In mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series.
A subset E of the circle is called a set of uniqueness, or a U-set, if any trigonometric expansion which converges to zero for
In the latter case, one needs to specify the order of summation, e.g. "a set of uniqueness with respect to summing over balls".
This was proved by Riemann, using a delicate technique of double formal integration; and showing that the resulting sum has some generalized kind of second derivative using Toeplitz operators.
This was disproved by Dimitrii E. Menshov who in 1916 constructed an example of a set of multiplicity which has measure zero.
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.
A crucial part in this research is played by the index of the set, which is an ordinal between 1 and ω1, first defined by Pyatetskii-Shapiro.