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A [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n0666301.png" /> consisting of functions defined on an open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n0666302.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n0666303.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n0666304.png" /> and having certain difference-differentiability properties characterized by a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n0666305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n0666306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n0666307.png" />, in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n0666308.png" />-metric, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n0666309.png" />. The concept was introduced by S.M. Nikol'skii.
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The Nikol'skii space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663010.png" /> can be described in terms of properties of the partial derivatives of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663011.png" /> in the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663014.png" /> is an integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663016.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663017.png" /> denotes the difference of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663018.png" /> and of step <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663019.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663020.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663021.png" />, then
+
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 +
{{TEX|done}}
  
<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/n/n066/n066630/n06663022.png" /></td> </tr></table>
+
A [[Banach space|Banach space]]  $  H _ {p}  ^ {r} ( \Omega ) $
 +
consisting of functions defined on an open set  $  \Omega $
 +
of an  $  n $-
 +
dimensional Euclidean space  $  \mathbf R  ^ {n} $
 +
and having certain difference-differentiability properties characterized by a vector  $  r = ( r _ {1} \dots r _ {n} ) $,
 +
$  r _ {i} > 0 $,
 +
$  i = 1 \dots n $,
 +
in the  $  L _ {p} $-
 +
metric,  $  1 \leq  p \leq  \infty $.
 +
The concept was introduced by S.M. Nikol'skii.
  
if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663023.png" /> has in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663024.png" /> generalized partial derivatives
+
The Nikol'skii space  $  H _ {p}  ^ {r} ( \Omega ) $
 +
can be described in terms of properties of the partial derivatives of order  $  r _ {i}  ^ {*} $
 +
in the variable  $  x _ {i} $,
 +
where  $  r _ {i} = r _ {i}  ^ {*} + \alpha _ {i} $,
 +
$  r _ {i}  ^ {*} $
 +
is an integer,  $  0 < \alpha _ {i} \leq  1 $,
 +
$  i = 1 \dots n $;
 +
if  $  \Delta _ {h _ {i}  }  ^ {s} $
 +
denotes the difference of order  $  s = 1 , 2 \dots $
 +
and of step  $  h _ {i} $
 +
with respect to  $  x _ {i} $
 +
of a function  $  f $,
 +
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/n/n066/n066630/n06663025.png" /></td> </tr></table>
+
$$
 +
f  \in  H _ {p}  ^ {r} ( M _ {1} \dots M _ {n} ; \Omega )
 +
,\ \
 +
M _ {i} > 0 ,
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663026.png" />, and if for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663027.png" />,
+
if and only if $  f $
 +
has in  $  \Omega $
 +
generalized partial derivatives
  
<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/n/n066/n066630/n06663028.png" /></td> </tr></table>
+
$$
 +
f _ {x _ {i}  } ^ { ( r _ {i}  ^ {*} ) }  = \
  
while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663029.png" />,
+
\frac{\partial  ^ {r _ {i}  ^ {*} } f }{\partial  x _ {i} ^ {r _ {i}  ^ {*} } }
 +
,
 +
$$
  
<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/n/n066/n066630/n06663030.png" /></td> </tr></table>
+
$  i = 1 \dots n $,
 +
and if for  $  0 < \alpha _ {i} < 1 $,
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663031.png" /> is the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663032.png" /> that are distant by more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663033.png" /> from the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663035.png" /> is arbitrary.
+
$$
 +
\| \Delta _ {h _ {i}  }  ^ {1}
 +
f _ {x _ {i}  } ^ { ( r _ {i}  ^ {*} ) } \| _ {L _ {p}  ( \Omega _ {| h _ {i}  | } ) }  \leq  \
 +
M _ {i} | h _ {i} | ^ {\alpha _ {i} } ,
 +
$$
  
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663036.png" /> is defined as the union of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663037.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663039.png" />.
+
while for $  \alpha _ {i} = 1 $,
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663040.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663043.png" />, the Nikol'skii space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663044.png" /> is not empty and contains functions that do not belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663045.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663046.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663047.png" />.
+
$$
 +
\| \Delta _ {h _ {i}  }  ^ {2}
 +
f _ {x _ {i}  } ^ { ( r _ {i}  ^ {*} ) } \| _ {L _ {p}  ( \Omega _ {2 | h _ {i}  | } ) }  \leq  \
 +
M _ {i} | h _ {i} | ,
 +
$$
  
