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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g0445801.png" /> is a complex summable function on a closed rectifiable Jordan curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g0445802.png" /> in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g0445803.png" />-plane, then a necessary and sufficient condition for the existence of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g0445804.png" />, regular in the interior of the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g0445805.png" /> bounded by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g0445806.png" /> and whose angular boundary values coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g0445807.png" /> almost-everywhere on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g0445808.png" />, is
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<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/g/g044/g044580/g0445809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|done}}
  
These conditions are known as the Golubev–Privalov conditions. That they are sufficient has been shown by V.V. Golubev [[#References|[1]]]; that they are necessary has been shown by I.I. Privalov [[#References|[2]]]. In other words, conditions (1) are necessary and sufficient for the integral of Cauchy–Lebesgue type (cf. [[Cauchy integral|Cauchy integral]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458010.png" /> constructed for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458011.png" /> and the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458012.png" />:
+
If  $  f( z) $
 +
is a complex summable function on a closed rectifiable Jordan curve  $  L $
 +
in the complex  $  z $-
 +
plane, then a necessary and sufficient condition for the existence of a function  $  F( z) $,
 +
regular in the interior of the domain  $  D $
 +
bounded by  $  L $
 +
and whose angular boundary values coincide with  $  f( z) $
 +
almost-everywhere on  $  L $,
 +
is
  
<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/g/g044/g044580/g04458013.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
\int\limits _ { L } z  ^ {n} f ( z)  dz  = 0,\ \
 +
n = 0, 1 , .  .  . .
 +
$$
 +
 
 +
These conditions are known as the Golubev–Privalov conditions. That they are sufficient has been shown by V.V. Golubev [[#References|[1]]]; that they are necessary has been shown by I.I. Privalov [[#References|[2]]]. In other words, conditions (1) are necessary and sufficient for the integral of Cauchy–Lebesgue type (cf. [[Cauchy integral|Cauchy integral]])  $  F( z) $
 +
constructed for the function  $  f( z) $
 +
and the curve  $  L $:
 +
 
 +
$$
 +
F ( z)  = \
 +
 
 +
\frac{1}{2 \pi i }
 +
 
 +
\int\limits _ { L }
 +
\frac{f ( \zeta )  d \zeta }{\zeta - z }
 +
,\ \
 +
z \in D,
 +
$$
  
 
to be a Cauchy–Lebesgue integral.
 
to be a Cauchy–Lebesgue integral.
  
In a more general formulation, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458014.png" /> be a complex [[Borel measure|Borel measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458015.png" />. Then the integral of Cauchy–Stieltjes type (cf. [[Cauchy integral|Cauchy integral]]),
+
In a more general formulation, let $  \mu $
 +
be a complex [[Borel measure|Borel measure]] on $  L $.  
 +
Then the integral of Cauchy–Stieltjes type (cf. [[Cauchy integral|Cauchy integral]]),
 +
 
 +
$$
 +
F ( z)  = \
  
<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/g/g044/g044580/g04458016.png" /></td> </tr></table>
+
\frac{1}{2 \pi i }
 +
\int\limits
 +
\frac{d \mu ( \zeta ) }{\zeta - z }
 +
,\ \
 +
z \in D,
 +
$$
  
 
is a Cauchy–Stieltjes integral if and only if the generalized Golubev–Privalov conditions
 
is a Cauchy–Stieltjes integral if and only if the generalized Golubev–Privalov conditions
  
<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/g/g044/g044580/g04458017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
\int\limits z  ^ {n}  d \mu ( z)  = 0,\ \
 +
n = 0, 1 \dots
 +
$$
  
 
are satisfied.
 
are satisfied.
  
In other words, conditions (2) are necessary and sufficient for the existence of a regular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458019.png" /> such that its angular boundary values coincide almost-everywhere (with respect to Lebesgue measure) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458020.png" /> with
+
In other words, conditions (2) are necessary and sufficient for the existence of a regular function $  F( z) $
 +
in $  D $
 +
such that its angular boundary values coincide almost-everywhere (with respect to Lebesgue measure) on $  L $
 +
with
  
<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/g/g044/g044580/g04458021.png" /></td> </tr></table>
+
$$
 +
e ^ {- i \phi ( z) } \mu  ^  \prime  ( z),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458022.png" /> is the angle between the positive direction of the abscissa axis and the tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458023.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458025.png" /> is the derivative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458026.png" /> with respect to Lebesgue measure (arc length) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044580/g04458027.png" />.
+
where $  \phi ( z) $
 +
is the angle between the positive direction of the abscissa axis and the tangent to $  L $
 +
at the point $  z \in L $
 +
and $  \mu  ^  \prime  ( z) $
 +
is the derivative of $  \mu $
 +
with respect to Lebesgue measure (arc length) on $  L $.
  
