Abstract
Characteristic functions that are radially symmetric have a dual interpretation, as they can be used as the isotropic correlation functions of spatial random fields. Extensions of isotropic correlation functions from balls into d-dimensional Euclidean spaces, , have been understood after Rudin. Yet, extension theorems on product spaces are elusive, and a counterexample provided by Rudin on rectangles suggests that the problem is challenging. This paper provides extension theorems for multiradial characteristic functions that are defined in balls embedded in cross, either or the unit sphere embedded in , for any two positive integers d and . We then examine Turning Bands operators that provide bijections between the class of multiradial correlation functions in given product spaces, and multiradial correlations in product spaces having different dimensions. The combination of extension theorems with Turning Bands provides a connection with random fields that are defined in balls cross linear or circular time.
Funding Statement
E. Porcu and S.F. Feng acknowledge this publication is based upon work supported by the Khalifa University of Science and Technology under Research Center Award No. 8474000331 (RDISC). X. Emery acknowledges the funding of the National Agency for Research and Development of Chile, through grants ANID FONDECYT Regular 1210050 and ANID PIA AFB180004. A.P. Peron was partially supported by Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP # 2021/04269-0.
Citation
Emilio Porcu. Samuel F. Feng. Xavier Emery. Ana P. Peron. "Rudin extension theorems on product spaces, turning bands, and random fields on balls cross time." Bernoulli 29 (2) 1464 - 1475, May 2023. https://doi.org/10.3150/22-BEJ1506