Stochastic representations of isotropic vector random fields on spheres

ABSTRACTA simple but efficient approach is proposed in this paper to construct the isotropic random field in (d ⩾ 2), whose univariate marginal distributions may be taken as any infinitely divisible distribution with finite variance. The three building blocks in our building tool box are a second-order Lévy process on the real line, a d-variate random vector uniformly distributed on the unit sphere, and a positive random variable that generates a Pólya-type function. The approach extends readily to the multivariate case and results in a vector random field in with isotropic direct covariance functions and with any specified infinitely divisible marginal distributions. A characterization of the turning bands simulation feature is also derived for the covariance matrix function of a Gaussian or elliptically contoured random field that is isotropic and mean square continuous in .

This paper is concerned with spherically invariant or elliptically contoured vector random fields in space and/or time, which are formulated as scale mixtures of vector Gaussian random fields. While a spherically invariant vector random field may or may not have second-order moments, a spherically invariant second-order vector random field is determined by its mean and covariance matrix functions, just like the Gaussian one. This paper explores basic properties of spherically invariant second-order vector random fields, and proposes an efficient approach to develop covariance matrix functions for such vector random fields.

This article introduces the hyperbolic vector random field whose finite-dimensional distributions are the generalized hyperbolic one, which is formulated as a scale mixture of Gaussian random fields and is thus an elliptically contoured (or spherically invariant) random field. Such a vector random field may or may not have second-order moments, while a second-order one is characterized by its mean function and its covariance matrix function, just as in a Gaussian case. Some covariance matrix structures of hyperbolic type are constructed in this paper for second-order hyperbolic vector random fields.

Let (P ν) be a sequence of monic polynomials orthogonal on the unit circle with respect to a nonnegative weight function, let (Ωυ) the monic associated polynomials of (P v), and letA andB be self-reciprocal polynomials. We show that the sequence of polynomials (APυλ+BΩυλ)/Aλ, λ stuitably determined, is a sequence of orthogonal polynomials having, up to a multiplicative complex constant, the same recurrence coefficients as theP ν's from a certain index value onward, and determine the orthogonality measure explicity. Conversely, it is also shown that every sequence of orthogonal polynomials on the unit circle having the same recurrence coefficients from a certain index value onward is of the above form. With the help of these results an explicit representation of the associated polynomials of arbitrary order ofP ν and of the corresponding orthogonality measure and Szego function is obtained. The asymptotic behavior of the associated polynomials is also studied. Finally necessary and suficient conditions are given such that the measure to which the above introduced polynomials are orthogonal is positive.

We present a brief summary of some results related to deriving orthogonal representations of second-order random fields and its application in solving linear prediction problems. In the homogeneous and/or isotropic case, the spectral theory provides an orthogonal expansion in terms of spherical harmonics, called spectral decomposition (Yadrenko 1983). A prediction formula based on this orthogonal representation is shown. Finally, an application of this formula in solving a real-data problem related to prospective geophysics techniques is presented.

RII type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle

Thousands of permanent GNSS stations sample nowadays the 3D deformation of the Earth’s surface. The temporal covariance structure of the field of GNSS station displacements is well characterized and modelled. On the other hand, there lacks a general agreed-upon model of its spatial covariance structure, in part because the theory of random vector fields on the sphere remains hardly developed.In this contribution, we show how the well-established theory of random isotropic scalar fields on the sphere generalizes to the case of vector fields. We derive in particular a spectral representation of random isotropic vector fields on the sphere in the domain of vector spherical harmonics, from which several properties of their covariance functions follow. We then present several parametric families of covariance functions which could be used to describe and model the covariance structure of vector fields on the sphere, such as GNSS station displacements.Although this presentation focuses on theoretical aspects, it is given with future practical applications in mind. A realistic spatio-temporal covariance model of GNSS station displacements could indeed benefit different problems such as the estimation of long-term GNSS station velocities, the identification and mitigation of offsets in GNSS station position time series, the filtering of spatial “common modes” to isolate local deformation, or the spatial interpolation of GNSS station displacements into global maps. Applications may also be found in other domains involving vector quantities distributed on a sphere, e.g., winds, ocean currents, magnetic anomalies, etc.

