Set of uniqueness
In mathematics, a set of uniqueness is a concept relevant to trigonometric expansions which are not necessarily Fourier series. Their study is a relatively pure branch of harmonic analysis.
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
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
In the theory of uniqueness the order is different:
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.
The empty set is a set of uniqueness. This is just a fancy way to say that if a trigonometric series converges to zero everywhere then it is trivial. 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. Later on, Cantor generalized Riemann's techniques to show that any countable, closed set is a set of uniqueness, a discovery which led him to the development of set theory. Interestingly, Paul Cohen, another great innovator in set theory, started his career with a thesis on sets of uniqueness.
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