From Schoenberg Coefficients to Schoenberg Functions

• Published in: Constructive approximation Vol. 45; no. 2; pp. 217 - 241
• Main Authors: Berg, Christian, Porcu, Emilio
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2017; Springer Nature B.V
• Subjects: Analysis; Continuity (mathematics); Covariance; Mathematics; Mathematics and Statistics; Numerical Analysis; Probability theory; Stochastic processes; Theorems; Primary 43A35; Positive definite; Secondary 33C55; Spherical harmonics; Space-time covariances
• ISSN: 0176-4276; 1432-0940
• DOI: 10.1007/s00365-016-9323-9

In his seminal paper, Schoenberg (Duke Math J 9:96–108, 1942 ) characterized the class P ( S d ) of continuous functions f : [ - 1 , 1 ] → R such that f ( cos θ ( ξ , η ) ) is positive definite on the product space S d × S d , with S d being the unit sphere of R d + 1 and θ ( ξ , η ) being the great circle distance between ξ , η ∈ S d . In the present paper, we consider the product space S d × G , for G a locally compact group, and define the class P ( S d , G ) of continuous functions f : [ - 1 , 1 ] × G → C such that f ( cos θ ( ξ , η ) , u - 1 v ) is positive definite on S d × S d × G × G . This offers a natural extension of Schoenberg’s theorem. Schoenberg’s second theorem corresponding to the Hilbert sphere S ∞ is also extended to this context. The case G = R is of special importance for probability theory and stochastic processes, because it characterizes completely the class of space-time covariance functions where the space is the sphere, being an approximation of planet Earth.