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''of functions''
 
''of functions''
  
A class of functions characterized by a uniqueness property: If two functions of the class coincide  "locally" , then they are identical. The simplest quasi-analytic class is the class of analytic functions on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q0763701.png" /> on the real axis (a function of this class is represented in a sufficiently small neighbourhood of each point of the interval by a Taylor series): If two analytic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q0763702.png" /> are equal on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q0763703.png" />, then they are identical (to coincide  "locally"  here means equality of the functions in the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q0763704.png" />). Coincidence  "locally"  for analytic functions can also mean equality of the functions together with all their derivatives at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q0763705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q0763706.png" />. Coincidence  "locally"  in this new sense also implies equality of the functions on the whole interval.
+
A class of functions characterized by a uniqueness property: If two functions of the class coincide  "locally" , then they are identical. The simplest quasi-analytic class is the class of analytic functions on an interval $  [ a , b ] $
 +
on the real axis (a function of this class is represented in a sufficiently small neighbourhood of each point of the interval by a Taylor series): If two analytic functions on $  [ a , b ] $
 +
are equal on an interval $  ( \alpha , \beta ) \subset  [ a , b ] $,  
 +
then they are identical (to coincide  "locally"  here means equality of the functions in the interior of $  ( \alpha , \beta ) $).  
 +
Coincidence  "locally"  for analytic functions can also mean equality of the functions together with all their derivatives at some point $  x _ {0} $,  
 +
$  0 \leq  x _ {0} \leq  b $.  
 +
Coincidence  "locally"  in this new sense also implies equality of the functions on the whole interval.
  
E. Borel discovered that the uniqueness property holds not merely for analytic functions. In this connection J. Hadamard, 1912, posed the following problem. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q0763707.png" /> be a sequence of positive numbers and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q0763708.png" /> be some interval on the real axis. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q0763709.png" /> be the set of infinitely-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637011.png" /> for which
+
E. Borel discovered that the uniqueness property holds not merely for analytic functions. In this connection J. Hadamard, 1912, posed the following problem. Let $  \{ M _ {n} \} $
 +
be a sequence of positive numbers and let $  [ a , b ] $
 +
be some interval on the real axis. Let $  C \{ M _ {n} \} $
 +
be the set of infinitely-differentiable functions $  f $
 +
on $  [ a , b ] $
 +
for which
  
<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/q/q076/q076370/q07637012.png" /></td> </tr></table>
+
$$
 +
| f ^ { ( n) } ( x) |  \leq  K  ^ {n} M _ {n} ,\ \
 +
a \leq  x \leq  b ,\ \
 +
n = 0 , 1 \dots
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637013.png" /> is a constant not dependent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637014.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637015.png" /> is analytic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637016.png" /> if and only if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637017.png" />,
+
where $  K = K ( f  ) $
 +
is a constant not dependent on $  n $.  
 +
A function $  f $
 +
is analytic on $  [ a , b ] $
 +
if and only if for some $  K = K ( f  ) $,
  
<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/q/q076/q076370/q07637018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
| f ^ { ( n) } ( x) |  < K  ^ {n} n ! ,\ \
 +
a \leq  x \leq  b ,\ \
 +
n = 0 , 1 ,\dots .
 +
$$
  
