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The case when the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a0130101.png" /> to be approximated depends on two or more variables:
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$#C+1 = 157 : ~/encyclopedia/old_files/data/A013/A.0103010 Approximation of functions of several real variables
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a0130102.png" /></td> </tr></table>
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(See [[Approximation of functions|Approximation of functions]].) In comparison with the one-dimensional case, the study of problems on the approximation of functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a0130103.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a0130104.png" />) variables is highly complicated because of essentially new circumstances that arise related to dimension. First of all this applies to the domain on which the approximation is carried out. Simply-connected compact sets (in the one-dimensional case intervals) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a0130105.png" /> (even in the plane) can have a large variety of shapes; therefore it becomes necessary to classify the domains according to, for instance, the smoothness properties of their boundaries. Complications also arise in the description of difference-differential properties of functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a0130106.png" /> variables. Generally speaking, these properties can be different in different directions, and their characterization ought to take into account both the geometry of the domain on which the function is defined and the behaviour of the function while approaching the boundary, so that the study of boundary properties of the function is of great importance. If one attempts to simplify the solution of an approximation problem by passing to a domain with a simpler structure, the problem that arises is to extend a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a0130107.png" /> from a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a0130108.png" /> into a canonical domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a0130109.png" /> (for instance a parallelepiped or the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301010.png" />) containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301011.png" /> while preserving certain smoothness properties (see [[Extension theorems|Extension theorems]]). This kind of problems is closely connected with [[Imbedding theorems|imbedding theorems]], and also with questions arising in the solution of the boundary value problems of mathematical physics.
+
The case when the function  $  f $
 +
to be approximated depends on two or more variables:
  
An increase in the number of independent variables complicates, of course, also the tools of approximation, because when the dimension is increased, with it, e.g., the degrees of the polynomials increase. An algebraic polynomial of degrees <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301012.png" /> in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301013.png" />, respectively, has the form
+
$$
 +
f (t)  =  f
 +
( t _ {1} \dots t _ {m} ) ,\ \
 +
m \geq  2 .
 +
$$
  
<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/a013/a013010/a01301014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
(See [[Approximation of functions|Approximation of functions]].) In comparison with the one-dimensional case, the study of problems on the approximation of functions of  $  m $(
 +
$  m \geq  2 $)
 +
variables is highly complicated because of essentially new circumstances that arise related to dimension. First of all this applies to the domain on which the approximation is carried out. Simply-connected compact sets (in the one-dimensional case intervals) in  $  \mathbf R  ^ {m} $(
 +
even in the plane) can have a large variety of shapes; therefore it becomes necessary to classify the domains according to, for instance, the smoothness properties of their boundaries. Complications also arise in the description of difference-differential properties of functions of  $  m $
 +
variables. Generally speaking, these properties can be different in different directions, and their characterization ought to take into account both the geometry of the domain on which the function is defined and the behaviour of the function while approaching the boundary, so that the study of boundary properties of the function is of great importance. If one attempts to simplify the solution of an approximation problem by passing to a domain with a simpler structure, the problem that arises is to extend a function  $  f $
 +
from a domain  $  Q \subset  \mathbf R  ^ {m} $
 +
into a canonical domain  $  Q _ {1} $(
 +
for instance a parallelepiped or the whole space  $  \mathbf R  ^ {m} $)
 +
containing  $  Q $
 +
while preserving certain smoothness properties (see [[Extension theorems|Extension theorems]]). This kind of problems is closely connected with [[Imbedding theorems|imbedding theorems]], and also with questions arising in the solution of the boundary value problems of mathematical physics.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301015.png" /> are real coefficients. The summation is carried out with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301017.png" />, from 0 to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301018.png" />, so that, e.g., the subspace of polynomials of degree 3 in each of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301019.png" /> variables has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301020.png" />. Sometimes the overall or total degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301021.png" /> of the polynomial is fixed; then the summation in (1) is carried out over all indices satisfying the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301022.png" />. A real trigonometric polynomial of degrees <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301023.png" /> in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301024.png" /> can be written in the form
+
An increase in the number of independent variables complicates, of course, also the tools of approximation, because when the dimension is increased, with it, e.g., the degrees of the polynomials increase. An algebraic polynomial of degrees $  n _ {1} \dots n _ {m} $
 +
in the variables $  t _ {1} \dots t _ {m} $,
 +
respectively, has the form
  