The weight functions, generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials

A recently introduced fast algorithm for the computation of the first N terms in an expansion of an analytic function into ultraspherical polynomials consists of three steps: Firstly, each expansion coefficient is represented as a linear combination of derivatives; secondly, it is represented, using the Cauchy integral formula, as a contour integral of the function multiplied by a kernel; finally, the integrand is transformed to accelerate the convergence of the Taylor expansion of the kernel, allowing for rapid computation using Fast Fourier Transform. In the current paper we demonstrate that the first two steps remain valid in the general setting of orthogonal polynomials on the real line with finite support, orthogonal polynomials on the unit circle and Laurent orthogonal polynomials on the unit circle.

We characterize the sequences of orthogonal polynomials on the unit circle whose derivatives are also orthogonal polynomials on the unit circle. Some relations for the sequences of derivatives of orthogonal polynomials are provided. Finally, we pose some problems about orthogonality-preserving maps and differential equations for orthogonal polynomials on the unit circle.

The purpose of this text is to present some new results in the spectral theory of orthogonal polynomials and Schrodinger operators. These results concern perturbations of the free Schrodinger operator and of the free case for orthogonal polynomials on the unit circle (which corresponds to Verblunsky coefficients equal to 0) and the real line (which corresponds to off-diagonal Jacobi coefficients equal to 1 and diagonal Jacobi coefficients equal to 0). The condition central to our results is that of generalized bounded variation. This class consists of finite linear combinations of sequences of rotated bounded variation with an L^1 perturbation. This generalizes both usual bounded variation and expressions of the form lambda(x) cos(phi x + alpha) with lambda(x) of bounded variation (and, in particular, with lambda(x) = x^gamma, Wigner-von Neumann potentials) as well as their finite linear combinations. Assuming generalized bounded variation and an L^p condition (with any finite p) on the perturbation, our results show preservation of absolutely continuous spectrum, absence of singular continuous spectrum, and that embedded pure points in the continuous spectrum can only occur in an explicit finite set.

The location of the zeros of discrete systems characteristic polynomials with respect to the unit circle is investigated. A new sequence of symmetric polynomials of descending degrees is defined for the characteristic polynomial and the number of zeros inside and outside the unit circle is shown to be related to a certain sign variation pattern of the polynomials in the sequence. A stability table based on this sequence is presented to obtain the sought distribution of zeros. The new table is close in appearance, size, and number of arithmetic operations to the Routh table used for continuous-time system polynomials. By comparison with the table of Jury, based on the theory of Marden, Cohn, and Schur, the new table involves about half the number of entries and a corresponding significant saving in computations. The study includes a detailed consideration of all possible cases of singular conditions so that the complete information on the number of zeros inside, outside, and on the unit circle is always obtained (including some additional information on possible reciprocal pairs of zeros). Necessary and sufficient conditions for a polynomial to have all its zeros inside the unit circle are obtained as a special outcome. Other additional necessary conditions, that are useful to shorten procedures when the table is used only for testing the stability of the system, are also given.

The digital control of a nonlinear system is analyzed in general terms. The analysis considers the case of a pair of branches of the root loci crossing the unit circle. The crossing occurs as the eigenvalues of the Jacobian of the discrete-time system are varied depending on, for example, the sampling period, and the crossing is assumed to take place anywhere on the unit circle except on the real axis. Results are presented for a particular example. >

The authors suggest a modification of the Toeplitz approximation method for estimating frequencies of multiple sinusoids from covariance measurements. The method constructs a state-feedback matrix following a low-rank approximation of the Toeplitz covariance matrix via singular-value decomposition. Ideally, the eigenvalues of this state-feedback matrix will be on the unit circle in the complex plane, and the angles that they make with the real axis will be equal to the unknown sinusoid frequencies. The modification proposed here exploits this prior knowledge of the modulus of the eigenvalues and guarantees that even in the presence of noise the eigenvalues of the estimated state-feedback matrix will lie on the unit circle. >

How to count the number of zeros that a polynomial has on the unit circle?