Thus, the class of analytic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637019.png" /> is the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637020.png" />. Hadamard's problem consists in determining conditions on the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637021.png" /> such that every function in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637022.png" /> vanishing together with each of its derivatives at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637024.png" />, is identically zero (or, what is the same, such that two functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637025.png" /> that are equal together with all their derivatives at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637026.png" /> are equal everywhere). A class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637027.png" /> with this property is called quasi-analytic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637028.png" />. According to what has been said above, the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637029.png" /> is quasi-analytic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637030.png" />.
+
Thus, the class of analytic functions on $  [ a , b ] $
 +
is the class $  C \{ n ! \} $.  
 +
Hadamard's problem consists in determining conditions on the numbers $  M _ {n} $
 +
such that every function in the class $  C \{ M _ {n} \} $
 +
vanishing together with each of its derivatives at some point $  \alpha _ {0} $,  
 +
$  a \leq  \alpha _ {0} \leq  b $,  
 +
is identically zero (or, what is the same, such that two functions in $  C \{ M _ {n} \} $
 +
that are equal together with all their derivatives at a point $  \alpha _ {0} $
 +
are equal everywhere). A class $  C \{ M _ {n} \} $
 +
with this property is called quasi-analytic on $  [ a , b ] $.  
 +
According to what has been said above, the class $  C \{ n ! \} $
 +
is quasi-analytic on $  [ a , b ] $.
  
 
A. Denjoy, 1921, gave sufficient conditions for quasi-analyticity. He pointed out that if
 
A. Denjoy, 1921, gave sufficient conditions for quasi-analyticity. He pointed out that if
  
<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/q/q076/q076370/q07637031.png" /></td> </tr></table>
+
$$
 +
M _ {n}  = n ! (  \mathop{\rm ln}  n )  ^ {n} ,\
 +
M _ {n}  = n ! (  \mathop{\rm ln}  n )  ^ {n} ( { \mathop{\rm ln}  \mathop{\rm ln} }  n )  ^ {n}
 +
\dots
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637032.png" /> is quasi-analytic (these classes are, by virtue of (1), wider than the class of analytic functions).
+
then $  C \{ M _ {n} \} $
 +
is quasi-analytic (these classes are, by virtue of (1), wider than the class of analytic functions).
  
T. Carleman completely solved Hadamard's problem by giving necessary and sufficient conditions for quasi-analyticity. These conditions were subsequently modified. The Denjoy–Carleman quasi-analyticity theorem is stated as follows: Each of the following conditions is necessary and sufficient for the quasi-analyticity of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637033.png" />:
+
T. Carleman completely solved Hadamard's problem by giving necessary and sufficient conditions for quasi-analyticity. These conditions were subsequently modified. The Denjoy–Carleman quasi-analyticity theorem is stated as follows: Each of the following conditions is necessary and sufficient for the quasi-analyticity of the class $  C \{ M _ {n} \} $:
  
 
a) if one sets
 
a) if one sets
  
<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/q/q076/q076370/q07637034.png" /></td> </tr></table>
+
$$
 +
\beta _ {n}  = \
 +
\inf _ {k \geq  n }  M _ {k}  ^ {1/k} ,
 +
$$
  
 
then
 
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/q/q076/q076370/q07637035.png" /></td> </tr></table>
+
$$
 +
\sum _ { n= } 1 ^  \infty 
 +
 
 +
\frac{1}{\beta _ {n} }
 +
  = \infty ;
 +
$$
  
 
b) if one sets
 
b) if one sets
  
<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/q/q076/q076370/q07637036.png" /></td> </tr></table>
+
$$
 +
T ( r)  = \sup _ {n \geq  1 } \
 +
 
 +
\frac{r  ^ {n} }{M _ {n} }
 +
,
 +
$$
  
 
then
 
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/q/q076/q076370/q07637037.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { 0 } ^  \infty 
 +
 
 +
\frac{ \mathop{\rm ln}  T ( r) }{r  ^ {2} }
 +
\
 +
d r  = \infty ;
 +
$$
  
 
c) either
 
c) either
  
<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/q/q076/q076370/q07637038.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\overline{ {n \rightarrow \infty }}\; } \
 +
M _ {n}  ^ {1/n}  < \infty ,
 +
$$
  
 
or
 
or
  
<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/q/q076/q076370/q07637039.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty }  M _ {n}  ^ {1/n}  = \infty
 +
$$
  
 
and
 
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/q/q076/q076370/q07637040.png" /></td> </tr></table>
+
$$
 +
\sum ^  \infty 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637041.png" /> is the convex regularization by means of logarithms of the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637042.png" />. Condition a) is called the Carleman condition, b) the Ostrowski condition, c) the Wang–Mandelbrojt condition.
+
\frac{M _ {n}  ^  \prime  }{M _ {n+} 1  ^  \prime  }
  
