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An analog of Rudin’s theorem for continuous radial positive definite functions of several variables

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Abstract

Let be the class of radial real-valued functions of m variables with support in the unit ball \(\mathbb{B}\) of the space ℝm that are continuous on the whole space ℝm and have a nonnegative Fourier transform. For m ≥ 3, it is proved that a function f from the class can be presented as the sum \(\sum {f_k \tilde *f_k } \) of at most countably many self-convolutions of real-valued functions f k with support in the ball of radius 1/2. This result generalizes the theorem proved by Rudin under the assumptions that the function f is infinitely differentiable and the functions f k are complex-valued.

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Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Ural Federal University, pr. Lenina 51, Yekaterinburg, 620000, Russia

    A. V. Efimov

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  1. A. V. Efimov
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Correspondence to A. V. Efimov.

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Original Russian Text © A.V. Efimov, 2012, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2012, Vol. 18, No. 4.

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Efimov, A.V. An analog of Rudin’s theorem for continuous radial positive definite functions of several variables. Proc. Steklov Inst. Math. 284 (Suppl 1), 79–86 (2014). https://doi.org/10.1134/S0081543814020072

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  • DOI: https://doi.org/10.1134/S0081543814020072

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