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663048.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663049.png" /> are not integers and the relevant derivatives are continuous, then a Nikol'skii space is a [[Hölder space|Hölder space]]. The concept of a Nikol'skii space generalizes to the case of functions that are defined on sufficiently smooth manifolds (see [[#References|[2]]]).
+
where  $  \Omega _  \eta  $
 +
is the set of points  $  x \in \Omega $
 +
that are distant by more than  $  \eta > 0 $
 +
from the boundary of  $  \Omega $
 +
and  $  h _ {i} $
 +
is arbitrary.
  
There is a description of the Nikol'skii space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663050.png" /> in terms of properties of the differences of the partial derivatives of orders less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663051.png" />; in particular, of those of sufficiently high order of the function itself.
+
The space $  H _ {p}  ^ {r} ( \Omega ) = H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $
 +
is defined as the union of all  $  H _ {p}  ^ {r} ( M _ {1} \dots M _ {n} ;  \Omega ) $
 +
for all  $  M _ {i} > 0 $,
 +
$  i = 1 \dots n $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663052.png" /> be an isotropic space, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663053.png" />. If the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663054.png" /> is such that any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663055.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663056.png" /> can be extended with preservation of the class to the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663057.png" />, that is, in such a way that the extended function belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663058.png" /> (this is always the case when the boundary of the domain is sufficiently smooth), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663059.png" /> if and only if for any non-negative integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663061.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663062.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663063.png" /> has partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663064.png" /> of all orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663065.png" /> and there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663066.png" /> such that
+
If  $  \Omega \neq \emptyset $,  
 +
then for any $  r _ {i} > 0 $,
 +
$  i = 1 \dots n $,
 +
$  1 \leq  p \leq  \infty $,
 +
the Nikol'skii space $  H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $
 +
is not empty and contains functions that do not belong to $  H _ {p} ^ {r _ {1} \dots r _ {i-} 1 , r _ {i} + \epsilon , r _ {i+} 1 \dots r _ {n} } $
 +
for any $  \epsilon > 0 $
 +
and any  $  i = 1 \dots n $.
  
<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/n/n066/n066630/n06663067.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
When  $  p = \infty $,
 +
the  $  r _ {i} $
 +
are not integers and the relevant derivatives are continuous, then a Nikol'skii space is a [[Hölder space|Hölder space]]. The concept of a Nikol'skii space generalizes to the case of functions that are defined on sufficiently smooth manifolds (see [[#References|[2]]]).
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663069.png" /> is the difference of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663070.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663071.png" /> with vectorial step <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663072.png" />. Condition (1) is equivalent to the analogous condition for the modulus of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663073.png" />: There is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663074.png" /> such that
+
There is a description of the Nikol'skii space  $  H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $
 +
in terms of properties of the differences of the partial derivatives of orders less than  $  r _ {i}  ^ {*} $;
 +
in particular, of those of sufficiently high order of the function itself.
  
<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/n/n066/n066630/n06663075.png" /></td> </tr></table>
+
Let  $  H _ {p}  ^ {r} ( \Omega ) $
 +
be an isotropic space, that is,  $  r _ {1} = \dots r _ {n} = r $.
 +
If the domain  $  \Omega $
 +
is such that any function  $  f $
 +
of class  $  H _ {p}  ^ {r} ( \Omega ) $
 +
can be extended with preservation of the class to the whole space  $  \mathbf R  ^ {n} $,
 +
that is, in such a way that the extended function belongs to  $  H _ {p}  ^ {r} ( \mathbf R  ^ {n} ) $(
 +
this is always the case when the boundary of the domain is sufficiently smooth), then  $  f \in H _ {p}  ^ {r} ( \Omega ) $
 +
if and only if for any non-negative integers  $  k $
 +
and  $  s $
 +
such that  $  0 < r - s < k $
 +
the function  $  f $
 +
has partial derivatives  $  f ^ { ( s) } $
 +
of all orders  $  s $
 +
and there is a constant  $  M > 0 $
 +
such that
 +
 
