Set of Uniqueness - Definition

Definition

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 is identically zero; that is, such that

c(n) = 0 for all n.

Otherwise E is a set of multiplicity (sometimes called an M-set or a Menshov set). Analogous definitions apply on the real line, and in higher dimensions. In the latter case one needs to specify the order of summation, e.g. "a set of uniqueness with respect to summing over balls".

To understand the importance of the definition it is important to get out of the Fourier mind-set. In Fourier analysis there is no question of uniqueness, since the coefficients c(n) are derived by integrating the function. Hence in Fourier analysis the order of actions is

• Start with a function f.
• Calculate the Fourier coefficients using

• Ask: does the sum converge to f? In which sense?

In the theory of uniqueness the order is different:

• Start with some coefficients c(n) for which the sum converge in some sense
• Ask: does this means that they are the Fourier coefficients of the function?

In effect, it is usually sufficiently interesting (as in the definition above) to assume that the sum converges to zero and ask if that means that all the c(n) must be zero. As is usual in analysis, the most interesting questions arise when one discusses pointwise convergence. Hence the definition above, which arose when it became clear that neither convergence everywhere nor convergence almost everywhere give a satisfactory answer.