Pseudo cross-variograms appear naturally in the context of multivariate Brown–Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram, and further provide a Schoenberg-type result connecting pseudo cross-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space–time covariance model of Gneiting to the multivariate case.

The Dagum family of isotropic covariance functions has two parameters that allow for decoupling of the fractal dimension and the Hurst effect for Gaussian random fields that are stationary and isotropic over Euclidean spaces. Sufficient conditions that allow for positive definiteness in $\mathbb{R}^d$ of the Dagum family have been proposed on the basis of the fact that the Dagum family allows for complete monotonicity under some parameter restrictions. The spectral properties of the Dagum family have been inspected to a very limited extent only, and this paper gives insight into this direction. Specifically, we study finite and asymptotic properties of the isotropic spectral density (intended as the Hankel transform) of the Dagum model. Also, we establish some closed-form expressions for the Dagum spectral density in terms of the Fox–Wright functions. Finally, we provide asymptotic properties for such a class of spectral densities.

In this paper, we propose a splitting positive definite mixed finite element method for the approximation of convex optimal control problems governed by linear parabolic equations, where the primal state variable y and its flux σ are approximated simultaneously. By using the first order necessary and sufficient optimality conditions for the optimization problem, we derive another pair of adjoint state variables z and ω, and also a variational inequality for the control variable u is derived. As we can see the two resulting systems for the unknown state variable y and its flux σ are splitting, and both symmetric and positive definite. Besides, the corresponding adjoint states z and ω are also decoupled, and they both lead to symmetric and positive definite linear systems. We give some a priori error estimates for the discretization of the states, adjoint states and control, where Ladyzhenkaya-Babuska-Brezzi consistency condition is not necessary for the approximation of the state variable y and its flux σ. Finally, numerical experiments are given to show the efficiency and reliability of the splitting positive definite mixed finite element method.

In a paper from 1976, Berg, Christensen, and Ressel prove that the closure of the cone of sums of squares $\sum{\mathbb{R}{{\left[ \underline{X} \right]}^{2}}}$ in the polynomial ring $\mathbb{R}\left[ \underline{X} \right]\,:=\,\mathbb{R}\left[ {{X}_{1}},\,.\,.\,.\,,\,{{X}_{n}} \right]$ in the topology induced by the $~{{\ell }_{1}}$-norm is equal to $\text{Pos}\left( {{\left[ -1,\,1 \right]}^{n}} \right)$, the cone consisting of all polynomials that are non-negative on the hypercube ${{\left[ -1,\,1 \right]}^{n}}$. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen, and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted $~{{\ell }_{1}}$-seminorm topology associated with an absolute value. In this paper we give a new proof of these results, which is based on Jacobi’s representation theorem from 2001. At the same time, we use Jacobi’s representation theorem to extend these results from sums of squares to sums of $2d$-powers, proving, in particular, that for any integer $d\,\ge \,1$, the closure of the cone of sums of $2d$-powers $\sum \mathbb{R}{{\left[ \underline{X} \right]}^{2d}}$ in the $\mathbb{R}\left[ \underline{X} \right]$ topology induced by the $~{{\ell }_{1}}$-norm is equal to $\text{Pos}\left( {{\left[ -1,\,1 \right]}^{n}} \right)$.

Variograms and covariance functions are the fundamental tools for modeling dependent data observed over time, space, or space-time. This paper aims at constructing nonseparable spatio-temporal variograms and covariance models. Special attention is paid to an intrinsically stationary spatio-temporal random field whose covariance function is of Schoenberg-Lévy type. The correlation structure is studied for its increment process and for its partial derivative with respect to the time lag, as well as for the superposition over time of a stationary spatio-temporal random field. As another approach, we investigate the permissibility of the linear combination of certain separable spatio-temporal covariance functions to be a valid covariance, and obtain a subclass of stationary spatio-temporal models isotropic in space.

In this paper we investigate when negative definite functions on commutative hypergroups satisfy the Schoenberg criterion.

This paper studies a class of stationary covariance models, in both the discrete- and the continuous-time domains, which possess a simple functional form γ(τ + τ0)+γ(τ − τ0)− 2γ(τ), where τ 0 is a fixed lag andγ(τ) is an intrinsically stationary variogram, and include the fractional Gaussian noise of Kolmogorov (1940) and a stochastic volatility model of Barndorff-Nielsen and Shephard (2001), (2002) as special cases. Properties of the class, and interesting special cases with long memory, are studied. We also characterize the covariance function via the variogram.

In this paper, we give some conditions to judge when a system of Hermitian quadratic forms has a real linear combination which is positive definite or positive semi-definite. We also study some related geometric and topological properties of the moduli space.

Let $H$ be a complex Hilbert space, and $\mathcal{H}\left( H \right)$ be the real linear space of bounded selfadjoint operators on $H$. We study linear maps $\phi :\,\mathcal{H}(H)\,\to \,\mathcal{H}(H)$ leaving invariant various properties such as invertibility, positive definiteness, numerical range, etc. The maps $\phi$ are not assumed a priori continuous. It is shown that under an appropriate surjective or injective assumption $\phi$ has the form $X\mapsto \xi TX{{T}^{*}}orX\mapsto \xi T{{X}^{t}}{{T}^{*}}$, for a suitable invertible or unitary $T$ and $\xi \,\in \,\{1,\,-1\}$, where ${{X}^{t}}$ stands for the transpose of $X$ relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on $H$. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form $\alpha I\,+\,K$, where $\alpha$ is a scalar and $K$ is compact.

Variograms and covariance functions are key tools in geostatistics. However, various properties, characterizations, and decomposition theorems have been established for covariance functions only. We present analogous results for variograms and explore the connections with covariance functions. Our findings include criteria for covariance functions on intervals, and we apply them to exponential models, fractional Brownian motion, and locally polynomial covariances. In particular, we characterize isotropic locally polynomial covariance functions of degree 3.

We discuss necessary and sufficient conditions for power-law and polynomial models to be correlation functions on bounded domains. These results date back to unpublished work by Matheron (1974) and generalize the findings of Gneiting (1999).

A popular procedure in spatial data analysis is to fit a line segment of the form c(x) = 1 - α ||x||, ||x|| < 1, to observed correlations at (appropriately scaled) spatial lag x in d-dimensional space. We show that such an approach is permissible if and only if

the upper bound depending on the spatial dimension d. The proof relies on Matheron's turning bands operator and an extension theorem for positive definite functions due to Rudin. Side results and examples include a general discussion of isotropic correlation functions defined on d-dimensional balls.