hyperinterpolation

This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires a positive-weight quadrature rule with exactness degree 2n. We examine the behavior of such approximation when the required exactness degree 2n is relaxed to \(n+k\) with \(0<k\le n\). Aided by the Marcinkiewicz–Zygmund inequality, we affirm that the \(L^2\) norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of n, and this approximation scheme is convergent as \(n\rightarrow \infty \) if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw–Curtis quadrature, and the spherical t-designs.

Approximation and localized polynomial frame on conic domains

Highly localized kernels constructed by orthogonal polynomials have been fundamental in the recent development of approximation and computational analysis on the unit sphere, unit ball, and several other regular domains. In this work, we first study homogeneous spaces that are assumed to contain highly localized kernels and establish a framework for approximation and localized tight frames in such spaces, which extends recent works on bounded regular domains. We then show that the framework is applicable to homogeneous spaces defined on bounded conic domains, which consists of conic surfaces and the solid domains bounded by such surfaces and hyperplanes. The highly localized kernels on conic domains require precise estimates that rely on recently discovered addition formulas for orthogonal polynomials with respect to special weight functions on each domain and an intrinsic distance that takes into account the boundary of the domain, the latter is not comparable to the Euclidean distance at around the apex of the cone.The main results provide construction of semi-discrete localized tight frame in weighted L2 norm and characterization of best approximation by polynomials on conic domains. The latter is achieved by using a K-functional, defined via the differential operator that has orthogonal polynomials as eigenfunctions, as well as a modulus of smoothness defined via a multiplier operator that is equivalent to the K-functional. Several intermediate results are of interest in their own right, including the Marcinkiewicz-Zygmund inequalities, positive cubature rules, Christoeffel functions, and Bernstein type inequalities. Moreover, although the highly localizable kernels hold only for special families of weight functions on each domain, many intermediate results are shown to hold for doubling weights defined via the intrinsic distance on the domain.

Optimal asymptotic bounds for designs on manifolds

We extend to the case of a $d$-dimensional compact connected oriented Riemannian manifold $\mathcal M$ the theorem of A. Bondarenko, D. Radchenko and M. Viazovska on the existence of $L$-designs consisting of $N$ nodes, for any $N\ge C_{\mathcal M} L^d$. For this, we need to prove a version of the Marcinkiewicz-Zygmund inequality for the gradient of diffusion polynomials.

On Marcinkiewicz–Zygmund Inequalities and $$A_p$$-Weights for $$L$$-Shape Arcs

Let $\Gamma$ be an $L$-shape arc consisting of 2 line segments that meet at an angle different from $\pi$ in the complex $z$-plane $\CC$. This paper is to investigate the behavior of the polynomial interpolants at the Fej\'er points, defined by $\{z_{n,k} = \psi(e^{i(2k\pi + \theta)/(n+1)})\}$ for any choice of $\theta$. In this regard, we recall that for the interval [-1, 1], the Fej\'er points $\{z_{n,k} = \psi^*(e^{i(2k+1)\pi/(n+1)})\}$ agree with the Chebyshev points and that the Chebyshev points are most commonly used as nodes for Lagrange polynomial interpolation. On the other hand, numerical experimentation demonstrates that for a typical open $L$-shape arc $\Gamma$, the Lebesgue constants tend to $\infty$ at the rate of $O((log(n))^2)$, as the polynomial degree $n$ increases, while the $A_{p}$-weight conditions for the Fej\'er points $\{z_{n,k}\}$ do not carry over from [-1, 1] to a truly $L$-shape arc. Further numerical experiments also demonstrate that the least upper bounds of the Marcinkiewicz-Zygmund inequalities for the canonical Lagrange interpolation polynomials at $\{z_{n,k}\}$ seem to grow at the rate of $n^{\beta}$, for some $\beta >0$ that depends on $p >1$.

Sampling, Marcinkiewicz–Zygmund inequalities, approximation, and quadrature rules

Given a sequence of Marcinkiewicz-Zygmund inequalities in L2, we derive approximation theorems and quadrature rules. The derivation is completely elementary and requires only the definition of Marcinkiewicz-Zygmund inequality, Sobolev spaces, and the solution of least square problems.

Discrepancy and Numerical Integration on Metric Measure Spaces

We study here the error of numerical integration on metric measure spaces adapted to a decomposition of the space into disjoint subsets. We consider both the error for a single given function, and the worst case error for all functions in a given class of potentials. The main tools are the classical Marcinkiewicz-Zygmund inequality and ad hoc definitions of function spaces on metric measure spaces. The same techniques are used to prove the existence of point distributions in metric measure spaces with small $L^p$ discrepancy with respect to certain classes of subsets, for example metric balls.

Asymptotically honest confidence regions for high dimensional parameters by the desparsified conservative Lasso

In this paper we consider the conservative Lasso which we argue penalizes more correctly than the Lasso and show how it may be desparsified in the sense of van de Geer et al. (2014) in order to construct asymptotically honest (uniform) confidence bands. In particular, we develop an oracle inequality for the conservative Lasso only assuming the existence of a certain number of moments. This is done by means of the Marcinkiewicz–Zygmund inequality. We allow for heteroskedastic non-subgaussian error terms and covariates. Next, we desparsify the conservative Lasso estimator and derive the asymptotic distribution of tests involving an increasing number of parameters. Our simulations reveal that the desparsified conservative Lasso estimates the parameters more precisely than the desparsified Lasso, has better size properties and produces confidence bands with superior coverage rates.

