# Uniqueness set

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-set

A set such that a trigonometric series that converges to zero at each point of is the zero series. A set that is not a -set is a called a set of non-uniqueness, or an -set. These concepts are related to the problem of the uniqueness of the representation of a function by a trigonometric series converging to it everywhere, except perhaps on a given set . G. Cantor (1872) showed that a finite set (including the empty set) is a set of uniqueness, and the extension of this result to infinite sets led him to the creation of set theory.

Sets of positive Lebesgue measure are always -sets. Any countable set is a -set. There exists perfect sets (cf. Perfect set) of measure zero that are -sets (D.E. Men'shov, 1916), and ones that are -sets (N.K. Bari, 1921); for example, the Cantor set with a constant rational ratio is a -set if and only is an integer, that is, whether a set of numbers is a -set or an -set depends on the arithmetical nature of the numbers forming it. However, there exist sets of full measure (so-called -sets) such that any trigonometric series that converges to zero at every point of and has coefficients that are , where , is the zero series.

The concepts of -sets and -sets can be generalized to Fourier–Stieltjes series.

#### References

 [1] N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) [2] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) [3] N.K. Bari, "The uniqueness problem of the representation of functions by trigonometric series" Transl. Amer. Math. Soc. (1) , 3 (1951) pp. 107–195 Uspekhi Mat. Nauk , 4 : 3 (1949) pp. 3–68