From Schoenberg Coefficients to Schoenberg Functions

By Christian Berg, Emilio Porcu. 2015

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In his seminal paper, Schoenberg (Duke Math J 9:96–108, 1942) characterized the class $$\mathcal P(\mathbb {S}^d)$$P(Sd) of continuous functions $$f:[-1,1] \rightarrow \mathbb {R}$$f:[-1,1]→R such that $$f(\cos \theta (\xi ,\eta ))$$f(cosθ(ξ,η)) is positive definite on the product space $$\mathbb {S}^d \times \mathbb {S}^d$$Sd×Sd, with $$\mathbb {S}^d$$Sd being the unit sphere of $$\mathbb {R}^{d+1}$$Rd+1 and $$\theta (\xi ,\eta )$$θ(ξ,η) being the great circle distance between $$\xi ,\eta \in \mathbb {S}^d$$ξ,η∈Sd. In the present paper, we consider the product space $$\mathbb {S}^d \times G$$Sd×G, for G a locally compact group, and define the class $$\mathcal P(\mathbb {S}^d, G)$$P(Sd,G) of continuous functions $$f:[-1,1]\times G \rightarrow \mathbb {C}$$f:[-1,1]×G→C such that $$f(\cos \theta (\xi ,\eta ), u^{-1}v)$$f(cosθ(ξ,η),u-1v) is positive definite on $$\mathbb {S}^d \times \mathbb {S}^d \times G \times G$$Sd×Sd×G×G. This offers a natural extension of Schoenberg’s theorem. Schoenberg’s second theorem corresponding to the Hilbert sphere $$\mathbb {S}^\infty $$S∞ is also extended to this context. The case $$G=\mathbb {R}$$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.

Subfields: Finance, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6ecab6e46a633d80ad5151d27de6dbacdcdfc375