Positive definite functions on spheres

By I. J. Schoenberg. 1942

How researchers see this paper

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Random Feature Maps for Dot Product Kernels (2012, background): “This is a classical result in harmonic analysis due to Schoenberg (1942), that characterizes positive definite functions on the unit sphere in a Hilbert space.”
On the Validity of Commonly Used Covariance and Variogram Functions on the Sphere (2011, background): “Therefore, the stable covariance function is not valid on the sphere, since a valid covariance function on the sphere will always be valid on the circle (Schoenberg 1942).”
Strictly Positive Definite Functions on a Real Inner Product Space (2004, unknown): “Schoenberg [7], Berg, Christensen, Ressel [1, p.”
Randomized numerical linear algebra: Foundations and algorithms (2020, background): “I. J. Schoenberg (1942), ‘Positive definite functions on spheres’, Duke Math.”
A class of Matérn-like covariance functions for smooth processes on a sphere (2015, background): “Schoenberg (1942) presented a classical representation for positive definite functions with great circle distance in terms of Gegenbauer polynomials.”
Multivariate Bayesian analysis of atmosphere–ocean general circulation models (2007, background): “Schoenberg (1942) showed that a function belongs to the class of positive definite functions on the sphere if and only if it can be represented as an infinite series of Legendre polynomials.”