<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/a013/a013010/a01301025.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
P _ {n _ {1}  \dots n _ {m} }
 +
( t _ {1} \dots t _ {m} )  = \
 +
\sum a _ {k _ {1}  \dots k _ {m} }
 +
t _ {1} ^ {k _ {1} } \dots
 +
t _ {m} ^ {k _ {m} } ,
 +
$$
  
where complex coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301026.png" /> with indices of opposite sign are complex conjugate, and the summation is carried out with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301028.png" />, from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301029.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301030.png" />. Such a polynomial can be represented in the form of a linear combination of all possible products of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301032.png" /> is either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301033.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301034.png" />) or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301035.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301036.png" />). Multi-dimensional splines have a wide field of applications; they consist of "pieces" of algebraic polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301037.png" /> variables "tied" together in order to obtain specified smoothness properties. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301038.png" />, splines of the simplest form are formed by polynomials "tied" together along straight lines parallel to the coordinate axes. As a tool of approximation functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301039.png" /> are used which are polynomials or splines only in one of the variables. For the approximation of non-periodic functions given on the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301040.png" /> (or on an unbounded subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301041.png" />) one may use entire functions of exponential type. These can be represented in the form of sums of absolutely convergent power series
+
where $  a _ {k _ {1}  \dots k _ {m} } $
 +
are real coefficients. The summation is carried out with respect to $  k _  \nu  $,  
 +
$  \nu = 1 \dots m $,  
 +
from 0 to $  n _  \nu  $,
 +
so that, e.g., the subspace of polynomials of degree 3 in each of the $  m $
 +
variables has dimension  $  4  ^ {m} $.  
 +
Sometimes the overall or total degree  $  n $
 +
of the polynomial is fixed; then the summation in (1) is carried out over all indices satisfying the inequality $ 0 \leq k _ {1} + \dots + k _ {m} \leq n $.  
 +
A real trigonometric polynomial of degrees $ n _ {1} \dots n _ {m} $
 +
in the variables $  t _ {1} \dots t _ {m} $
 +
can be written in the form
  
<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/a013/a013010/a01301042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$
 +
T _ {n _ {1}  \dots n _ {m} }
 +
( t _ {1} \dots t _ {m} )  = \
 +
\sum a _ {k _ {1}  \dots k _  \nu  }
 +
\mathop{\rm exp}
 +
\left ( i
 +
\sum _ {\nu = 1 } ^ { {m }  }
 +
k _  \nu  t _  \nu  \right ) ,
 +
$$
  
under the condition that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301043.png" /> and all complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301044.png" /> one has
+
where complex coefficients  $  a _ {k _ {1}  \dots k _ {n} } $
 +
with indices of opposite sign are complex conjugate, and the summation is carried out with respect to  $  k _  \nu  $,
 +
$  \nu = 1 \dots m $,
 +
from  $  - n _  \nu  $
 +
to  $  n _  \nu  $.
 +
Such a polynomial can be represented in the form of a linear combination of all possible products of the form  $  \phi _ {k _ {1}  } ( t _ {1} ) \dots \phi _ {k _ {m}  } ( t _ {m} ) $,
 +
where  $  \phi _ {k _  \nu  } ( t _  \nu  ) $
 +
is either  $  \sin  k _  \nu  t _  \nu  $(
 +
$  0 \leq  k _  \nu  \leq  n _  \nu  $)
 +
or  $  \cos  k _  \nu  t _  \nu  $(
 +
0 \leq  k _  \nu  \leq  n _  \nu  $).  
 +
Multi-dimensional splines have a wide field of applications; they consist of  "pieces" of algebraic polynomials in  $  m $
 +
variables  "tied" together in order to obtain specified smoothness properties. In the case  $  m = 2 $,
 +
splines of the simplest form are formed by polynomials  "tied" together along straight lines parallel to the coordinate axes. As a tool of approximation functions  $  g ( t _ {1} \dots t _ {m} ) $
 +
are used which are polynomials or splines only in one of the variables. For the approximation of non-periodic functions given on the whole space  $  \mathbf R  ^ {m} $(
 +
or on an unbounded subset of  $  \mathbf R  ^ {m} $)
 +
one may use entire functions of [[Function of exponential type|exponential type]]. These can be represented in the form of sums of absolutely convergent power series
  
<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/a013/a013010/a01301045.png" /></td> </tr></table>
+
$$ \tag{2 }
 +
G _ {n _ {1}  \dots n _ {m} }
 +
( t _ {i} \dots t _ {m} )  = \
 +
\sum _ {\begin{array}{c}
  
where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301046.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301047.png" /> (see [[#References|[1]]]). It should be noted that, unlike polynomials, the function (2) is determined by an infinite number of parameters.
+
k _  \nu  \geq  0 ,
 +
\\
  