For the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637043.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637044.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637045.png" />), one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637046.png" />, condition a) holds, and again one finds that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637047.png" /> is a quasi-analytic class. For the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637048.png" />, one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637049.png" />, condition a) holds, and hence the Denjoy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637050.png" /> is quasi-analytic. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637052.png" />, one has
+
= \infty ,
 +
$$
  
<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/q/q076/q076370/q07637053.png" /></td> </tr></table>
+
where  $  \{ M _ {n}  ^  \prime  \} $
 +
is the convex regularization by means of logarithms of the sequence  $  \{ M _ {n} \} $.  
 +
Condition a) is called the Carleman condition, b) the Ostrowski condition, c) the Wang–Mandelbrojt condition.
  
as a result of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637054.png" /> is not quasi-analytic.
+
For the case  $  M _ {n} = n ! = n  ^ {n} / ( e + \epsilon _ {n} )  ^ {n} $(
 +
$  \epsilon _ {n} \rightarrow 0 $
 +
as  $  n \rightarrow \infty $),
 +
one has  $  \beta _ {n} \approx n / e $,
 +
condition a) holds, and again one finds that  $  C \{ n ! \} $
 +
is a quasi-analytic class. For the case  $  M _ {n} = n ! (  \mathop{\rm ln}  n )  ^ {n} $,
 +
one has  $  \beta _ {n} \approx ( n  \mathop{\rm ln}  n ) / e $,
 +
condition a) holds, and hence the Denjoy class  $  C \{ n (  \mathop{\rm ln}  n )  ^ {n} \} $
 +
is quasi-analytic. In the case  $  M = n ! (  \mathop{\rm ln} ^ {1 + \epsilon }  n )  ^ {n} $,
 +
$  \epsilon > 0 $,
 +
one has
  
S.N. Bernshtein introduced other quasi-analytic classes of functions. He showed that a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637055.png" /> is analytic on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637056.png" /> if and only if
+
$$
 +
\beta _ {n}  \approx \
  
<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/q/q076/q076370/q07637057.png" /></td> </tr></table>
+
\frac{n  \mathop{\rm ln} ^ {1 + \epsilon }  n }{e}
 +
,\ \
 +
\sum _ { n= } 1 ^  \infty 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637059.png" /> do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637060.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637061.png" /> is the best approximation error of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637062.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637063.png" /> by polynomials of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637064.png" />. With this in mind, he considered the class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637065.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637066.png" /> satisfying the condition
+
\frac{1}{\beta _ {n} }
 +
  < \infty ,
 +
$$
  
<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/q/q076/q076370/q07637067.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
as a result of which  $  C \{ M _ {n} \} $
 +
is not quasi-analytic.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637068.png" /> is an infinite increasing sequence of integers, and showed that if a function of this class vanishes on some interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637069.png" />, then it is identically zero. A class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637070.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637071.png" /> is called (Bernshtein) quasi-analytic if two functions of this class that are equal on some segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637072.png" /> are necessarily equal on the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637073.png" />. The class (2) is quasi-analytic in this sense. It should be noted that (2) does not imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637074.png" /> is infinitely differentiable (there are appropriate examples).
+
S.N. Bernshtein introduced other quasi-analytic classes of functions. He showed that a function $  f $
 +
is analytic on an interval $  [ a , b ] $
 +
if and only if
  
Other problems of quasi-analyticity have also been studied. For example, the question has been solved concerning the rate of decrease of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637076.png" /> in the series
+
$$
 +
E _ {n} ( f  )  <  M \rho  ^ {n} ,\ \
 +
n = 0 , 1 \dots \  \rho < 1 ,
 +
$$
  
