Namespaces
Variants
Actions

Difference between revisions of "Abstract analytic function"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
 +
<!--
 +
a0104601.png
 +
$#A+1 = 91 n = 0
 +
$#C+1 = 91 : ~/encyclopedia/old_files/data/A010/A.0100460 Abstract analytic function,
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
''analytic mapping of Banach spaces''
 
''analytic mapping of Banach spaces''
  
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a0104601.png" /> acting from some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a0104602.png" /> of a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a0104603.png" /> into a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a0104604.png" /> that is differentiable according to Fréchet everywhere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a0104605.png" />, i.e. is such that for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a0104606.png" /> there exists a bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a0104607.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a0104608.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a0104609.png" /> for which the following relation is true:
+
A function $  f(x) $
 +
acting from some domain $  D $
 +
of a Banach space $  X $
 +
into a Banach space $  Y $
 +
that is differentiable according to Fréchet everywhere in $  D $,  
 +
i.e. is such that for any point a \in D $
 +
there exists a bounded linear operator $  \delta f ( a , \cdot ) $
 +
from $  X $
 +
into $  Y $
 +
for which the following relation is true:
  
<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/a/a010/a010460/a01046010.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\| h \| \rightarrow 0 } \
 +
\| h \|  ^ {-1}  \cdot \
 +
\| f ( a + h ) - f ( a ) - \delta f
 +
( a , h ) \|  = 0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046011.png" /> denotes the norm on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046012.png" /> or on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046013.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046014.png" /> is called the Fréchet differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046015.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046016.png" />.
+
where $  \| \cdot \| $
 +
denotes the norm on $  X $
 +
or on $  Y $;  
 +
$  \delta f ( a , h ) $
 +
is called the Fréchet differential of $  f $
 +
at a $.
  
Another approach to the notion of an abstract analytic function is based on differentiability according to Gâteaux. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046017.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046018.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046019.png" /> is weakly analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046020.png" />, or differentiable according to Gâteaux in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046021.png" />, if for each continuous linear functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046023.png" /> and each element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046024.png" /> the complex function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046025.png" /> is a holomorphic function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046026.png" /> in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046028.png" />. Any abstract analytic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046029.png" /> is continuous and weakly analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046030.png" />. The converse proposition is also true, and the continuity condition can be replaced by local boundedness or by continuity according to Baire.
+
Another approach to the notion of an abstract analytic function is based on differentiability according to Gâteaux. A function $  f(x) $
 +
from $  D $
 +
into $  Y $
 +
is weakly analytic in $  D $,  
 +
or differentiable according to Gâteaux in $  D $,  
 +
if for each continuous linear functional $  y  ^  \prime  $
 +
on $  Y $
 +
and each element $  h \in X $
 +
the complex function $  y  ^  \prime  ( f ( x + \xi h ) ) $
 +
is a holomorphic function of the complex variable $  \xi $
 +
in the disc $  | \xi | < \rho ( x , h ) $,  
 +
where $  \rho ( x, h ) = \sup \{ {| \xi | } : {x + \xi h \in D } \} $.  
 +
Any abstract analytic function in a domain $  D $
 +
is continuous and weakly analytic in $  D $.  
 +
The converse proposition is also true, and the continuity condition can be replaced by local boundedness or by continuity according to Baire.
  
