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Difference between revisions of "Luzin-Privalov theorems"

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''in the theory of functions of a complex variable''
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Classical results of N.N. Luzin and I.I. Privalov that clarify the character of a boundary uniqueness property of analytic functions (cf. [[Uniqueness properties of analytic functions|Uniqueness properties of analytic functions]]) (see [[#References|[1]]]).
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The Luzin–Privalov theorems in the theory of functions of a complex variable are classical results of N.N. Luzin and I.I. Privalov that clarify the character of a boundary uniqueness property of analytic functions (cf. [[Uniqueness properties of analytic functions]]) (see {{Cite|LuPr}}).
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l0610601.png" /> be a meromorphic function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l0610602.png" /> in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l0610603.png" /> with rectifiable boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l0610604.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l0610605.png" /> takes angular boundary values zero on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l0610606.png" /> of positive Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l0610607.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l0610608.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l0610609.png" />. There is no function meromorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106010.png" /> that has infinite angular boundary values on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106011.png" /> of positive measure.
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1) Let $f(z)$ be a meromorphic function of the complex variable $z$ in a simply-connected domain $D$ with rectifiable boundary $\Gamma$. If $f(z)$ takes angular boundary values zero on a set $E\subset\Gamma$ of positive Lebesgue measure on $\Gamma$, then $f(z)=0$ in $D$. There is no function meromorphic in $D$ that has infinite angular boundary values on a set $E\subset\Gamma$ of positive measure.
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106012.png" /> be a meromorphic function in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106013.png" /> other than a constant and having radial boundary values (finite or infinite) on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106014.png" /> situated on an arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106015.png" /> of the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106016.png" /> that is metrically dense and of the second Baire category (cf. [[Baire classes|Baire classes]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106017.png" />. Then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106018.png" /> of its radial boundary values on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106019.png" /> contains at least two distinct points. Metric density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106020.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106021.png" /> means that every [[Portion|portion]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106023.png" /> has positive measure. This implies that if the radial boundary values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106024.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106025.png" /> of the given type are equal to zero, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106027.png" />. Moreover, there is no meromorphic function in the unit disc that takes infinite radial boundary values on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l061/l061060/l06106028.png" /> of the given type.
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2) Let $w=f(z)$ be a meromorphic function in the unit disc $D=\{z:\left|z\right|<1\}$ other than a constant and having radial boundary values (finite or infinite) on a set $E$ situated on an arc $\sigma$ of the unit circle $\Gamma=\{z:\left|z\right|=1\}$ that is metrically dense and of the second Baire category (cf. [[Baire     classes]]) on $\sigma$. Then the set $W$ of its radial boundary values on $E$ contains at least two distinct points. Metric density of $E$ on $\sigma$ means that every [[Portion|portion]] of $E$ on $\sigma$ has positive measure. This implies that if the radial boundary values of $f(z)$ on a set $E$ of the given type are equal to zero, then $f(z)=0$ in $D$. Moreover, there is no meromorphic function in the unit disc that takes infinite radial boundary values on a set $E$ of the given type.
  
Luzin and Privalov (see [[#References|[1]]], [[#References|[2]]]) constructed examples to show that neither metric density nor second Baire category are by themselves sufficient for the assertion in 2 to hold.
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Luzin and Privalov (see {{Cite|LuPr}}, {{Cite|Pr}}) constructed examples to show that neither metric density nor second Baire category are by themselves sufficient for the assertion in 2 to hold.
  
See also [[Boundary properties of analytic functions|Boundary properties of analytic functions]]; [[Luzin examples|Luzin examples]]; [[Cluster set|Cluster set]]; [[Privalov theorem|Privalov theorem]]; [[Riesz theorem(2)|Riesz theorem]].
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See also [[Boundary properties of analytic functions]]; [[Luzin examples]]; [[Cluster set]]; [[Privalov theorem]]; [[Riesz theorem(2)|Riesz theorem]].
  