<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/q/q076/q076370/q07637077.png" /></td> </tr></table>
+
where  $  M = M ( f  ) $
 +
and  $  \rho = \rho ( f  ) $
 +
do not depend on  $  n $,
 +
and  $  E _ {n} ( f  ) $
 +
is the best approximation error of  $  f $
 +
on  $  [ a , b ] $
 +
by polynomials of degree  $  n $.  
 +
With this in mind, he considered the class of functions  $  f $
 +
on  $  [ a , b ] $
 +
satisfying the condition
  
under which the class of such functions is quasi-analytic; conditions have been found on the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637078.png" /> such that functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637079.png" /> that are analytic in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637080.png" />, are infinitely differentiable on the closed disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637081.png" /> and satisfy the conditions
+
$$ \tag{2 }
 +
E _ {n} ( f  )  < M \rho  ^ {n} ,\  \rho < 1 ,\  n= n _ {1} , n _ {2} \dots
 +
$$
  
<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/q/q076/q076370/q07637082.png" /></td> </tr></table>
+
where  $  n _ {1} , n _ {2} ,\dots $
 +
is an infinite increasing sequence of integers, and showed that if a function of this class vanishes on some interval  $  ( \alpha , \beta ) \subset  [ a , b ] $,
 +
then it is identically zero. A class  $  C $
 +
defined on  $  [ a , b ] $
 +
is called (Bernshtein) quasi-analytic if two functions of this class that are equal on some segment  $  ( \alpha , \beta ) \subset  [ a , b ] $
 +
are necessarily equal on the whole of  $  [ a , b ] $.
 +
The class (2) is quasi-analytic in this sense. It should be noted that (2) does not imply that  $  f $
 +
is infinitely differentiable (there are appropriate examples).
 +
 
 +
Other problems of quasi-analyticity have also been studied. For example, the question has been solved concerning the rate of decrease of the coefficients  $  a _ {n} $
 +
and  $  b _ {n} $
 +
in the series
 +
 
 +
$$
 +
f ( x)  = \sum _ { n= } 0 ^  \infty 
 +
( a _ {n}  \cos  n x + b _ {n}  \sin  n x )
 +
$$
 +
 
 +
under which the class of such functions is quasi-analytic; conditions have been found on the numbers  $  M _ {n} $
 +
such that functions  $  f ( z) $
 +
that are analytic in the disc  $  | z | < 1 $,
 +
are infinitely differentiable on the closed disc  $  | z | \leq  1 $
 +
and satisfy the conditions
 +
 
 +
$$
 +
| f ^ { ( n) } ( x) |  \leq  K  ^ {n} M _ {n} ,\ \
 +
n = 0 , 1 \dots \  | z | \leq  1 ,
 +
$$
  
 
form a quasi-analytic class; etc.
 
form a quasi-analytic class; etc.
Line 79: Line 217:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.N. Bernshtein,  "Collected works" , '''2''' , Moscow  (1964)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Mandelbrojt,  "Séries de Fourier et classes quasi-analytiques de fonctions" , Gauthier-Villars  (1935)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Mandelbrojt,  "Séries adhérentes, régularisations des suites, applications" , Gauthier-Villars  (1952)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.N. Bernshtein,  "Collected works" , '''2''' , Moscow  (1964)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Mandelbrojt,  "Séries de Fourier et classes quasi-analytiques de fonctions" , Gauthier-Villars  (1935)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Mandelbrojt,  "Séries adhérentes, régularisations des suites, applications" , Gauthier-Villars  (1952)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The original paper by Denjoy is [[#References|[a4]]], for Carleman's work see also [[#References|[a5]]]. For a neat proof of the Denjoy–Carleman theorem see [[#References|[a1]]].
 
The original paper by Denjoy is [[#References|[a4]]], for Carleman's work see also [[#References|[a5]]]. For a neat proof of the Denjoy–Carleman theorem see [[#References|[a1]]].
  