The term  "abstract analytic function"  is sometimes employed in a narrower sense, when it means a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046031.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046032.png" /> with values in a Banach space or even in a locally convex linear topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046033.png" />. In such a case any weakly analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046034.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046035.png" /> of the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046036.png" /> is an abstract analytic function. One can also say that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046037.png" /> is an abstract analytic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046038.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046039.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046040.png" /> and if for any simple closed rectifiable contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046041.png" /> the integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046042.png" /> vanishes. For an abstract analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046043.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046044.png" /> the Cauchy formula (cf. [[Cauchy integral|Cauchy integral]]) is valid.
+
The term  "abstract analytic function"  is sometimes employed in a narrower sense, when it means a function $  f(z) $
 +
of a complex variable $  z $
 +
with values in a Banach space or even in a locally convex linear topological space $  Y $.  
 +
In such a case any weakly analytic function $  f(z) $
 +
in a domain $  D $
 +
of the complex plane $  \mathbf C $
 +
is an abstract analytic function. One can also say that a function $  f(z) $
 +
is an abstract analytic function in a domain $  D \subset  \mathbf C $
 +
if and only if $  f(z) $
 +
is continuous in $  D $
 +
and if for any simple closed rectifiable contour $  L \subset  D $
 +
the integral $  \int _ {L} f(z)  dz $
 +
vanishes. For an abstract analytic function $  f(z) $
 +
of a complex variable $  z $
 +
the Cauchy formula (cf. [[Cauchy integral|Cauchy integral]]) is valid.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046045.png" /> be a weakly analytic function in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046046.png" /> of a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046047.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046048.png" />, as a function of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046049.png" />, has derivatives of all orders in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046051.png" />, these derivatives being abstract analytic functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046052.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046053.png" />. If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046054.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046055.png" />, then
+
Let $  f(x) $
 +
be a weakly analytic function in a domain $  D $
 +
of a Banach space $  X $.  
 +
Then $  f ( x + \xi h ) $,  
 +
as a function of the complex variable $  \xi $,  
 +
has derivatives of all orders in the domain $  \widetilde{D}  = \{  \xi  : {x + \xi h \in D } \} $,  
 +
$  h \in X $,  
 +
these derivatives being abstract analytic functions from $  \widetilde{D}  $
 +
into $  Y $.  
 +
If the set $  \{ {x + \xi h } : {| \xi | \leq  1 } \} $
 +
belongs to $  D $,  
 +
then
  
<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/a/a010/a010460/a01046056.png" /></td> </tr></table>
+
$$
 +
f ( x + h )  = \sum _ {n = 0 } ^  \infty 
 +
 
 +
\frac{1}{n!}
 +
\delta  ^ {n} f ( x , h ) ,
 +
$$
  
 
where the series converges in norm, and
 
where the series converges in norm, and
  
<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/a/a010/a010460/a01046057.png" /></td> </tr></table>
+
$$
 +
\left .
 +
\delta  ^ {n} f ( x , h )  =
 +
\frac{d  ^ {n} }{d \xi  ^ {n} }
 +
 
 +
f ( x + \xi h ) \right | _ {\xi = 0 }  =
 +
$$
 +
 
 +
$$
 +
= \
  
<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/a/a010/a010460/a01046058.png" /></td> </tr></table>
+
\frac{1}{2 \pi i }
 +
\int\limits _ {| \xi | = 1 } f ( x + \xi h ) \xi ^ {- n - 1 }  d \xi .
 +
$$
  
A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046059.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046060.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046061.png" /> is called a polynomial with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046062.png" /> of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046063.png" /> if, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046064.png" /> and for all complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046065.png" />, one has
+
A function $  y = P(x) $
 +
from $  X $
 +
into $  Y $
 +
is called a polynomial with respect to the variable $  x $
 +
of degree at most $  m $
 +
if, for all $  x , h \in X $
 +
and for all complex $  \xi $,  
 +
one has
  
<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/a/a010/a010460/a01046066.png" /></td> </tr></table>
+
$$
 +
P ( x + \xi h )  = \sum _ {\nu = 0 } ^ { m }  P _  \nu  ( x , h ) \xi  ^  \nu  ,
 +
$$
  
where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046067.png" /> are independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046068.png" />. The degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046069.png" /> is exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046070.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046071.png" />. A power series is a series of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046072.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046073.png" /> are homogeneous polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046074.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046076.png" />, for all complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046077.png" />. An arbitrary weakly convergent power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046078.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046079.png" /> converges in norm towards some weakly analytic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046080.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046081.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046082.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046083.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046084.png" /> is an abstract analytic function if and only if it can be developed in a power series in a neighbourhood of all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046085.png" />
+
where the functions $  P _  \nu  (x, h ) $
 +
are independent of $  \xi $.  
 +
The degree of $  P(x) $
 +
is exactly $  m $
 +
if $  P _ {m} ( x , h ) \neq 0 $.  
 +
A power series is a series of the form $  \sum _ {n=0}  ^  \infty  P _ {n} (x) $
 +
where $  P _ {n} (x) $
 +
are homogeneous polynomials of degree $  n $
 +
so that $  P _ {n} ( \alpha x ) = \alpha  ^ {n} P _ {n} ( x ) $,  
 +
$  x \in X $,  
 +
for all complex $  \alpha $.  
 +
An arbitrary weakly convergent power series $  \sum _ {n=0}  ^  \infty  P _ {n} (x) $
 +
in a domain $  D $
 +
converges in norm towards some weakly analytic function $  f(x) $
 +
in $  D $,  
 +
and $  P _ {n} (x) = \delta  ^ {n} f(0, x)/n! $,  
 +
0 \in D $.  
 +
A function $  f(x) $
 +
is an abstract analytic function if and only if it can be developed in a power series in a neighbourhood of all points a \in D $
  
