Abstract
Positive-definite functions or distributions appear naturally in the theory of homogeneous random fields and, in particular, in the definition of their correlation functionals. We consider the problem of extending a radial positive-definite function, or distribution, defined in a ball centered at the origin of ℜn to one defined in the whole space. Such an extension was shown to exist by W. Rudin (in the case of a continuous function) and, later on, by A.E. Nussbaum (in the case of a distribution). Our goal in this paper is to explain how the maximum entropy principle can be used to obtain explicit solutions of the n-dimensional radial extension problem via a reduction to a one-dimensional one. We also investigate the question of uniqueness of the extension.
The author was supported by NSERC grant OGP0036564.543
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© 1992 Springer Science+Business Media Dordrecht
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Gabardo, JP. (1992). Extension of radial positive-definite distributions in ℜn and maximum entropy. In: Byrnes, J.S., Byrnes, J.L., Hargreaves, K.A., Berry, K. (eds) Probabilistic and Stochastic Methods in Analysis, with Applications. NATO ASI Series, vol 372. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2791-2_25
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DOI: https://doi.org/10.1007/978-94-011-2791-2_25
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