====References====
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====References====  
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> N.N. [N.N. Luzin] Lusin,   I.I. [I.I. Privalov] Priwaloff,   "Sur l'unicité et la multiplicité des fonctions analytiques" ''Ann. Sci. Ecole Norm. Sup. (3)'' , '''42''' (1925) pp. 143–191</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  A. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR></table>
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|valign="top"|{{Ref|CoLo}}||valign="top"| E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets", Cambridge Univ. Press (1966) pp. Chapt. 9
 
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====Comments====
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|valign="top"|{{Ref|Lo}}||valign="top"| A. Lohwater, "The boundary behaviour of analytic functions" ''Itogi Nauk. i Tekhn. Mat. Anal.'', '''10''' (1973) pp. 99–259 (In Russian)
 
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|valign="top"|{{Ref|LuPr}}||valign="top"| N.N. [N.N.   Luzin] Lusin, I.I. [I.I. Privalov] Priwaloff, "Sur l'unicité et la multiplicité des fonctions analytiques" ''Ann. Sci. Ecole Norm. Sup. (3)'', '''42''' (1925) pp. 143–191
====References====
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<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>
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|valign="top"|{{Ref|Pr}}||valign="top"| I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen", Deutsch. Verlag Wissenschaft.  (1956) (Translated from Russian)
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Latest revision as of 23:56, 24 July 2012

2010 Mathematics Subject Classification: Primary: 30D40 [MSN][ZBL]

The Luzin–Privalov theorems in the theory of functions of a complex variable are classical results of N.N. Luzin and I.I. Privalov that clarify the character of a boundary uniqueness property of analytic functions (cf. Uniqueness properties of analytic functions) (see [LuPr]).

1) Let $f(z)$ be a meromorphic function of the complex variable $z$ in a simply-connected domain $D$ with rectifiable boundary $\Gamma$. If $f(z)$ takes angular boundary values zero on a set $E\subset\Gamma$ of positive Lebesgue measure on $\Gamma$, then $f(z)=0$ in $D$. There is no function meromorphic in $D$ that has infinite angular boundary values on a set $E\subset\Gamma$ of positive measure.

2) Let $w=f(z)$ be a meromorphic function in the unit disc $D=\{z:\left|z\right|<1\}$ other than a constant and having radial boundary values (finite or infinite) on a set $E$ situated on an arc $\sigma$ of the unit circle $\Gamma=\{z:\left|z\right|=1\}$ that is metrically dense and of the second Baire category (cf. Baire classes) on $\sigma$. Then the set $W$ of its radial boundary values on $E$ contains at least two distinct points. Metric density of $E$ on $\sigma$ means that every portion of $E$ on $\sigma$ has positive measure. This implies that if the radial boundary values of $f(z)$ on a set $E$ of the given type are equal to zero, then $f(z)=0$ in $D$. Moreover, there is no meromorphic function in the unit disc that takes infinite radial boundary values on a set $E$ of the given type.

Luzin and Privalov (see [LuPr], [Pr]) constructed examples to show that neither metric density nor second Baire category are by themselves sufficient for the assertion in 2 to hold.

See also Boundary properties of analytic functions; Luzin examples; Cluster set; Privalov theorem; Riesz theorem.

References

[CoLo] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets", Cambridge Univ. Press (1966) pp. Chapt. 9
[Lo] A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. i Tekhn. Mat. Anal., 10 (1973) pp. 99–259 (In Russian)
[LuPr] N.N. [N.N. Luzin] Lusin, I.I. [I.I. Privalov] Priwaloff, "Sur l'unicité et la multiplicité des fonctions analytiques" Ann. Sci. Ecole Norm. Sup. (3), 42 (1925) pp. 143–191
[Pr] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen", Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
How to Cite This Entry:
Luzin-Privalov theorems. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Luzin-Privalov_theorems&oldid=27205
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article