Quasi-analytic classes have also been introduced on certain arcs in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076370/q07637083.png" />, and the sufficiency part of the Denjoy–Carleman survives in this setting, cf. [[#References|[a3]]].
+
Quasi-analytic classes have also been introduced on certain arcs in $  \mathbf C $,  
 +
and the sufficiency part of the Denjoy–Carleman survives in this setting, cf. [[#References|[a3]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1987)  pp. 24</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)  pp. Chapt. 1</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Zeinstra,  "Müntz–Szász approximation on curves and area problems for zero sets" , Univ. Amsterdam  (1985)  (Thesis)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Denjoy,  "Sur les fonctions quasi-analytiques de variable réelle"  ''C.R. Acad. Sci. Paris'' , '''173'''  (1921)  pp. 1329–1331</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Carleman,  "Les fonctions quasi-analytiques" , Gauthier-Villars  (1926)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Mandelbrojt,  "Analytic functions and classes of infinitely differentiable functions" , ''Pamphlet'' , '''29''' :  1 , Rice Univ.  (1942)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Beurling,  "Quasi-analyticity" , ''Collected works'' , '''I''' , Birkhäuser  (1989)  pp. 396–431</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Real and complex analysis" , McGraw-Hill  (1987)  pp. 24</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.V. Hörmander,  "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)  pp. Chapt. 1</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Zeinstra,  "Müntz–Szász approximation on curves and area problems for zero sets" , Univ. Amsterdam  (1985)  (Thesis)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Denjoy,  "Sur les fonctions quasi-analytiques de variable réelle"  ''C.R. Acad. Sci. Paris'' , '''173'''  (1921)  pp. 1329–1331</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Carleman,  "Les fonctions quasi-analytiques" , Gauthier-Villars  (1926)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Mandelbrojt,  "Analytic functions and classes of infinitely differentiable functions" , ''Pamphlet'' , '''29''' :  1 , Rice Univ.  (1942)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Beurling,  "Quasi-analyticity" , ''Collected works'' , '''I''' , Birkhäuser  (1989)  pp. 396–431</TD></TR></table>

Latest revision as of 08:09, 6 June 2020


of functions

A class of functions characterized by a uniqueness property: If two functions of the class coincide "locally" , then they are identical. The simplest quasi-analytic class is the class of analytic functions on an interval $ [ a , b ] $ on the real axis (a function of this class is represented in a sufficiently small neighbourhood of each point of the interval by a Taylor series): If two analytic functions on $ [ a , b ] $ are equal on an interval $ ( \alpha , \beta ) \subset [ a , b ] $, then they are identical (to coincide "locally" here means equality of the functions in the interior of $ ( \alpha , \beta ) $). Coincidence "locally" for analytic functions can also mean equality of the functions together with all their derivatives at some point $ x _ {0} $, $ 0 \leq x _ {0} \leq b $. Coincidence "locally" in this new sense also implies equality of the functions on the whole interval.

E. Borel discovered that the uniqueness property holds not merely for analytic functions. In this connection J. Hadamard, 1912, posed the following problem. Let $ \{ M _ {n} \} $ be a sequence of positive numbers and let $ [ a , b ] $ be some interval on the real axis. Let $ C \{ M _ {n} \} $ be the set of infinitely-differentiable functions $ f $ on $ [ a , b ] $ for which

$$ | f ^ { ( n) } ( x) | \leq K ^ {n} M _ {n} ,\ \ a \leq x \leq b ,\ \ n = 0 , 1 \dots $$

where $ K = K ( f ) $ is a constant not dependent on $ n $. A function $ f $ is analytic on $ [ a , b ] $ if and only if for some $ K = K ( f ) $,

$$ \tag{1 } | f ^ { ( n) } ( x) | < K ^ {n} n ! ,\ \ a \leq x \leq b ,\ \ n = 0 , 1 ,\dots . $$