 +
$$ \tag{1 }
 +
\| \Delta _ {h}  ^ {k} f ^ { ( s) } \| _ {L _ {p}  (
 +
\Omega _ {k | h | }  ) }
 +
\leq  M | h |  ^ {r-} s ,
 +
$$
 +
 
 +
where  $  h =( h _ {1} \dots h _ {n} ) $
 +
and  $  \Delta _ {h}  ^ {k} f ^ { ( s) } $
 +
is the difference of order  $  k $
 +
of  $  f ^ { ( s) } $
 +
with vectorial step  $  h $.  
 +
Condition (1) is equivalent to the analogous condition for the modulus of continuity of  $  f ^ { ( s) } $:
 +
There is an  $  M > 0 $
 +
such that
 +
 
 +
$$
 +
\Omega  ^ {k} ( f ^ { ( s) } , \delta )  \leq  \
 +
M \delta  ^ {r-} s ,\  \delta > 0 ,
 +
$$
  
 
where
 
where
  
<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/n/n066/n066630/n06663076.png" /></td> </tr></table>
+
$$
 +
\Omega  ^ {k} ( f ^ { ( s) } , \delta )  = \
 +
\sup _ {| h | = 1 } \
 +
\sup _ {0 \leq  t \leq  \delta } \
 +
\| \Delta _ {th}  ^ {k} f ^ { ( s) } \| _ {L _ {p}  ( \Omega _ {kt} ) }  \leq  M \delta  ^ {r-} s .
 +
$$
 +
 
 +
If  $  M _ {f} $,
 +
for  $  f \in H _ {p}  ^ {r} ( \Omega ) $,
 +
denotes the infimum of all  $  M $
 +
for which (1) holds for all  $  h \in \mathbf R  ^ {n} $
 +
and all partial derivatives of an admissible order  $  s $,
 +
then
 +
 
 +
$$
 +
\| f \|  =  \| f \| _ {L _ {p}  ( \Omega ) } + M _ {f}  $$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663077.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663078.png" />, denotes the infimum of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663079.png" /> for which (1) holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663080.png" /> and all partial derivatives of an admissible order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663081.png" />, then
+
is a norm in  $  H _ {p}  ^ {r} ( \Omega ) $
 +
and the norms obtained for distinct admissible pairs  $  k , s $
 +
are equivalent.
  
<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/n/n066/n066630/n06663082.png" /></td> </tr></table>
+
A Nikol'skii space consisting of functions defined on the whole space  $  \mathbf R  ^ {n} $
 +
can be characterized in terms of best approximations of the functions in this space by entire functions of exponential type. Let  $  E _ {v _ {1}  \dots v _ {n} } ( f  ) _ {p} $
 +
be the best approximation (error) in the  $  L _ {p} ( \mathbf R  ^ {n} ) $-
 +
metric of an  $  f \in L _ {p} ( \mathbf R  ^ {n} ) $
 +
by entire functions  $  q _ {v _ {1}  \dots v _ {n} } ( x _ {1} \dots x _ {n} ) \in L _ {p} ( \mathbf R  ^ {n} ) $
 +
of exponential type and of order  $  v _ {i} $
 +
in  $  x _ {i} $,
 +
$  i = 1 \dots n $.  
 +
The following direct and inverse theorems of Bernshtein, Jackson and Zygmund type hold for Nikol'skii functions.
  
is a norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663083.png" /> and the norms obtained for distinct admissible pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663084.png" /> are equivalent.
+
If  $  f \in H _ {p} ^ {r _ {1} \dots r _ {n} } ( M _ {1} \dots M _ {n} ;  \mathbf R  ^ {n} ) $,
 +
then for any  $  v _ {i} > 0 $,
  
A Nikol'skii space consisting of functions defined on the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663085.png" /> can be characterized in terms of best approximations of the functions in this space by entire functions of exponential type. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663086.png" /> be the best approximation (error) in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663087.png" />-metric of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663088.png" /> by entire functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663089.png" /> of exponential type and of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663090.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663092.png" />. The following direct and inverse theorems of Bernshtein, Jackson and Zygmund type hold for Nikol'skii functions.
+
$$ \tag{2 }
 +
E _ {v _ {1}  \dots v _ {n} } ( ) \leq  c
 +
\sum _ { i= } 1 ^ { n }
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663093.png" />, then for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663094.png" />,
+
\frac{M _ {i} }{v _ {i} ^ {r _ {i} } }
  
<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/n/n066/n066630/n06663095.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
  
(the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663096.png" /> does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663097.png" />).
+
(the constant $  c > 0 $
 +
does not depend on $  f  $).
  