In the multi-dimensional case one has Weierstrass' theorem on the possibility of approximating a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301048.png" />, i.e. continuous on a bounded and closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301049.png" />, or a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301050.png" />, continuous on the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301051.png" /> with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301052.png" /> in each variable, with arbitrary accuracy by means of algebraic (respectively, trigonometric) polynomials. A similar result holds in the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301054.png" /> (in the periodic case) (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301055.png" />). General results and theorems concerning properties of the best approximation, the existence and uniqueness, the characteristic properties of functions of best approximation, and general relations of duality when approximating by (means of) a convex set (of functions) and, in particular, by a subspace, can be extended to normed linear spaces of functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301056.png" /> variables (see [[#References|[3]]] and [[#References|[4]]]). However, to obtain explicit formulations of these theorems, taking into account a specific metric and specific properties of the approximating subspace in the multi-dimensional case, involves great difficulties.
+
\nu = 1 \dots m
 +
 +
\end{array}
 +
}
 +
a _ {k _ {1}  \dots k _ {m} }
 +
t _ {1} ^ {k _ {1} } \dots
 +
t _ {m} ^ {k _ {m} }
 +
$$
 +
 
 +
under the condition that for any  $  \epsilon > 0 $
 +
and all complex  $  t _ {1} \dots t _ {m} $
 +
one has
 +
 
 +
$$
 +
| G _ {n _ {1}  \dots n _ {m} }
 +
( t _ {1} \dots t _ {m} ) |
 +
\leq  M _  \epsilon  \
 +
\mathop{\rm exp} \
 +
\sum _ {\nu = 1 } ^ { m }
 +
( n _  \nu  + \epsilon )
 +
| t _  \nu  | ,
 +
$$
 +
 
 +
where the constant  $  M _  \epsilon  $
 +
depends only on  $  \epsilon $(
 +
see [[#References|[1]]]). It should be noted that, unlike polynomials, the function (2) is determined by an infinite number of parameters.
 +
 
 +
In the multi-dimensional case one has Weierstrass' theorem on the possibility of approximating a function $  f \in C (Q) $,  
 +
i.e. continuous on a bounded and closed set $  Q \subset  \mathbf R  ^ {m} $,  
 +
or a function $  f \in \widetilde{C}  ( \mathbf R  ^ {m} ) $,  
 +
continuous on the whole space $  \mathbf R  ^ {m} $
 +
with period $  2 \pi $
 +
in each variable, with arbitrary accuracy by means of algebraic (respectively, trigonometric) polynomials. A similar result holds in the spaces $  L _ {p} (Q) $
 +
and $  \widetilde{L}  _ {p} ( \mathbf R  ^ {m} ) $(
 +
in the periodic case) ( $  1 \leq  p < \infty $).  
 +
General results and theorems concerning properties of the best approximation, the existence and uniqueness, the characteristic properties of functions of best approximation, and general relations of duality when approximating by (means of) a convex set (of functions) and, in particular, by a subspace, can be extended to normed linear spaces of functions of $  m $
 +
variables (see [[#References|[3]]] and [[#References|[4]]]). However, to obtain explicit formulations of these theorems, taking into account a specific metric and specific properties of the approximating subspace in the multi-dimensional case, involves great difficulties.
  
 
Quite thorough investigations were carried out on questions concerning the connection between the smoothness properties of a function of several variables and the rate of decrease of its best approximation by algebraic and trigonometric polynomials, as well as by entire functions.
 
Quite thorough investigations were carried out on questions concerning the connection between the smoothness properties of a function of several variables and the rate of decrease of its best approximation by algebraic and trigonometric polynomials, as well as by entire functions.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301057.png" /> be an arbitrary open set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301058.png" /> (in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301059.png" />), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301060.png" /> be a unit vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301061.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301062.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301063.png" /> be the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301064.png" /> such that the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301065.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301067.png" />, then the quantity
+
Let $  Q $
 +
be an arbitrary open set in $  \mathbf R  ^ {m} $(
 +
in particular, $  Q = \mathbf R  ^ {m} $),  
 +
let $  \mathbf e $
 +
be a unit vector in $  \mathbf R  ^ {m} $,  
 +
let $  h > 0 $
 +
and let $  Q _ {h \mathbf e }  $
 +
be the set of points $  t \in Q $
 +
such that the segment $  [ t , t + h \mathbf e ] \in Q $.  
 +
If $  f \in L _ {p} (Q) $
 +
and  $  1 \leq  p \leq  \infty $,
 +
then the quantity
 +
 
 +
$$
 +
\omega _ {\mathbf e }
 +
( f ;  \delta ) _ {L _ {p}  (Q) }  = \
 +
\sup _
 +
{h \leq  \delta } \
 +
\| f ( t + h \mathbf e ) - f
 +
(t) \| _ {L _ {p} ( Q _ {h \mathbf e }  ) }
 +
$$
 +
 
 +
is called the modulus of continuity of the function  $  f ( t _ {1} \dots t _ {m} ) $
 +
in the direction  $  \mathbf e $
 +
with respect to the metric of  $  L _ {p} (Q) $.  
 +
The quantity
 +
 
