The rearrangement of functions and maximization of a convolution integral

Citation

Welch, Lloyd Richard (1958) The rearrangement of functions and maximization of a convolution integral. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/B59P-3R31. https://resolver.caltech.edu/CaltechETD:etd-01192006-081053

Abstract

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For the additive group of the real numbers, the value, at the origin, of the convolution of three non-negative integrable functions is less than or equal to the corresponding value obtained from the symmetric rearrangements of those functions. That is [...]. This problem is investigated on general groups and it is shown that under a wide interpretation of the notion of symmetric rearrangement the validity of the inequality implies serious restrictions on the group.

Both new and known results in [...] are developed using methods which in modified form yield the answer to the similar problem on the sphere.

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