Conversely, if (2) holds for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663098.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n06663099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630100.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630102.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630103.png" /> is an entire function of order 1 in each variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630104.png" /> for which
+
Conversely, if (2) holds for a function $  f \in L _ {p} ( \mathbf R  ^ {n} ) $
 +
for $  v _ {i} = a _ {i}  ^ {k} $,
 +
$  k = 0 , 1 \dots $
 +
$  a _ {i} > 1 $,  
 +
$  i = 1 \dots n $,  
 +
and if $  q $
 +
is an entire function of order 1 in each variable $  x _ {1} \dots x _ {n} $
 +
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/n/n066/n066630/n066630105.png" /></td> </tr></table>
+
$$
 +
\| f - q \| _ {L _ {p}  ( \mathbf R  ^ {n} ) }
 +
\leq  c \sum _ { i= } 1 ^ { n }  M _ {i}  $$
  
(which exist for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630106.png" />, by (2)), then
+
(which exist for $  k = 0 $,  
 +
by (2)), 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/n/n066/n066630/n066630107.png" /></td> </tr></table>
+
$$
 +
f - q  \in \
 +
H _ {p} ^ {r _ {1} \dots r _ {n} }
 +
( M _ {1}  ^ {*} \dots M _ {n}  ^ {*} ; \mathbf R  ^ {n} ) ,
 +
$$
  
 
where
 
where
  
<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/n/n066/n066630/n066630108.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
M _ {i}  ^ {*}  = c _ {i} \sum _ { j= } 1 ^ { n }  M _ {j} ,
 +
$$
  
and the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630109.png" /> in (2) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630110.png" /> in (3) do not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630112.png" />.
+
and the constants $  c > 0 $
 +
in (2) and $  c _ {i} > 0 $
 +
in (3) do not depend on $  M _ {i} $,  
 +
$  i = 1 \dots n $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630113.png" /> is periodic in all variables, then a similar description of a Nikol'skii space can be given by means of best approximations of the functions by trigonometric polynomials instead of entire functions of exponential type (see [[#References|[1]]], [[#References|[4]]]).
+
If $  f $
 +
is periodic in all variables, then a similar description of a Nikol'skii space can be given by means of best approximations of the functions by trigonometric polynomials instead of entire functions of exponential type (see [[#References|[1]]], [[#References|[4]]]).
  
 
Nikol'skii spaces can be described by means of a Bessel–Macdonald operator applied to some class of generalized functions (see [[Imbedding theorems|Imbedding theorems]]).
 
Nikol'skii spaces can be described by means of a Bessel–Macdonald operator applied to some class of generalized functions (see [[Imbedding theorems|Imbedding theorems]]).
  
For the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630114.png" /> Nikol'skii has proved transitive imbedding theorems for various dimensions and metrics (see [[#References|[3]]] and [[Imbedding theorems|Imbedding theorems]]), which were subsequently carried over to more general classes of functions. These theorems show that Nikol'skii spaces form a closed system relative to the boundary values of the functions occurring in them: The traces of functions in Nikol'skii spaces on smooth manifolds can in a certain sense be completely described in terms of Nikol'skii spaces.
+
For the space $  H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $
 +
Nikol'skii has proved transitive imbedding theorems for various dimensions and metrics (see [[#References|[3]]] and [[Imbedding theorems|Imbedding theorems]]), which were subsequently carried over to more general classes of functions. These theorems show that Nikol'skii spaces form a closed system relative to the boundary values of the functions occurring in them: The traces of functions in Nikol'skii spaces on smooth manifolds can in a certain sense be completely described in terms of Nikol'skii spaces.
  
