Zoltán Sasvári

Analogies and correspondences between variograms and covariance functions

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.

The characterization problem for isotropic covariance functions

Isotropic covariance functions are successfully used to model spatial continuity in a multitude of scientific disciplines. Nevertheless, a satisfactory characterization of the class of permissible isotropic covariance models has been missing. The intention of this note is to review, complete, and extend the existing literature on the problem. As it turns out, a famous conjecture of Schoenberg (1938) holds true: any measurable, isotropic covariance function on ℝd (d ≥ 2) admits a decomposition as the sum of a pure nugget effect and a continuous covariance function. Moreover, any measurable, isotropic covariance function defined on a ball in ℝd can be extended to an isotropic covariance function defined on the entire space ℝd.

An Elementary Proof of Binet's Formula for the Gamma Function

(1999). An Elementary Proof of Binets Formula for the Gamma Function. The American Mathematical Monthly: Vol. 106, No. 2, pp. 156-158.

Functions with a finite number of negative squares on semigroups

We consider indefinite functions on semigroups and their relation to representations in spaces with an indefinite metric. Special attention is given to functions of finite rank, where the space of representation is of finite dimension, and to functions for which the corresponding representation consists of bounded operators in Pontrjagin spaces.

On norm dependent positive definite functions

In the first part of the paper we prove a decomposition theorem for positive definite functions (Theorem 2.3) generalizing a result of de Leeuw and Glicksberg. Using this theorem, we then show (Theorem 3.1) that certain norm dependent positive definite functions are automatically continuous at every point different from zero.

On bounded functions with a finite number of negative squares

In the case of a locally compact abelian groupG we give a characterization of bounded continuous functions onG which have a finite number of negative squares. We show that these functions can be regarded in a certain sense as a special case of positive definite functions.

Indefinite functions on commutative groups

The purpose of this paper is to study indefinite functions of orderk on commutative locally compact groups. In section 3 we establish some fundamental properties of these functions. Using the results of this section we get a characterization for indefinite functions of order one (section 4). In section 5 we give a decomposition of measurable indefinite functions.

On a Theorem of Karhunen and Related Moment Problems and Quadrature Formulae

In the present paper we give a refinement of a classical result by Karhunen concerning spectral representations of second-order random fields. We also investigate some related questions dealing with moment problems and quadrature formulae.S ome of these questions are closely related to Heinz Langer’s work.

Decomposition of positive definite functions defined on a neighbourhood of the identity

LetV be a symmetric open neighbourhood of the identity of a topological groupG. We show that every positive definite functionf onV can be written asf=fc+fs wherefc andfs are positive definite functions onV, fc is continuous andfs averages to zero. IfG is locally compact with Haar measuremG andf ismG-measurable thenfs=0mG-almost everywhere.

Correlation Functions of Intrinsically Stationary Random Fields

In the introduction we give a short historical survey on the theory of correlation functions of intrinsically stationary random fields. We then prove the existence of generalized correlation functions for intrinsically stationary fields on ℝ d as well as an integral representation for these functions. At the end of the paper we show that intrinsically stationary fields are related to unitary operators in Pontryagin spaces in a similar way as stationary fields are related to unitary operators in Hilbert spaces.