Thus, the class of analytic functions on $ [ a , b ] $ is the class $ C \{ n ! \} $. Hadamard's problem consists in determining conditions on the numbers $ M _ {n} $ such that every function in the class $ C \{ M _ {n} \} $ vanishing together with each of its derivatives at some point $ \alpha _ {0} $, $ a \leq \alpha _ {0} \leq b $, is identically zero (or, what is the same, such that two functions in $ C \{ M _ {n} \} $ that are equal together with all their derivatives at a point $ \alpha _ {0} $ are equal everywhere). A class $ C \{ M _ {n} \} $ with this property is called quasi-analytic on $ [ a , b ] $. According to what has been said above, the class $ C \{ n ! \} $ is quasi-analytic on $ [ a , b ] $.

A. Denjoy, 1921, gave sufficient conditions for quasi-analyticity. He pointed out that if

$$ M _ {n} = n ! ( \mathop{\rm ln} n ) ^ {n} ,\ M _ {n} = n ! ( \mathop{\rm ln} n ) ^ {n} ( { \mathop{\rm ln} \mathop{\rm ln} } n ) ^ {n} \dots $$

then $ C \{ M _ {n} \} $ is quasi-analytic (these classes are, by virtue of (1), wider than the class of analytic functions).

T. Carleman completely solved Hadamard's problem by giving necessary and sufficient conditions for quasi-analyticity. These conditions were subsequently modified. The Denjoy–Carleman quasi-analyticity theorem is stated as follows: Each of the following conditions is necessary and sufficient for the quasi-analyticity of the class $ C \{ M _ {n} \} $:

a) if one sets

$$ \beta _ {n} = \ \inf _ {k \geq n } M _ {k} ^ {1/k} , $$

then

$$ \sum _ { n= } 1 ^ \infty \frac{1}{\beta _ {n} } = \infty ; $$

b) if one sets

$$ T ( r) = \sup _ {n \geq 1 } \ \frac{r ^ {n} }{M _ {n} } , $$

then

$$ \int\limits _ { 0 } ^ \infty \frac{ \mathop{\rm ln} T ( r) }{r ^ {2} } \ d r = \infty ; $$

c) either

$$ \lim\limits _ {\overline{ {n \rightarrow \infty }}\; } \ M _ {n} ^ {1/n} < \infty , $$

or

$$ \lim\limits _ {n \rightarrow \infty } M _ {n} ^ {1/n} = \infty $$

and

$$ \sum ^ \infty \frac{M _ {n} ^ \prime }{M _ {n+} 1 ^ \prime } = \infty , $$

where $ \{ M _ {n} ^ \prime \} $ is the convex regularization by means of logarithms of the sequence $ \{ M _ {n} \} $. Condition a) is called the Carleman condition, b) the Ostrowski condition, c) the Wang–Mandelbrojt condition.

For the case $ M _ {n} = n ! = n ^ {n} / ( e + \epsilon _ {n} ) ^ {n} $( $ \epsilon _ {n} \rightarrow 0 $ as $ n \rightarrow \infty $), one has $ \beta _ {n} \approx n / e $, condition a) holds, and again one finds that $ C \{ n ! \} $ is a quasi-analytic class. For the case $ M _ {n} = n ! ( \mathop{\rm ln} n ) ^ {n} $, one has $ \beta _ {n} \approx ( n \mathop{\rm ln} n ) / e $, condition a) holds, and hence the Denjoy class $ C \{ n ( \mathop{\rm ln} n ) ^ {n} \} $ is quasi-analytic. In the case $ M = n ! ( \mathop{\rm ln} ^ {1 + \epsilon } n ) ^ {n} $, $ \epsilon > 0 $, one has

$$ \beta _ {n} \approx \ \frac{n \mathop{\rm ln} ^ {1 + \epsilon } n }{e} ,\ \ \sum _ { n= } 1 ^ \infty \frac{1}{\beta _ {n} } < \infty , $$

as a result of which $ C \{ M _ {n} \} $ is not quasi-analytic.

