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''in the theory of functions of a complex variable''
 
''in the theory of functions of a complex variable''
  
 
Examples that characterize boundary [[Uniqueness properties of analytic functions|uniqueness properties of analytic functions]] (see [[#References|[1]]], [[#References|[2]]]).
 
Examples that characterize boundary [[Uniqueness properties of analytic functions|uniqueness properties of analytic functions]] (see [[#References|[1]]], [[#References|[2]]]).
  
1) For any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610301.png" /> of measure zero on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610302.png" />, N.N. Luzin constructed (1919, see [[#References|[1]]]) a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610303.png" /> that is regular, analytic and bounded in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610304.png" /> and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610305.png" /> does not have radial boundary values along each of the radii that end at points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610306.png" />.
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1) For any set $  E $
 +
of measure zero on the unit circle $  \Gamma = \{ {z } : {| z | = 1 } \} $,  
 +
N.N. Luzin constructed (1919, see [[#References|[1]]]) a function $  f ( z) $
 +
that is regular, analytic and bounded in the unit disc $  D = \{ {z } : {| z | < 1 } \} $
 +
and is such that $  f ( z) $
 +
does not have radial boundary values along each of the radii that end at points of $  E $.
  
 
A similar example of Luzin and I.I. Privalov (1925, see [[#References|[2]]], [[#References|[3]]]) differs only by insignificant changes.
 
A similar example of Luzin and I.I. Privalov (1925, see [[#References|[2]]], [[#References|[3]]]) differs only by insignificant changes.
  
2) Luzin also constructed (1925, see [[#References|[2]]]) regular analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610308.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l0610309.png" /> that tend, respectively, to infinity and zero along all radii that end at points of some set of full measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l06103010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l06103011.png" />. This set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l06103012.png" /> is of the first Baire category (cf. [[Baire classes|Baire classes]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061030/l06103013.png" />.
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2) Luzin also constructed (1925, see [[#References|[2]]]) regular analytic functions $  f _ {1} ( z) $
 +
and $  f _ {2} ( z) \not\equiv 0 $
 +
in $  D $
 +
that tend, respectively, to infinity and zero along all radii that end at points of some set of full measure $  2 \pi $
 +
on $  \Gamma $.  
 +
This set $  E $
 +
is of the first Baire category (cf. [[Baire classes|Baire classes]]) on $  \Gamma $.
  
 
See also [[Boundary properties of analytic functions|Boundary properties of analytic functions]]; [[Luzin–Privalov theorems|Luzin–Privalov theorems]]; [[Cluster set|Cluster set]].
 
See also [[Boundary properties of analytic functions|Boundary properties of analytic functions]]; [[Luzin–Privalov theorems|Luzin–Privalov theorems]]; [[Cluster set|Cluster set]].
Line 13: Line 36:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Luzin,  , ''Collected works'' , '''1''' , Moscow  (1953)  pp. 267–269  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. Luzin,  , ''Collected works'' , '''1''' , Moscow  (1953)  pp. 280–318  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauki i Tekhn. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N.N. Luzin,  , ''Collected works'' , '''1''' , Moscow  (1953)  pp. 267–269  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.N. Luzin,  , ''Collected works'' , '''1''' , Moscow  (1953)  pp. 280–318  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauki i Tekhn. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR></table>

Latest revision as of 04:11, 6 June 2020


in the theory of functions of a complex variable

Examples that characterize boundary uniqueness properties of analytic functions (see [1], [2]).

1) For any set $ E $ of measure zero on the unit circle $ \Gamma = \{ {z } : {| z | = 1 } \} $, N.N. Luzin constructed (1919, see [1]) a function $ f ( z) $ that is regular, analytic and bounded in the unit disc $ D = \{ {z } : {| z | < 1 } \} $ and is such that $ f ( z) $ does not have radial boundary values along each of the radii that end at points of $ E $.

A similar example of Luzin and I.I. Privalov (1925, see [2], [3]) differs only by insignificant changes.

2) Luzin also constructed (1925, see [2]) regular analytic functions $ f _ {1} ( z) $ and $ f _ {2} ( z) \not\equiv 0 $ in $ D $ that tend, respectively, to infinity and zero along all radii that end at points of some set of full measure $ 2 \pi $ on $ \Gamma $. This set $ E $ is of the first Baire category (cf. Baire classes) on $ \Gamma $.

See also Boundary properties of analytic functions; Luzin–Privalov theorems; Cluster set.

References

[1] N.N. Luzin, , Collected works , 1 , Moscow (1953) pp. 267–269 (In Russian)
[2] N.N. Luzin, , Collected works , 1 , Moscow (1953) pp. 280–318 (In Russian)
[3] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[4] A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauki i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)

Comments

References

[a1] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9
How to Cite This Entry:
Luzin examples. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Luzin_examples&oldid=18316
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article