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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954402.png" />-set''
+
{{TEX|done}}
  
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954403.png" /> such that a [[Trigonometric series|trigonometric series]] that converges to zero at each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954404.png" /> is the zero series. A set that is not a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954405.png" />-set is a called a set of non-uniqueness, or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954407.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954408.png" />. 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|set theory]].
+
'' $  U $-
 +
set''
  
Sets of positive [[Lebesgue measure|Lebesgue measure]] are always <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u0954409.png" />-sets. Any countable set is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544010.png" />-set. There exists perfect sets (cf. [[Perfect set|Perfect set]]) of measure zero that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544011.png" />-sets (D.E. Men'shov, 1916), and ones that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544012.png" />-sets (N.K. Bari, 1921); for example, the [[Cantor set|Cantor set]] with a constant rational ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544013.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544014.png" />-set if and only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544015.png" /> is an integer, that is, whether a set of numbers is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544016.png" />-set or an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544017.png" />-set depends on the arithmetical nature of the numbers forming it. However, there exist sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544018.png" /> of full measure (so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544020.png" />-sets) such that any trigonometric series that converges to zero at every point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544021.png" /> and has coefficients that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544023.png" />, is the zero series.
+
A set  $  E \subset [ 0 ,\  2 \pi ] $
 +
such that a [[Trigonometric series|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|set theory]].
  
The concepts of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544024.png" />-sets and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544025.png" />-sets can be generalized to Fourier–Stieltjes series.
+
Sets of positive [[Lebesgue measure|Lebesgue measure]] are always  $  M $-
 +
sets. Any countable set is a  $  U $-
 +
set. There exists perfect sets (cf. [[Perfect set|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|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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544026.png" />-sets are also called sets of multiplicity. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544027.png" /> such that a [[Fourier–Stieltjes series|Fourier–Stieltjes series]] that converges to zero at each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544028.png" /> is the zero series, is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544030.png" />-set, or a set of extended uniqueness. A set that is not a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544031.png" />-set is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544033.png" />-set, or a set of restricted multiplicity. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544034.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544035.png" />-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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544036.png" />-sets play a more prominent role than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544037.png" />-sets. In 1983, R. Lyons proved that the Rajchman measures are exactly the measures that annihilate all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544038.png" />-sets. In [[#References|[a1]]]–[[#References|[a3]]] many more results are given, e.g. relating uniqueness sets with Helson sets and sets of spectral synthesis (cf. [[Harmonic analysis, abstract|Harmonic analysis, abstract]]).
+
$  M $-
 +
sets are also called sets of multiplicity. A set $  E \subset [ 0,\  2 \pi ] $
 +
such that a [[Fourier–Stieltjes series|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 [[#References|[a1]]]–[[#References|[a3]]] many more results are given, e.g. relating uniqueness sets with Helson sets and sets of spectral synthesis (cf. [[Harmonic analysis, abstract|Harmonic analysis, abstract]]).
  
Consider a closed interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544039.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544040.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544042.png" />, be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544043.png" /> numbers and consider the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544044.png" /> closed intervals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544046.png" /> 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
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544048.png" /></td> </tr></table>
+
$$
 +
[ d ; \  \alpha (1) \dots \alpha (d) ; \  \eta ] .
 +
$$
  
Now start with any interval of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544049.png" />. Perform a dissection of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544050.png" />, perform a dissection of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544051.png" /> on each of the intervals obtained, etc. After <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544052.png" /> iterations one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544053.png" /> intervals, each of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544054.png" />, and as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544055.png" /> the final result is a closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544056.png" /> of measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544057.png" /> (the limit exists). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544058.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544059.png" />, the resulting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544060.png" /> is perfect (cf. [[Perfect set|Perfect set]]) and non-dense. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544064.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544065.png" />, one obtains the [[Cantor set|Cantor set]]. Taking successive dissections of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544066.png" /> yields a so-called set of Cantor type. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544067.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095440/u09544068.png" />, one speaks of a set of Cantor type of constant ratio (of dissection). Cf. [[#References|[2]]], pp. 194ff, for more details.
+
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|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|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. [[#References|[2]]], pp. 194ff, for more details.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.C. Graham,  O.C. McGehee,  "Essays in commutative harmonic analysis" , Springer  (1979)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.-P. Kahane,  "Séries de Fourier absolument convergentes" , Springer  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.S. Kechris,  A. Louveau,  "Descriptive set theory and the structure of sets of uniqueness" , Cambridge Univ. Press  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.C. Graham,  O.C. McGehee,  "Essays in commutative harmonic analysis" , Springer  (1979)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.-P. Kahane,  "Séries de Fourier absolument convergentes" , Springer  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.S. Kechris,  A. Louveau,  "Descriptive set theory and the structure of sets of uniqueness" , Cambridge Univ. Press  (1987)</TD></TR></table>

Latest revision as of 22:26, 29 January 2020


$ 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)
How to Cite This Entry:
Uniqueness set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniqueness_set&oldid=44370
This article was adapted from an original article by V.F. Emel'yanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article