Positive Definite Functions and Sharp Inequalities for Periodic Functions

Let φ be a positive definite and continuous function on R, and let μ be the corresponding Bochner measure. For fixed ε,τ∈R, ε≠0, we consider a linear operator Aε,τ generated by the function φ: Aε,τ(f)(t):=∫Re−iuτf(t+εu)dμ(u),t∈R,f∈C(T). Let J be a convex and nondecreasing function on [0,+∞). In this paper, we prove the inequalities ∥Aε,τ(f)∥p⩽φ(0)∥f∥p,∫TJ(|Aε,τ(f)(t)|)dt≤∫TJ(φ(0)|f(t)|)dt for p∈[1,∞] and f∈C(T) and obtain criteria of extremal function. We study in more detail the case in which ε=1/n, n∈N, τ=1, and φ(x)≡eiβxψ(x), where β∈R and the function ψ is 2-periodic and positive definite. In turn, we consider in more detail the case where the 2-periodic function ψ is constructed by means of a finite positive definite function g. As a particular case, we obtain the Bernstein–Szegő inequality for the derivative in the Weyl–Nagy sense of trigonometric polynomials. In one of our results, we consider the case of the family of functions g1/n,h(x):=hg(x)+(1−1/n−h)g(nx), where n∈N, n≥2, −1/n≤h≤1−1/n, and the function g∈C(R) is even, nonnegative, decreasing, and convex on (0,+∞) with suppg⊂[−1,1]. This case is related to the positive definiteness of piecewise linear functions. We also obtain some general interpolation formulas for periodic functions and trigonometric polynomials which include the known interpolation formulas of M. Riesz, of G. Szegő, and of A.I. Kozko for trigonometric polynomials.I express my gratitude to the Organizing Committee of the International S.B. Stechkin Summer Workshop-Conference on Function Theory for the given opportunity to participate in this event and to all participants for creating a warm and working atmosphere. Inaddition, I would like to thank Professor V.V. Arestov and P.Yu. Glazyrina for useful discussions and sharing information and Professor A.A. Kovalevskii for the translation of my article into English

Stein hypothesis and screening effect for covariances with compact support

In spatial statistics, the screening effect historically refers to the situation when the observations located far from the predictand receive a small (ideally, zero) kriging weight. Several factors play a crucial role in this phenomenon: among them, the spatial design, the dimension of the spatial domain where the observations are defined, the mean-square properties of the underlying random field and its covariance function or, equivalently, its spectral density. The tour de force by Michael L. Stein provides a formal definition of the screening effect and puts emphasis on the Matérn covariance function, advocated as a good covariance function to yield such an effect. Yet, it is often recommended not to use covariance functions with a compact support. This paper shows that some classes of covariance functions being compactly supported allow for a screening effect according to Stein’s definition, in both regular and irregular settings of the spatial design. Further, numerical experiments suggest that the screening effect under a class of compactly supported covariance functions is even stronger than the screening effect under a Matérn model

Covariance tapering for multivariate Gaussian random fields estimation

In recent literature there has been a growing interest in the construction of covariance models for multivariate Gaussian random fields. However, effective estimation methods for these models are somehow unexplored. The maximum likelihood method has attractive features, but when we deal with large data sets this solution becomes impractical, so computationally efficient solutions have to be devised. In this paper we explore the use of the covariance tapering method for the estimation of multivariate covariance models. In particular, through a simulation study, we compare the use of simple separable tapers with more flexible multivariate tapers recently proposed in the literature and we discuss the asymptotic properties of the method under increasing domain asymptotics