 +
$$
 +
\omega ( f ;  \delta ) _ {L _ {p}  (Q) }  = \
 +
\sup _
 +
{\mathbf e } \
 +
\omega _ {\mathbf e }
 +
( f ;  \delta ) _ {L _ {p}  (Q) }
 +
$$
 +
 
 +
is called the modulus of continuity of  $  f $
 +
in  $  L _ {p} (Q) $.
 +
 
 +
In the periodic case, the best approximation  $  \widetilde{E}  _ {n _ {1}  \dots n _ {m} } ( f _ {\widetilde{L}  _ {p}  } ( \mathbf R  ^ {m} ) ) $
 +
of a function  $  f \in \widetilde{L}  _ {p} ( \mathbf R  ^ {m} ) $
 +
having (Sobolev generalized) partial derivatives
 +
 
 +
$$
 +
D _  \nu  ^ {r _  \nu  } 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/a/a013/a013010/a01301068.png" /></td> </tr></table>
+
\frac{\partial  ^ {r _  \nu  } }{\partial  t _  \nu  ^ {r _  \nu  } }
  
is called the modulus of continuity of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301070.png" /> in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301071.png" /> with respect to the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301072.png" />. The quantity
+
f  \in \widetilde{L}  _ {p} ( \mathbf R  ^ {m} )
 +
$$
  
<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/a013/a013010/a01301073.png" /></td> </tr></table>
+
(where the  $  r _  \nu  \geq  0 $
 +
are integers and  $  D  ^ {0} f = f $,
 +
$  \nu = 1 \dots m $)
 +
by trigonometric polynomials  $  T _ {n _ {1}  \dots n _ {m} } $,
 +
satisfies the inequality
  
is called the modulus of continuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301075.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301076.png" />.
+
$$ \tag{3 }
 +
\widetilde{E}  _ {n _ {1}  \dots n _ {m} }
 +
(f) _ {\widetilde{L}  _ {p} ( \mathbf R  ^ {m}  ) }  \leq  M
 +
\sum _ {\nu = 1 } ^ { m }
 +
n _  \nu  ^ {- r _  \nu  }
 +
\omega _ {\mathbf e _  \nu  }
 +
( D _  \nu  ^ {r _  \nu  }
 +
f ;  n _  \nu  ^ {-1} ) _ {\widetilde{L}  _ {p} ( \mathbf R  ^ {m}  ) } ,
 +
$$
  
In the periodic case, the best approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301077.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301078.png" /> having (Sobolev generalized) partial derivatives
+
where  $  \mathbf e _  \nu  $
 +
is the unit vector directed along  $  t _  \nu  $,  
 +
and where the constant  $  M $
 +
depends neither on  $  f $
 +
nor on  $  n _  \nu  $.  
 +
For a function $  f \in \widetilde{L}  _ {p} ( \mathbf R  ^ {m} ) $
 +
having 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/a/a013/a013010/a01301079.png" /></td> </tr></table>
+
$$
 +
D ^ {\mathbf r } f  = \
  
(where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301080.png" /> are integers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301082.png" />) by trigonometric polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301083.png" />, satisfies the inequality
+
\frac{\partial  ^ {r} }{\partial  t _ {1} ^ {r _ {1} } \dots
 +
\partial  t _ {m} ^ {r _ {m} } }
  
<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/a013/a013010/a01301084.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
f  \in  \widetilde{L}  _ {p} ( \mathbf R  ^ {m} )
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301085.png" /> is the unit vector directed along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301086.png" />, and where the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301087.png" /> depends neither on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301088.png" /> nor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301089.png" />. For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301090.png" /> having generalized partial derivatives
+
of order  $  r = r _ {1} + \dots + r _ {m} $(
 +
where $  \mathbf r = ( r _ {1} \dots r _ {m} ) $),  
 +
the following inequalities hold:
  
<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/a013/a013010/a01301091.png" /></td> </tr></table>
+
$$ \tag{4 }
 +
\widetilde{E}  _ {n \dots n }
 +
(f) _ {\widetilde{L}  _ {p} ( \mathbf R  ^ {m}  ) }  \leq  \
  
of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301092.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301093.png" />), the following inequalities hold:
+
\frac{M}{n  ^ {r} }
  
<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/a013/a013010/a01301094.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
\sum _ {r _ {1} + \dots
 +
+ r _ {m} =r } \omega
 +
( D ^ {\mathbf r } f ; \
 +
n  ^ {-1} ) _ {\widetilde{L}  _ {p}  ( \mathbf R  ^ {m} ) } .
 +
$$
  