The properties of Nikol'skii spaces make it possible to obtain necessary and sufficient conditions for the solvability of the [[Dirichlet problem|Dirichlet problem]] in appropriate Nikol'skii spaces in terms of membership of the boundary function to a certain Nikol'skii space: A harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630115.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630117.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630118.png" /> is a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630119.png" /> with a sufficiently smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630120.png" />, if and only if the boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630121.png" /> belong to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630122.png" />. This implies for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630123.png" />, in particular, that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630124.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630125.png" />, then the [[Dirichlet integral|Dirichlet integral]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630126.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630127.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630128.png" /> is finite, therefore, the Dirichlet problem can be solved by a direct variational method. From imbedding theorems for Nikol'skii spaces it follows that if the Dirichlet integral of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630129.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630130.png" /> is finite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630131.png" /> (see [[#References|[6]]]). A generalization of Nikol'skii spaces are the Besov spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066630/n066630132.png" />.
+
The properties of Nikol'skii spaces make it possible to obtain necessary and sufficient conditions for the solvability of the [[Dirichlet problem|Dirichlet problem]] in appropriate Nikol'skii spaces in terms of membership of the boundary function to a certain Nikol'skii space: A harmonic function $  u $
 +
belongs to the class $  H _ {p}  ^ {r} ( \Omega ) $,
 +
$  r > 1 / p $,  
 +
where $  \Omega $
 +
is a bounded domain in $  \mathbf R  ^ {n} $
 +
with a sufficiently smooth boundary $  \partial  \Omega $,  
 +
if and only if the boundary values $  u \mid  _ {\partial  \Omega }  $
 +
belong to the class $  H _ {p} ^ {r - 1/p } ( \partial  \Omega ) $.  
 +
This implies for $  p = 2 $,  
 +
in particular, that if $  u \mid  _ {\partial  \Omega }  \in H _ {2}  ^  \rho  ( \partial  \Omega ) $,
 +
$  \rho > 1/2 $,  
 +
then the [[Dirichlet integral|Dirichlet integral]] $  D ( u) $
 +
of $  u $
 +
over $  \Omega $
 +
is finite, therefore, the Dirichlet problem can be solved by a direct variational method. From imbedding theorems for Nikol'skii spaces it follows that if the Dirichlet integral of $  u $
 +
over $  \Omega $
 +
is finite, then $  u \mid  _ {\partial  \Omega }  \in H _ {2}  ^ {1/2} ( \partial  \Omega ) $(
 +
see [[#References|[6]]]). A generalization of Nikol'skii spaces are the Besov spaces $  B _ {p \theta }  ^ {r _ {1} \dots r _ {n} } $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.M. Nikol'skii,  "Inequalities for entire functions of finite order and their application to the theory of differentiable functions in several variables"  ''Trudy Mat. Inst. Steklov.'' , '''38'''  (1951)  pp. 244–278  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "Properties of certain classes of functions of several variables on a differentiable manifold"  ''Mat. Sb.'' , '''33''' :  2  (1953)  pp. 261–326  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  "Imbedding theorems for functions with partial derivatives, considered in differential metrics"  ''Dokl. Akad. Nauk SSSR'' , '''118''' :  1  (1958)  pp. 35–37  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  O.V. Besov,  V.P. Il'in,  S.M. Nikol'skii,  "Integral representations of functions and imbedding theorems" , Wiley  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.M. Nikol'skii,  "On the solution of the polyharmonic equation by a variational method"  ''Dokl. Akad. Nauk SSSR'' , '''88''' :  3  (1953)  pp. 409–411  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.M. Nikol'skii,  "Inequalities for entire functions of finite order and their application to the theory of differentiable functions in several variables"  ''Trudy Mat. Inst. Steklov.'' , '''38'''  (1951)  pp. 244–278  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.M. Nikol'skii,  "Properties of certain classes of functions of several variables on a differentiable manifold"  ''Mat. Sb.'' , '''33''' :  2  (1953)  pp. 261–326  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.M. Nikol'skii,  "Imbedding theorems for functions with partial derivatives, considered in differential metrics"  ''Dokl. Akad. Nauk SSSR'' , '''118''' :  1  (1958)  pp. 35–37  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables and imbedding theorems" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  O.V. Besov,  V.P. Il'in,  S.M. Nikol'skii,  "Integral representations of functions and imbedding theorems" , Wiley  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.M. Nikol'skii,  "On the solution of the polyharmonic equation by a variational method"  ''Dokl. Akad. Nauk SSSR'' , '''88''' :  3  (1953)  pp. 409–411  (In Russian)</TD></TR></table>

