Uniqueness set

-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

Comments

-sets are also called sets of multiplicity. A set such that a Fourier–Stieltjes series that converges to zero at each point of is the zero series, is called a -set, or a set of extended uniqueness. A set that is not a -set is called an -set, or a set of restricted multiplicity. A set is a -set if and only if it does not support a non-zero Rajchman measure, that is, a measure whose Fourier–Stieltjes coefficients tend to zero at infinity. In the modern theory, -sets play a more prominent role than -sets. In 1983, R. Lyons proved that the Rajchman measures are exactly the measures that annihilate all -sets. In [a1]–[a3] many more results are given, e.g. relating uniqueness sets with Helson sets and sets of spectral synthesis (cf. Harmonic analysis, abstract).

Consider a closed interval of length . Let , , be numbers and consider the closed intervals , where is small enough so that the intervals have no points in common. Retain only these intervals (and throw the complementary intervals away). This is referred to as performing a dissection of type

Now start with any interval of length . Perform a dissection of type , perform a dissection of type on each of the intervals obtained, etc. After iterations one has intervals, each of length , and as the final result is a closed set of measure (the limit exists). If for all , the resulting is perfect (cf. Perfect set) and non-dense. For and , , for all , one obtains the Cantor set. Taking successive dissections of type yields a so-called set of Cantor type. If for all , one speaks of a set of Cantor type of constant ratio (of dissection). Cf. [2], pp. 194ff, for more details.

References

[a1]	C.C. Graham, O.C. McGehee, "Essays in commutative harmonic analysis" , Springer (1979) pp. Chapt. 5
[a2]	J.-P. Kahane, "Séries de Fourier absolument convergentes" , Springer (1970)
[a3]	A.S. Kechris, A. Louveau, "Descriptive set theory and the structure of sets of uniqueness" , Cambridge Univ. Press (1987)