 
If
 
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/a/a013/a013010/a01301095.png" /></td> </tr></table>
+
$$
 +
\omega ( D ^ {\mathbf r }
 +
f ; \delta ) _ {\widetilde{L}  _ {p}  ( \mathbf R  ^ {m} ) }
 +
\leq  K \delta  ^  \alpha  ,\ \
 +
0 < \alpha < 1 ,
 +
$$
  
i.e. if the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301096.png" /> satisfies a [[Hölder condition|Hölder condition]], then
+
i.e. if the function $  D ^ {\mathbf r } f $
 +
satisfies a [[Hölder condition|Hölder condition]], 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/a013/a013010/a01301097.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
$$ \tag{5 }
 +
\widetilde{E}  _ {n \dots n }
 +
(f) _ {\widetilde{L}  _ {p} ( \mathbf R  ^ {m}  ) }  \leq  \
  
In the last case the inverse theorem asserts that if for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301098.png" /> inequality (5) holds for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a01301099.png" /> then the derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010100.png" /> exist and satisfy for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010101.png" /> the inequalities
+
\frac{M}{n ^ {r + \alpha } }
 +
,\ \
 +
r = 0 , 1 ,\dots; \ \
 +
0 < \alpha \leq  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/a/a013/a013010/a013010102.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
In the last case the inverse theorem asserts that if for a function  $  f \in \widetilde{L}  _ {p} ( \mathbf R  ^ {m} ) $
 +
inequality (5) holds for all  $  n = 1 , 2 \dots $
 +
then the derivatives  $  D ^ {\mathbf r } f \in \widetilde{L}  _ {p} ( \mathbf R  ^ {m} ) $
 +
exist and satisfy for any  $  \mathbf h \in \mathbf R  ^ {m} $
 +
the inequalities
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010103.png" />; whereas
+
$$ \tag{6 }
 +
\| D ^ {\mathbf r } f
 +
( \mathbf t + \mathbf h ) - D ^ {\mathbf r }
 +
f ( \mathbf t ) \| _ {\widetilde{L}  _ {p}  ( \mathbf R  ^ {m} ) }  \leq  \
 +
K  | \mathbf h |  ^  \alpha
 +
$$
  
<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/a013/a013010/a013010104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
for  $  0 < \alpha < 1 $;  
 +
whereas
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010105.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010106.png" /> does not depend on the length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010107.png" /> of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010108.png" />.
+
$$ \tag{7 }
 +
\| D ^ {\mathbf r } f
 +
( \mathbf t + \mathbf h ) - 2 D  ^ {\mathbf r}
 +
f ( \mathbf t) - D ^ {\mathbf r }
 +
f ( \mathbf t - \mathbf h ) \| _ {\widetilde{L}  _ {p}  ( \mathbf R  ^ {m} ) }
 +
\leq  K  | \mathbf h |
 +
$$
  
Theorems analogous to those given above also hold for non-periodic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010109.png" /> if entire functions of exponential type are used as tools of approximation. The above-mentioned results can be extended also to classes of functions whose smoothness is described in terms of moduli of continuity (moduli of smoothness) of a higher order (see [[#References|[1]]]).
+
for $  \alpha = 1 $.
 +
Here  $  K $
 +
does not depend on the length  $  | \mathbf h | = ( h _ {1}  ^ {2} + \dots + h _ {m} ) ^ {1/2} $
 +
of the vector  $  \mathbf h = ( h _ {1} \dots h _ {m} ) $.
  
In the case of approximating functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010110.png" /> by algebraic polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010111.png" /> on a bounded parallelepiped (as well on some other bounded sets) direct theorems similar to (3), (4) and (5) have been proved. Inverses of these theorems, like in the case of functions defined on a bounded interval, are possible only on a proper subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010112.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010113.png" />. Inverse theorems are known assuming a better order of approximation in the neighbourhood of boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010114.png" /> (see [[#References|[13]]]); there are also direct theorems asserting the possibility of achieving such an improvement in a neighbourhood of angular points (see [[#References|[14]]]). Necessary and sufficient conditions for a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010115.png" /> to belong to a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010116.png" /> (defined in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010117.png" />) by conditions similar to the conditions (6) and (7)) at the cost of raising the order of approximation in a neighbourhood of the boundary (like in the one-dimensional case) are unknown (1983). However, one has the following results of a negative character (see [[#References|[13]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010118.png" />. There does not exist a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010119.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010120.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010121.png" />, of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010122.png" /> having the two following properties:
+
Theorems analogous to those given above also hold for non-periodic functions $  f \in L _ {p} ( \mathbf R  ^ {m} ) $
 +
if entire functions of exponential type are used as tools of approximation. The above-mentioned results can be extended also to classes of functions whose smoothness is described in terms of moduli of continuity (moduli of smoothness) of a higher order (see [[#References|[1]]]).
  
