Uniqueness set

$ U $- set

A set $ E \subset [ 0 ,\ 2 \pi ] $ such that a trigonometric series that converges to zero at each point of $ ( 0 ,\ 2 \pi ] \setminus E $ is the zero series. A set that is not a $ U $- set is a called a set of non-uniqueness, or an $ M $- 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 $ E $. 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 $ M $- sets. Any countable set is a $ U $- set. There exists perfect sets (cf. Perfect set) of measure zero that are $ M $- sets (D.E. Men'shov, 1916), and ones that are $ U $- sets (N.K. Bari, 1921); for example, the Cantor set with a constant rational ratio $ \theta $ is a $ U $- set if and only $ 1 / \theta $ is an integer, that is, whether a set of numbers is a $ U $- set or an $ M $- set depends on the arithmetical nature of the numbers forming it. However, there exist sets $ E \subset [ 0 ,\ 2 \pi ] $ of full measure (so-called $ U ( \epsilon ) $- sets) such that any trigonometric series that converges to zero at every point of $ [ 0 ,\ 2 \pi ] \setminus E $ and has coefficients that are $ O ( \epsilon _{n} ) $, where $ \epsilon _{n} \downarrow 0 $, is the zero series.

The concepts of $ U $- sets and $ M $- 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

$ M $- sets are also called sets of multiplicity. A set $ E \subset [ 0,\ 2 \pi ] $ such that a Fourier–Stieltjes series that converges to zero at each point of $ ( 0,\ 2 \pi ] \setminus E $ is the zero series, is called a $ U _{0} $- set, or a set of extended uniqueness. A set that is not a $ U _{0} $- set is called an $ M _{0} $- set, or a set of restricted multiplicity. A set $ E $ is a $ U _{0} $- 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, $ U _{0} $- sets play a more prominent role than $ U $- sets. In 1983, R. Lyons proved that the Rajchman measures are exactly the measures that annihilate all $ U _{0} $- 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 $ [ x,\ x+l ] $ of length $ l $. Let $ \alpha (1) \dots \alpha (d) $, $ 0 \leq \alpha (1) < \alpha (2) < \dots < \alpha (d) < 1 $, be $ d $ numbers and consider the $ d $ closed intervals $ [ x+ \alpha (j) l,\ x + \alpha (j)l + \eta ] $, where $ \eta $ 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

$$ [ d ; \ \alpha (1) \dots \alpha (d) ; \ \eta ] . $$

Now start with any interval of length $ m $. Perform a dissection of type $ [ d _{1} ; \ \alpha _{1} (1) \dots \alpha _{1} (d _{1} ) ; \ \eta _{1} ] $, perform a dissection of type $ [ d _{2} ; \ \alpha _{2} (1) \dots \alpha _{2} (d _{2} ) ; \ \eta _{2} ] $ on each of the intervals obtained, etc. After $ p $ iterations one has $ d _{1} \dots d _{p} $ intervals, each of length $ \eta _{1} \dots \eta _{p} m $, and as $ p \rightarrow \infty $ the final result is a closed set $ P $ of measure $ m \ \lim\limits _{p} \ d _{1} \dots d _{p} \eta _{1} \dots \eta _{p} $( the limit exists). If $ d _{p} \geq 2 $ for all $ p $, the resulting $ P $ is perfect (cf. Perfect set) and non-dense. For $ d _{p} = 2 $ and $ \alpha _{p} (1) = 0 $, $ \alpha _{p} (2) = 2/3 $, $ \eta _{p} = 1/3 $ for all $ p $, one obtains the Cantor set. Taking successive dissections of type $ [ 2 ; \ 0,\ 1 - \xi _{k} ; \ \xi _{k} ] $ yields a so-called set of Cantor type. If $ \xi _{k} = \xi $ for all $ k $, 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)