Complete convergence and complete moment convergence for extended negatively dependent random variables

In this paper, we provide some probability and moment inequalities (especially the Marcinkiewicz-Zygmund type inequality) for extended negatively dependent (END, in short) random variables. By using the Marcinkiewicz-Zygmund type inequality and the truncation method, we investigate the complete convergence for sums and weighted sums of arrays of rowwise END random variables. In addition, the complete moment convergence for END random variables is obtained. Our results generalize and improve the corresponding ones of Wang et al. [18] and Baek and Park [2].

EXTENSIONS OF SEVERAL CLASSICAL RESULTS FOR INDEPENDENT AND IDENTICALLY DISTRIBUTED RANDOM VARIABLES TO CONDITIONAL CASES

Extensions of the Kolmogorov convergence criterion and the Marcinkiewicz-Zygmund inequalities from independent random variables to conditional independent ones are derived. As their applications, a conditional version of the Marcinkiewicz-Zygmund strong law of large numbers and a result on convergence in <TEX>$L^p$</TEX> for conditionally independent and conditionally identically distributed random variables are established, respectively.

Marcinkiewicz–Zygmund inequalities and interpolation by spherical polynomials with respect to doubling weights

We find necessary density conditions for Marcinkiewicz–Zygmund inequalities and interpolation for spaces of spherical polynomials in the weighted Lp (0<p<∞) space with respect to a doubling weight. Moreover, we show that these conditions are sharp.

On sharp constants in Marcinkiewicz-Zygmund and Plancherel-Polya inequalities

The Plancherel-Polya inequalities assert that for 1 < p <∞, and entire functions f of exponential type at most π, Ap ∞ ∑ j=−∞ |f (j)| ≤ ∫ ∞ −∞ |f | ≤ Bp ∞ ∑ j=−∞ |f (j)| . The Marcinkiewicz-Zygmund inequalities assert that for n ≥ 1, and polynomials P of degree ≤ n− 1, Ap n n ∑ j=1 ∣∣∣P (e2πij/n)∣∣∣p ≤ ∫ 1 0 ∣∣P (e2πit)∣∣p dt ≤ B′ p n n ∑ j=1 ∣∣∣P (e2πij/n)∣∣∣p . We show that the sharp constants in both inequalities are the same, that is Ap = Ap and Bp = B ′ p. Moreover, the two inequalities are equivalent. We also discuss the case p ≤ 1.

Optimal constants in the Marcinkiewicz–Zygmund inequalities

We give the optimal constants in the Marcinkiewicz–Zygmund inequalities for symmetric summands. As an application we substantially improve the estimates of Ren and Liang (2001) in the Marcinkiewicz–Zygmund–Hölder inequality and identify the best possible constants in the symmetric case.

Marcinkiewicz–Zygmund inequality for nonnegative [formula omitted]-demimartingales and related results

In this paper the Marcinkiewicz–Zygmund inequality for mean zero independent random variables is extended to the case of nonnegative N -demimartingales. This result can be used for obtaining large deviation inequalities for nonnegative N -demimartingales as well as related complete convergence results.

Scattered Data Approximation on the Bisphere and Application to Texture Analysis

The paper deals with the approximation and optimal interpolation of functions defined on the bisphere \(\mathbb {S}^{2}\times \mathbb {S}^{2}\) from scattered data. We demonstrate how the least square approximation to the function can be computed in a stable and efficient manner. The analysis of this problem is based on Marcinkiewicz–Zygmund inequalities for scattered data which we present here for the bisphere. The complementary problem of optimal interpolation is also solved by using well-localized kernels for our setting. Finally, we discuss the application of the developed methods to problems of texture analysis in material science.

Probabilistic Marcinkiewicz–Zygmund Inequalities on the Rotation Group

Approximation problems on the rotation group SO(3) naturally arise in various fields, like crystallography, chemistry, and biology. For example, in crystallographic texture analysis one is confronted with the problem of evaluating so-called orientation density functions (ODFs). In many situations one only has a finite number of measurements at scattered sampling nodes. In order to reconstruct ODFs over all rotations, so-called Marcinkiewicz–Zygmund inequalities on the rotation group are an important tool. These inequalities provide norm equivalences between polynomials on SO(3) and their sample values. Recently shown equivalences depend on a density parameter of the sampling set and the proven inequalities hold true for polynomials on SO(3) whose degree does not exceed an upper bound which is determined by this density parameter. In this paper, we show that we can enlarge this upper bound for the polynomial degree significantly if we are satisfied by such norm equivalences that hold with a given probability only. Moreover, we show that there are fixed sampling sets for which we get probabilistic Marcinkiewicz–Zygmund inequalities that hold for polynomials on SO(3) of all degrees.

A SURVEY OF POLYNOMIAL APPROXIMATION ON THE SPHERE

The main purpose of this paper is to survey some of the work on spherical approximation done by the BNU group under the direction of Professor Sun. The equiconvergent operators of Cesàro means, and their interesting applications are described. The Jackson inequality for spherical polynomials and some moduli of smoothness on the sphere are investigated. The equivalence between moduli of smoothness and K-functionals is also discussed. We also describe several weighted polynomial inequalities on the sphere, including the Remez-type and the Nikolskii-type inequalities, the Marcinkiewicz–Zygmund inequality, the Bernstein-type and the Schur-type inequalities. Positive cubature formulas on the sphere, and their relation to the Marcinkiewicz–Zygmund inequality are also discussed. A survey on recent results on asymptotic orders of the n-widths of Sobolev's classes on the sphere is also given.

Multivariate irregular sampling theorem

In this paper, we prove a Marcinkiewicz-Zygmund type inequality for multivariate entire functions of exponential type with non-equidistant spaced sampling points. And from this result, we establish a multivariate irregular Whittaker-Kotelnikov-Shannon type sampling theorem.