1) For any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010123.png" /> there are a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010124.png" /> and a sequence of polynomials
+
In the case of approximating functions  $  f \in L _ {p} (Q) $
 +
by algebraic polynomials  $  P _ {n _ {1}  \dots n _ {m} } $
 +
on a bounded parallelepiped (as well on some other bounded sets) direct theorems similar to (3), (4) and (5) have been proved. Inverses of these theorems, like in the case of functions defined on a bounded interval, are possible only on a proper subset  $  Q _ {1} $
 +
of  $  Q $.  
 +
Inverse theorems are known assuming a better order of approximation in the neighbourhood of boundary of  $  Q $(
 +
see [[#References|[13]]]); there are also direct theorems asserting the possibility of achieving such an improvement in a neighbourhood of angular points (see [[#References|[14]]]). Necessary and sufficient conditions for a function  $  f $
 +
to belong to a class  $  H _ {C} ^ {r + \alpha } (Q) $(
 +
defined in the metric of  $  C (Q) $)
 +
by conditions similar to the conditions (6) and (7)) at the cost of raising the order of approximation in a neighbourhood of the boundary (like in the one-dimensional case) are unknown (1983). However, one has the following results of a negative character (see [[#References|[13]]]). Let  $  Q = \{ {\mathbf t } : {\mathbf t \in \mathbf R  ^ {2} ,  | \mathbf t | \leq  1 } \} $.
 +
There does not exist a sequence $  \lambda _ {n} ^ {( \alpha ) } ( | \mathbf t | ) $,
 +
$  n = 1 , 2 ,\dots $;
 +
$  0 < \alpha < 1 $,
 +
of functions in  $  Q $
 +
having the two following properties:
  
<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/a013/a013010/a013010125.png" /></td> </tr></table>
+
1) For any function  $  f \in H _ {C}  ^  \alpha  (Q) $
 +
there are a constant  $  M $
 +
and a sequence of polynomials
 +
 
 +
$$
 +
P _ {n} ( \mathbf t )  = \
 +
\sum _
 +
{0 \leq  k _ {1} + k _ {2} \leq  n }
 +
a _ {k _ {1}  , k _ {2} }
 +
^ {(n)} t _ {1} ^ {k _ {1} }
 +
t _ {2} ^ {k _ {2} } ,\ \
 +
n = 1 , 2 \dots
 +
$$
  
 
such that
 
such that
  
<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/a013/a013010/a013010126.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
$$ \tag{8 }
 +
| f ( \mathbf t ) - P _ {n} ( \mathbf t ) |
 +
\leq  M \lambda _ {n} ^ {( \alpha ) }
 +
( | \mathbf t | ) ,\ \
 +
\mathbf t \in Q .
 +
$$
  
2) The existence of a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010127.png" /> and of a sequence of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010128.png" /> satisfying (8) implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010129.png" /> for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010130.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010131.png" />.
+
2) The existence of a constant $  M > 0 $
 +
and of a sequence of polynomials $  P _ {n} ( \mathbf t ) $
 +
satisfying (8) implies that $  f \in H _ {C}  ^  \alpha  (Q) $
 +
for any function $  f $
 +
defined on $  Q $.
  
 
To illustrate the specific nature of approximation of functions of several variables, one can mention the following results.
 
To illustrate the specific nature of approximation of functions of several variables, one can mention the following results.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010132.png" /> be the best approximation of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010133.png" />-periodic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010134.png" /> of two variables by trigonometric polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010135.png" /> in the metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010136.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010137.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010138.png" />), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010139.png" /> be the best approximation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010140.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010141.png" /> by functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010142.png" />, i.e. by trigonometric polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010143.png" /> in the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010144.png" /> and with coefficients that are functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010145.png" />. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010146.png" /> is defined similarly.
+
Let $  \widetilde{E}  _ {n _ {1}  , n _ {2} } (f) _ {\widetilde{X}  }  $
 +
be the best approximation of a $  2 \pi $-
 +
periodic function $  f $
 +
of two variables by trigonometric polynomials $  T _ {n _ {1}  , n _ {2} } $
 +
in the metric of $  \widetilde{X}  $(
 +
$  \widetilde{X}  = \widetilde{C}  ( \mathbf R  ^ {2} ) $
 +
or $  \widetilde{X}  = \widetilde{L}  _ {p} ( \mathbf R  ^ {2} ) $),  
 +
and let $  E _ {n _ {1}  , \infty } (f) _ {\widetilde{X}  }  $
 +
be the best approximation of $  f $
 +
in $  \widetilde{X}  $
 +
by functions $  T _ {n _ {1}  , \infty } $,  
 +
i.e. by trigonometric polynomials of degree at most $  n _ {1} $
 +
in the variable $  t _ {1} $
 +
and with coefficients that are functions of $  t _ {2} $.  
 +
The quantity $  \widetilde{E}  _ {\infty , n _ {2}  } (f) _ {\widetilde{X}  }  $
 +
is defined similarly.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010147.png" />, one has the inequalities
+
If $  1 < p < \infty $,  
 +
one has the inequalities
  