 
The Golubev–Privalov theorem is of importance in the theory of [[Boundary properties of analytic functions|boundary properties of analytic functions]].
 
The Golubev–Privalov theorem is of importance in the theory of [[Boundary properties of analytic functions|boundary properties of analytic functions]].
Line 29: Line 86:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Golubev,  "Univalent analytic functions with perfect sets of singular points" , Moscow  (1916)  (In Russian)  (See also: V.V. Golubev, Single-valued analytic functions. Automorphic functions, Moscow, 1961 (in Russian))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. Privalov,  "The Cauchy integral" , Saratov  (1918)  (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></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.V. Golubev,  "Univalent analytic functions with perfect sets of singular points" , Moscow  (1916)  (In Russian)  (See also: V.V. Golubev, Single-valued analytic functions. Automorphic functions, Moscow, 1961 (in Russian))</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. Privalov,  "The Cauchy integral" , Saratov  (1918)  (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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR></table>

Revision as of 19:42, 5 June 2020


If $ f( z) $ is a complex summable function on a closed rectifiable Jordan curve $ L $ in the complex $ z $- plane, then a necessary and sufficient condition for the existence of a function $ F( z) $, regular in the interior of the domain $ D $ bounded by $ L $ and whose angular boundary values coincide with $ f( z) $ almost-everywhere on $ L $, is

$$ \tag{1 } \int\limits _ { L } z ^ {n} f ( z) dz = 0,\ \ n = 0, 1 , . . . . $$

These conditions are known as the Golubev–Privalov conditions. That they are sufficient has been shown by V.V. Golubev [1]; that they are necessary has been shown by I.I. Privalov [2]. In other words, conditions (1) are necessary and sufficient for the integral of Cauchy–Lebesgue type (cf. Cauchy integral) $ F( z) $ constructed for the function $ f( z) $ and the curve $ L $:

$$ F ( z) = \ \frac{1}{2 \pi i } \int\limits _ { L } \frac{f ( \zeta ) d \zeta }{\zeta - z } ,\ \ z \in D, $$

to be a Cauchy–Lebesgue integral.

In a more general formulation, let $ \mu $ be a complex Borel measure on $ L $. Then the integral of Cauchy–Stieltjes type (cf. Cauchy integral),

$$ F ( z) = \ \frac{1}{2 \pi i } \int\limits \frac{d \mu ( \zeta ) }{\zeta - z } ,\ \ z \in D, $$

is a Cauchy–Stieltjes integral if and only if the generalized Golubev–Privalov conditions

$$ \tag{2 } \int\limits z ^ {n} d \mu ( z) = 0,\ \ n = 0, 1 \dots $$

are satisfied.

In other words, conditions (2) are necessary and sufficient for the existence of a regular function $ F( z) $ in $ D $ such that its angular boundary values coincide almost-everywhere (with respect to Lebesgue measure) on $ L $ with

$$ e ^ {- i \phi ( z) } \mu ^ \prime ( z), $$

where $ \phi ( z) $ is the angle between the positive direction of the abscissa axis and the tangent to $ L $ at the point $ z \in L $ and $ \mu ^ \prime ( z) $ is the derivative of $ \mu $ with respect to Lebesgue measure (arc length) on $ L $.

The Golubev–Privalov theorem is of importance in the theory of boundary properties of analytic functions.

References

[1] V.V. Golubev, "Univalent analytic functions with perfect sets of singular points" , Moscow (1916) (In Russian) (See also: V.V. Golubev, Single-valued analytic functions. Automorphic functions, Moscow, 1961 (in Russian))
[2] I.I. Privalov, "The Cauchy integral" , Saratov (1918) (In Russian)
[3] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)

Comments

References

[a1] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
How to Cite This Entry:
Golubev-Privalov theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Golubev-Privalov_theorem&oldid=22521
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article