Revision as of 08:02, 6 June 2020


A Banach space $ H _ {p} ^ {r} ( \Omega ) $ consisting of functions defined on an open set $ \Omega $ of an $ n $- dimensional Euclidean space $ \mathbf R ^ {n} $ and having certain difference-differentiability properties characterized by a vector $ r = ( r _ {1} \dots r _ {n} ) $, $ r _ {i} > 0 $, $ i = 1 \dots n $, in the $ L _ {p} $- metric, $ 1 \leq p \leq \infty $. The concept was introduced by S.M. Nikol'skii.

The Nikol'skii space $ H _ {p} ^ {r} ( \Omega ) $ can be described in terms of properties of the partial derivatives of order $ r _ {i} ^ {*} $ in the variable $ x _ {i} $, where $ r _ {i} = r _ {i} ^ {*} + \alpha _ {i} $, $ r _ {i} ^ {*} $ is an integer, $ 0 < \alpha _ {i} \leq 1 $, $ i = 1 \dots n $; if $ \Delta _ {h _ {i} } ^ {s} $ denotes the difference of order $ s = 1 , 2 \dots $ and of step $ h _ {i} $ with respect to $ x _ {i} $ of a function $ f $, then

$$ f \in H _ {p} ^ {r} ( M _ {1} \dots M _ {n} ; \Omega ) ,\ \ M _ {i} > 0 , $$

if and only if $ f $ has in $ \Omega $ generalized partial derivatives

$$ f _ {x _ {i} } ^ { ( r _ {i} ^ {*} ) } = \ \frac{\partial ^ {r _ {i} ^ {*} } f }{\partial x _ {i} ^ {r _ {i} ^ {*} } } , $$

$ i = 1 \dots n $, and if for $ 0 < \alpha _ {i} < 1 $,

$$ \| \Delta _ {h _ {i} } ^ {1} f _ {x _ {i} } ^ { ( r _ {i} ^ {*} ) } \| _ {L _ {p} ( \Omega _ {| h _ {i} | } ) } \leq \ M _ {i} | h _ {i} | ^ {\alpha _ {i} } , $$

while for $ \alpha _ {i} = 1 $,

$$ \| \Delta _ {h _ {i} } ^ {2} f _ {x _ {i} } ^ { ( r _ {i} ^ {*} ) } \| _ {L _ {p} ( \Omega _ {2 | h _ {i} | } ) } \leq \ M _ {i} | h _ {i} | , $$

where $ \Omega _ \eta $ is the set of points $ x \in \Omega $ that are distant by more than $ \eta > 0 $ from the boundary of $ \Omega $ and $ h _ {i} $ is arbitrary.

The space $ H _ {p} ^ {r} ( \Omega ) = H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $ is defined as the union of all $ H _ {p} ^ {r} ( M _ {1} \dots M _ {n} ; \Omega ) $ for all $ M _ {i} > 0 $, $ i = 1 \dots n $.

If $ \Omega \neq \emptyset $, then for any $ r _ {i} > 0 $, $ i = 1 \dots n $, $ 1 \leq p \leq \infty $, the Nikol'skii space $ H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $ is not empty and contains functions that do not belong to $ H _ {p} ^ {r _ {1} \dots r _ {i-} 1 , r _ {i} + \epsilon , r _ {i+} 1 \dots r _ {n} } $ for any $ \epsilon > 0 $ and any $ i = 1 \dots n $.

When $ p = \infty $, the $ r _ {i} $ are not integers and the relevant derivatives are continuous, then a Nikol'skii space is a Hölder space. The concept of a Nikol'skii space generalizes to the case of functions that are defined on sufficiently smooth manifolds (see [2]).

There is a description of the Nikol'skii space $ H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $ in terms of properties of the differences of the partial derivatives of orders less than $ r _ {i} ^ {*} $; in particular, of those of sufficiently high order of the function itself.