<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/a/a010/a010460/a01046086.png" /></td> </tr></table>
+
$$
 +
f ( a + h )  = \sum _ {n = 0 } ^  \infty  P _ {n} ( h ) ,
 +
$$
  
where all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046087.png" /> are continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046088.png" />.
+
where all $  P _ {n} (h) $
 +
are continuous in $  X $.
  
Many fundamental results in the classical theory of analytic functions — such as the [[Maximum-modulus principle|maximum-modulus principle]], the uniqueness theorems, the [[Vitali theorem|Vitali theorem]], the [[Liouville theorems|Liouville theorem]], etc. — are applicable to abstract analytic functions if suitable changes are introduced. The set of all analytic functions in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046089.png" /> forms a linear space.
+
Many fundamental results in the classical theory of analytic functions — such as the [[Maximum-modulus principle|maximum-modulus principle]], the uniqueness theorems, the [[Vitali theorem|Vitali theorem]], the [[Liouville theorems|Liouville theorem]], etc. — are applicable to abstract analytic functions if suitable changes are introduced. The set of all analytic functions in a domain $  D $
 +
forms a linear space.
  
The notion of an abstract analytic function can be generalized to wider classes of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010460/a01046091.png" />, such as locally convex topological spaces, Banach spaces over an arbitrary complete valuation field, etc.
+
The notion of an abstract analytic function can be generalized to wider classes of spaces $  X $
 +
and $  Y $,  
 +
such as locally convex topological spaces, Banach spaces over an arbitrary complete valuation field, etc.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Schwartz,  "Cours d'analyse" , '''2''' , Hermann  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hille,  R.S. Phillips,  "Functional analysis and semi-groups" , Amer. Math. Soc.  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Schwartz,  "Cours d'analyse" , '''2''' , Hermann  (1967)</TD></TR></table>

Latest revision as of 16:08, 1 April 2020


analytic mapping of Banach spaces

A function $ f(x) $ acting from some domain $ D $ of a Banach space $ X $ into a Banach space $ Y $ that is differentiable according to Fréchet everywhere in $ D $, i.e. is such that for any point $ a \in D $ there exists a bounded linear operator $ \delta f ( a , \cdot ) $ from $ X $ into $ Y $ for which the following relation is true:

$$ \lim\limits _ {\| h \| \rightarrow 0 } \ \| h \| ^ {-1} \cdot \ \| f ( a + h ) - f ( a ) - \delta f ( a , h ) \| = 0 , $$

where $ \| \cdot \| $ denotes the norm on $ X $ or on $ Y $; $ \delta f ( a , h ) $ is called the Fréchet differential of $ f $ at $ a $.

Another approach to the notion of an abstract analytic function is based on differentiability according to Gâteaux. A function $ f(x) $ from $ D $ into $ Y $ is weakly analytic in $ D $, or differentiable according to Gâteaux in $ D $, if for each continuous linear functional $ y ^ \prime $ on $ Y $ and each element $ h \in X $ the complex function $ y ^ \prime ( f ( x + \xi h ) ) $ is a holomorphic function of the complex variable $ \xi $ in the disc $ | \xi | < \rho ( x , h ) $, where $ \rho ( x, h ) = \sup \{ {| \xi | } : {x + \xi h \in D } \} $. Any abstract analytic function in a domain $ D $ is continuous and weakly analytic in $ D $. The converse proposition is also true, and the continuity condition can be replaced by local boundedness or by continuity according to Baire.