<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/a013/a013010/a013010148.png" /></td> </tr></table>
+
$$
 +
\widetilde{E}  _ {n _ {1}  , n _ {2} }
 +
(f) _ {\widetilde{L}  _ {p} ( \mathbf R  ^ {2}  ) }  \leq  \
 +
A _ {p} \{ \widetilde{E}  _ {n _ {1}  , \infty }
 +
(f) _ {\widetilde{L}  _ {p} ( \mathbf R  ^ {2}  ) } +
 +
E _ {\infty , n _ {2}  }
 +
(f) _ {\widetilde{L}  _ {p} ( \mathbf R  ^ {2}  ) } \} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010149.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010150.png" />.
+
where $  A _ {p} $
 +
depends only on $  p $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010151.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010152.png" />, then
+
If $  \widetilde{X}  = \widetilde{L}  _ {1} ( \mathbf R  ^ {2} ) $
 +
or $  \widetilde{X}  = \widetilde{C}  ( \mathbf R  ^ {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/a/a013/a013010/a013010153.png" /></td> <td valign="top" style="width:5%;text-align:right;">(9)</td></tr></table>
+
$$ \tag{9 }
 +
\widetilde{E}  _ {n _ {1}  , n _ {2} }
 +
(f) _ {\widetilde{X}  }  \leq
 +
$$
  
<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/a013/a013010/a013010154.png" /></td> </tr></table>
+
$$
 +
\leq  \
 +
A  \mathop{\rm ln} ( 2 +  \mathop{\rm min} \{ n _ {1} , n _ {2} \} ) ( \widetilde{E}  _ {n _ {1}  , \infty } (f) _ {\widetilde{X}  }  + \widetilde{E}  _ {\infty , n _ {2}  } (f) _ {\widetilde{X}  }  ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010155.png" /> is an absolute constant, and the factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010156.png" /> in (9) cannot be replaced by another factor tending slower to infinity as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010157.png" /> (see [[#References|[15]]]).
+
where $  A $
 +
is an absolute constant, and the factor $  \mathop{\rm ln} ( 2 +  \mathop{\rm min} \{ n _ {1} , n _ {2} \} ) $
 +
in (9) cannot be replaced by another factor tending slower to infinity as $  \mathop{\rm min} \{ n _ {1} , n _ {2} \} \rightarrow \infty $(
 +
see [[#References|[15]]]).
  
Fundamental peculiarities arise in problems of interpolation for functions of several variables. For instance, contrary to the one-dimensional case, the existence of an algebraic interpolation polynomial depends substantially on the position of the interpolation nodes. Nevertheless, effective methods of constructing polynomials and splines interpolating a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010158.png" /> on a distribution of nodes chosen in a definite way, have been established (see [[Interpolation|Interpolation]]). For multi-dimensional interpolation by splines, in a number of cases order estimates of the approximation errors, for the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013010/a013010159.png" /> as well as for its partial derivatives, have been found; in this direction, two-dimensional splines of low degree, as well as local (Hermitian) splines of arbitrary degree, have been studied in more detail (see [[#References|[7]]], [[#References|[10]]]–[[#References|[12]]]). Among the other linear methods for approximating functions of several variables, multiple Fourier sums and their various means have been studied most extensively. Here, order estimates of the approximation with respect to classes of functions are known, and in some case asymptotically exact results were obtained (see [[#References|[5]]], [[#References|[6]]] and [[#References|[8]]]).
+
Fundamental peculiarities arise in problems of interpolation for functions of several variables. For instance, contrary to the one-dimensional case, the existence of an algebraic interpolation polynomial depends substantially on the position of the interpolation nodes. Nevertheless, effective methods of constructing polynomials and splines interpolating a function $  f ( t _ {1} \dots t _ {m} ) $
 +
on a distribution of nodes chosen in a definite way, have been established (see [[Interpolation|Interpolation]]). For multi-dimensional interpolation by splines, in a number of cases order estimates of the approximation errors, for the function $  f $
 +
as well as for its partial derivatives, have been found; in this direction, two-dimensional splines of low degree, as well as local (Hermitian) splines of arbitrary degree, have been studied in more detail (see [[#References|[7]]], [[#References|[10]]]–[[#References|[12]]]). Among the other linear methods for approximating functions of several variables, multiple Fourier sums and their various means have been studied most extensively. Here, order estimates of the approximation with respect to classes of functions are known, and in some case asymptotically exact results were obtained (see [[#References|[5]]], [[#References|[6]]] and [[#References|[8]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  R.S. Guter,  L.D. Kudryavtsev,  B.M. Levitan,  "Elements of the theory of functions" , Moscow  (1963)  pp. 106–198  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.P. Korneichuk,  "Extremal problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.M. Tikhomirov,  "Some problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.K. Dzyadyk,  "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.F. Timan,  "Theory of approximation of functions of a real variable" , Pergamon  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.B. Stechkin,  Yu.N. Subbotin,  "Splines in numerical mathematics" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.I. Stepanets,  "Uniform approximation by trigonometric polynomials. Linear methods" , Kiev  (1980)  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  P.J. Laurent,  "Approximation et optimisation" , Hermann  (1972)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.H. Ahlberg,  E.N. Nilson,  J.F. Walsh,  "Theory of splines and their applications" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  V.L. Miroshichenko,  "Methods of spline functions" , Moscow  (1980)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  R.S. Varga,  "Functional analysis and approximation theory in numerical analysis" , ''Reg. Conf. Ser. Appl. Math.'' , '''3''' , SIAM  (1971)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables by polynomials"  ''Siberian Math. J.'' , '''10''' :  4  (1969)  pp. 792–799  ''Sibirsk. Mat. Zh.'' , '''10'''  (1969)  pp. 1075–1083</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  Yu.A. Brudnyi,  "Approximation of functions defined in a convex polyhedron"  ''Soviet Math. Doklady'' , '''11''' :  6  (1970)  pp. 1587–1590  ''Dokl. Akad. Nauk SSSR'' , '''195'''  (1970)  pp. 1007–1009</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  V.N. Teml'yakov,  "Best approximations for functions of two variables"  ''Soviet Math. Doklady'' , '''16''' :  4  (1975)  pp. 1051–1055  ''Dokl. Akad. Nauk SSSR'' , '''223'''  (1975)  pp. 1079–1082</TD></TR></table>
 