Let $ H _ {p} ^ {r} ( \Omega ) $ be an isotropic space, that is, $ r _ {1} = \dots r _ {n} = r $. If the domain $ \Omega $ is such that any function $ f $ of class $ H _ {p} ^ {r} ( \Omega ) $ can be extended with preservation of the class to the whole space $ \mathbf R ^ {n} $, that is, in such a way that the extended function belongs to $ H _ {p} ^ {r} ( \mathbf R ^ {n} ) $( this is always the case when the boundary of the domain is sufficiently smooth), then $ f \in H _ {p} ^ {r} ( \Omega ) $ if and only if for any non-negative integers $ k $ and $ s $ such that $ 0 < r - s < k $ the function $ f $ has partial derivatives $ f ^ { ( s) } $ of all orders $ s $ and there is a constant $ M > 0 $ such that

$$ \tag{1 } \| \Delta _ {h} ^ {k} f ^ { ( s) } \| _ {L _ {p} ( \Omega _ {k | h | } ) } \leq M | h | ^ {r-} s , $$

where $ h =( h _ {1} \dots h _ {n} ) $ and $ \Delta _ {h} ^ {k} f ^ { ( s) } $ is the difference of order $ k $ of $ f ^ { ( s) } $ with vectorial step $ h $. Condition (1) is equivalent to the analogous condition for the modulus of continuity of $ f ^ { ( s) } $: There is an $ M > 0 $ such that

$$ \Omega ^ {k} ( f ^ { ( s) } , \delta ) \leq \ M \delta ^ {r-} s ,\ \delta > 0 , $$

where

$$ \Omega ^ {k} ( f ^ { ( s) } , \delta ) = \ \sup _ {| h | = 1 } \ \sup _ {0 \leq t \leq \delta } \ \| \Delta _ {th} ^ {k} f ^ { ( s) } \| _ {L _ {p} ( \Omega _ {kt} ) } \leq M \delta ^ {r-} s . $$

If $ M _ {f} $, for $ f \in H _ {p} ^ {r} ( \Omega ) $, denotes the infimum of all $ M $ for which (1) holds for all $ h \in \mathbf R ^ {n} $ and all partial derivatives of an admissible order $ s $, then

$$ \| f \| = \| f \| _ {L _ {p} ( \Omega ) } + M _ {f} $$

is a norm in $ H _ {p} ^ {r} ( \Omega ) $ and the norms obtained for distinct admissible pairs $ k , s $ are equivalent.

A Nikol'skii space consisting of functions defined on the whole space $ \mathbf R ^ {n} $ can be characterized in terms of best approximations of the functions in this space by entire functions of exponential type. Let $ E _ {v _ {1} \dots v _ {n} } ( f ) _ {p} $ be the best approximation (error) in the $ L _ {p} ( \mathbf R ^ {n} ) $- metric of an $ f \in L _ {p} ( \mathbf R ^ {n} ) $ by entire functions $ q _ {v _ {1} \dots v _ {n} } ( x _ {1} \dots x _ {n} ) \in L _ {p} ( \mathbf R ^ {n} ) $ of exponential type and of order $ v _ {i} $ in $ x _ {i} $, $ i = 1 \dots n $. The following direct and inverse theorems of Bernshtein, Jackson and Zygmund type hold for Nikol'skii functions.

If $ f \in H _ {p} ^ {r _ {1} \dots r _ {n} } ( M _ {1} \dots M _ {n} ; \mathbf R ^ {n} ) $, then for any $ v _ {i} > 0 $,

$$ \tag{2 } E _ {v _ {1} \dots v _ {n} } ( f ) \leq c \sum _ { i= } 1 ^ { n } \frac{M _ {i} }{v _ {i} ^ {r _ {i} } } $$

(the constant $ c > 0 $ does not depend on $ f $).

