Singular Distributions
A closed set is a set of uniqueness if and only if there exists a distribution S supported on the set (so in particular it must be singular) such that
( here are the Fourier coefficients). In all early examples of sets of uniqueness the distribution in question was in fact a measure. In 1954, though, Ilya Piatetski-Shapiro constructed an example of a set of uniqueness which does not support any measure with Fourier coefficients tending to zero. In other words, the generalization of distribution is necessary.
Read more about this topic: Set Of Uniqueness
Famous quotes containing the word singular:
“Singularity is only pardonable in old age and retirement; I may now be as singular as I please, but you may not.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)