The term "abstract analytic function" is sometimes employed in a narrower sense, when it means a function $ f(z) $ of a complex variable $ z $ with values in a Banach space or even in a locally convex linear topological space $ Y $. In such a case any weakly analytic function $ f(z) $ in a domain $ D $ of the complex plane $ \mathbf C $ is an abstract analytic function. One can also say that a function $ f(z) $ is an abstract analytic function in a domain $ D \subset \mathbf C $ if and only if $ f(z) $ is continuous in $ D $ and if for any simple closed rectifiable contour $ L \subset D $ the integral $ \int _ {L} f(z) dz $ vanishes. For an abstract analytic function $ f(z) $ of a complex variable $ z $ the Cauchy formula (cf. Cauchy integral) is valid.

Let $ f(x) $ be a weakly analytic function in a domain $ D $ of a Banach space $ X $. Then $ f ( x + \xi h ) $, as a function of the complex variable $ \xi $, has derivatives of all orders in the domain $ \widetilde{D} = \{ \xi : {x + \xi h \in D } \} $, $ h \in X $, these derivatives being abstract analytic functions from $ \widetilde{D} $ into $ Y $. If the set $ \{ {x + \xi h } : {| \xi | \leq 1 } \} $ belongs to $ D $, then

$$ f ( x + h ) = \sum _ {n = 0 } ^ \infty \frac{1}{n!} \delta ^ {n} f ( x , h ) , $$

where the series converges in norm, and

$$ \left . \delta ^ {n} f ( x , h ) = \frac{d ^ {n} }{d \xi ^ {n} } f ( x + \xi h ) \right | _ {\xi = 0 } = $$

$$ = \ \frac{1}{2 \pi i } \int\limits _ {| \xi | = 1 } f ( x + \xi h ) \xi ^ {- n - 1 } d \xi . $$

A function $ y = P(x) $ from $ X $ into $ Y $ is called a polynomial with respect to the variable $ x $ of degree at most $ m $ if, for all $ x , h \in X $ and for all complex $ \xi $, one has

$$ P ( x + \xi h ) = \sum _ {\nu = 0 } ^ { m } P _ \nu ( x , h ) \xi ^ \nu , $$

where the functions $ P _ \nu (x, h ) $ are independent of $ \xi $. The degree of $ P(x) $ is exactly $ m $ if $ P _ {m} ( x , h ) \neq 0 $. A power series is a series of the form $ \sum _ {n=0} ^ \infty P _ {n} (x) $ where $ P _ {n} (x) $ are homogeneous polynomials of degree $ n $ so that $ P _ {n} ( \alpha x ) = \alpha ^ {n} P _ {n} ( x ) $, $ x \in X $, for all complex $ \alpha $. An arbitrary weakly convergent power series $ \sum _ {n=0} ^ \infty P _ {n} (x) $ in a domain $ D $ converges in norm towards some weakly analytic function $ f(x) $ in $ D $, and $ P _ {n} (x) = \delta ^ {n} f(0, x)/n! $, $ 0 \in D $. A function $ f(x) $ is an abstract analytic function if and only if it can be developed in a power series in a neighbourhood of all points $ a \in D $

$$ f ( a + h ) = \sum _ {n = 0 } ^ \infty P _ {n} ( h ) , $$

where all $ P _ {n} (h) $ are continuous in $ X $.

Many fundamental results in the classical theory of analytic functions — such as the maximum-modulus principle, the uniqueness theorems, the Vitali theorem, the Liouville theorem, etc. — are applicable to abstract analytic functions if suitable changes are introduced. The set of all analytic functions in a domain $ D $ forms a linear space.

The notion of an abstract analytic function can be generalized to wider classes of spaces $ X $ and $ Y $, such as locally convex topological spaces, Banach spaces over an arbitrary complete valuation field, etc.

References

[1] E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957)
[2] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)
[3] L. Schwartz, "Cours d'analyse" , 2 , Hermann (1967)
How to Cite This Entry:
Abstract analytic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Abstract_analytic_function&oldid=17260
This article was adapted from an original article by A.A. DanilevichE.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article