S.N. Bernshtein introduced other quasi-analytic classes of functions. He showed that a function $ f $ is analytic on an interval $ [ a , b ] $ if and only if

$$ E _ {n} ( f ) < M \rho ^ {n} ,\ \ n = 0 , 1 \dots \ \rho < 1 , $$

where $ M = M ( f ) $ and $ \rho = \rho ( f ) $ do not depend on $ n $, and $ E _ {n} ( f ) $ is the best approximation error of $ f $ on $ [ a , b ] $ by polynomials of degree $ n $. With this in mind, he considered the class of functions $ f $ on $ [ a , b ] $ satisfying the condition

$$ \tag{2 } E _ {n} ( f ) < M \rho ^ {n} ,\ \rho < 1 ,\ n= n _ {1} , n _ {2} \dots $$

where $ n _ {1} , n _ {2} ,\dots $ is an infinite increasing sequence of integers, and showed that if a function of this class vanishes on some interval $ ( \alpha , \beta ) \subset [ a , b ] $, then it is identically zero. A class $ C $ defined on $ [ a , b ] $ is called (Bernshtein) quasi-analytic if two functions of this class that are equal on some segment $ ( \alpha , \beta ) \subset [ a , b ] $ are necessarily equal on the whole of $ [ a , b ] $. The class (2) is quasi-analytic in this sense. It should be noted that (2) does not imply that $ f $ is infinitely differentiable (there are appropriate examples).

Other problems of quasi-analyticity have also been studied. For example, the question has been solved concerning the rate of decrease of the coefficients $ a _ {n} $ and $ b _ {n} $ in the series

$$ f ( x) = \sum _ { n= } 0 ^ \infty ( a _ {n} \cos n x + b _ {n} \sin n x ) $$

under which the class of such functions is quasi-analytic; conditions have been found on the numbers $ M _ {n} $ such that functions $ f ( z) $ that are analytic in the disc $ | z | < 1 $, are infinitely differentiable on the closed disc $ | z | \leq 1 $ and satisfy the conditions

$$ | f ^ { ( n) } ( x) | \leq K ^ {n} M _ {n} ,\ \ n = 0 , 1 \dots \ | z | \leq 1 , $$

form a quasi-analytic class; etc.

References

[1] S.N. Bernshtein, "Collected works" , 2 , Moscow (1964) (In Russian)
[2] S. Mandelbrojt, "Séries de Fourier et classes quasi-analytiques de fonctions" , Gauthier-Villars (1935)
[3] S. Mandelbrojt, "Séries adhérentes, régularisations des suites, applications" , Gauthier-Villars (1952)

Comments

The original paper by Denjoy is [a4], for Carleman's work see also [a5]. For a neat proof of the Denjoy–Carleman theorem see [a1].

Quasi-analytic classes have also been introduced on certain arcs in $ \mathbf C $, and the sufficiency part of the Denjoy–Carleman survives in this setting, cf. [a3].

References

[a1] W. Rudin, "Real and complex analysis" , McGraw-Hill (1987) pp. 24
[a2] L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983) pp. Chapt. 1
[a3] R. Zeinstra, "Müntz–Szász approximation on curves and area problems for zero sets" , Univ. Amsterdam (1985) (Thesis)
[a4] A. Denjoy, "Sur les fonctions quasi-analytiques de variable réelle" C.R. Acad. Sci. Paris , 173 (1921) pp. 1329–1331
[a5] T. Carleman, "Les fonctions quasi-analytiques" , Gauthier-Villars (1926)
[a6] S. Mandelbrojt, "Analytic functions and classes of infinitely differentiable functions" , Pamphlet , 29 : 1 , Rice Univ. (1942)
[a7] A. Beurling, "Quasi-analyticity" , Collected works , I , Birkhäuser (1989) pp. 396–431
How to Cite This Entry:
Quasi-analytic class. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-analytic_class&oldid=48375
This article was adapted from an original article by A.F. Leont'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article