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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.

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|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]]]).

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 [[Function of exponential type|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 [[#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]]).

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  $  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====
<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>