Conversely, if (2) holds for a function $ f \in L _ {p} ( \mathbf R ^ {n} ) $ for $ v _ {i} = a _ {i} ^ {k} $, $ k = 0 , 1 \dots $ $ a _ {i} > 1 $, $ i = 1 \dots n $, and if $ q $ is an entire function of order 1 in each variable $ x _ {1} \dots x _ {n} $ for which

$$ \| f - q \| _ {L _ {p} ( \mathbf R ^ {n} ) } \leq c \sum _ { i= } 1 ^ { n } M _ {i} $$

(which exist for $ k = 0 $, by (2)), then

$$ f - q \in \ H _ {p} ^ {r _ {1} \dots r _ {n} } ( M _ {1} ^ {*} \dots M _ {n} ^ {*} ; \mathbf R ^ {n} ) , $$

where

$$ \tag{3 } M _ {i} ^ {*} = c _ {i} \sum _ { j= } 1 ^ { n } M _ {j} , $$

and the constants $ c > 0 $ in (2) and $ c _ {i} > 0 $ in (3) do not depend on $ M _ {i} $, $ i = 1 \dots n $.

If $ f $ is periodic in all variables, then a similar description of a Nikol'skii space can be given by means of best approximations of the functions by trigonometric polynomials instead of entire functions of exponential type (see [1], [4]).

Nikol'skii spaces can be described by means of a Bessel–Macdonald operator applied to some class of generalized functions (see Imbedding theorems).

For the space $ H _ {p} ^ {r _ {1} \dots r _ {n} } ( \Omega ) $ Nikol'skii has proved transitive imbedding theorems for various dimensions and metrics (see [3] and Imbedding theorems), which were subsequently carried over to more general classes of functions. These theorems show that Nikol'skii spaces form a closed system relative to the boundary values of the functions occurring in them: The traces of functions in Nikol'skii spaces on smooth manifolds can in a certain sense be completely described in terms of Nikol'skii spaces.

The properties of Nikol'skii spaces make it possible to obtain necessary and sufficient conditions for the solvability of the Dirichlet problem in appropriate Nikol'skii spaces in terms of membership of the boundary function to a certain Nikol'skii space: A harmonic function $ u $ belongs to the class $ H _ {p} ^ {r} ( \Omega ) $, $ r > 1 / p $, where $ \Omega $ is a bounded domain in $ \mathbf R ^ {n} $ with a sufficiently smooth boundary $ \partial \Omega $, if and only if the boundary values $ u \mid _ {\partial \Omega } $ belong to the class $ H _ {p} ^ {r - 1/p } ( \partial \Omega ) $. This implies for $ p = 2 $, in particular, that if $ u \mid _ {\partial \Omega } \in H _ {2} ^ \rho ( \partial \Omega ) $, $ \rho > 1/2 $, then the Dirichlet integral $ D ( u) $ of $ u $ over $ \Omega $ is finite, therefore, the Dirichlet problem can be solved by a direct variational method. From imbedding theorems for Nikol'skii spaces it follows that if the Dirichlet integral of $ u $ over $ \Omega $ is finite, then $ u \mid _ {\partial \Omega } \in H _ {2} ^ {1/2} ( \partial \Omega ) $( see [6]). A generalization of Nikol'skii spaces are the Besov spaces $ B _ {p \theta } ^ {r _ {1} \dots r _ {n} } $.

References

[1] S.M. Nikol'skii, "Inequalities for entire functions of finite order and their application to the theory of differentiable functions in several variables" Trudy Mat. Inst. Steklov. , 38 (1951) pp. 244–278 (In Russian)
[2] S.M. Nikol'skii, "Properties of certain classes of functions of several variables on a differentiable manifold" Mat. Sb. , 33 : 2 (1953) pp. 261–326 (In Russian)
[3] S.M. Nikol'skii, "Imbedding theorems for functions with partial derivatives, considered in differential metrics" Dokl. Akad. Nauk SSSR , 118 : 1 (1958) pp. 35–37 (In Russian)
[4] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[5] O.V. Besov, V.P. Il'in, S.M. Nikol'skii, "Integral representations of functions and imbedding theorems" , Wiley (1978) (Translated from Russian)
[6] S.M. Nikol'skii, "On the solution of the polyharmonic equation by a variational method" Dokl. Akad. Nauk SSSR , 88 : 3 (1953) pp. 409–411 (In Russian)
How to Cite This Entry:
Nikol'skii space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nikol%27skii_space&oldid=19306
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article