<table><TR><TD valign="top">[1]</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">[2]</TD> <TD valign="top">  R.S. Guter,  L.D. Kudryavtsev,  B.M. Levitan,  "Elements of the theory of functions" , Moscow  (1963)  pp. 106–198  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N.P. Korneichuk,  "Extremal problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.M. Tikhomirov,  "Some problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.K. Dzyadyk,  "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow  (1977)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.F. Timan,  "Theory of approximation of functions of a real variable" , Pergamon  (1963)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  S.B. Stechkin,  Yu.N. Subbotin,  "Splines in numerical mathematics" , Moscow  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.I. Stepanets,  "Uniform approximation by trigonometric polynomials. Linear methods" , Kiev  (1980)  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  P.J. Laurent,  "Approximation et optimisation" , Hermann  (1972)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  J.H. Ahlberg,  E.N. Nilson,  J.F. Walsh,  "Theory of splines and their applications" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  V.L. Miroshichenko,  "Methods of spline functions" , Moscow  (1980)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  R.S. Varga,  "Functional analysis and approximation theory in numerical analysis" , ''Reg. Conf. Ser. Appl. Math.'' , '''3''' , SIAM  (1971)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  S.M. Nikol'skii,  "Approximation of functions of several variables by polynomials"  ''Siberian Math. J.'' , '''10''' :  4  (1969)  pp. 792–799  ''Sibirsk. Mat. Zh.'' , '''10'''  (1969)  pp. 1075–1083</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  Yu.A. Brudnyi,  "Approximation of functions defined in a convex polyhedron"  ''Soviet Math. Doklady'' , '''11''' :  6  (1970)  pp. 1587–1590  ''Dokl. Akad. Nauk SSSR'' , '''195'''  (1970)  pp. 1007–1009</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  V.N. Teml'yakov,  "Best approximations for functions of two variables"  ''Soviet Math. Doklady'' , '''16''' :  4  (1975)  pp. 1051–1055  ''Dokl. Akad. Nauk SSSR'' , '''223'''  (1975)  pp. 1079–1082</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,  "Four lectures on multivariate approximation"  S.P. Singh (ed.)  J.H.W. Burry (ed.)  B. Watson (ed.) , ''Approximation theory and spline functions'' , Reidel  (1984)  pp. 65–87</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Schempp (ed.)  K. Zeller (ed.) , ''Multivariate approximation theory'' , Birkhäuser  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Schempp (ed.)  K. Zeller (ed.) , ''Multivariate approximation theory'' , '''II''' , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Schempp (ed.)  K. Zeller (ed.) , ''Multivariate approximation theory'' , '''III''' , Birkhäuser  (1985)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Schempp (ed.)  K. Zeller (ed.) , ''Constructive theory of functions of several variables'' , Springer  (1977)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G.G. Lorentz,  "Approximation of functions" , Holt, Rinehart &amp; Winston  (1966)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Golomb,  "Approximation by functions of fewer variables"  R.E. Langer (ed.) , ''On numerical approximation'' , Univ. of Wisconsin Press  (1959)  pp. 275–327</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L.L. Schumaker,  "Spline functions, basic theory" , Wiley  (1981)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,  "Four lectures on multivariate approximation"  S.P. Singh (ed.)  J.H.W. Burry (ed.)  B. Watson (ed.) , ''Approximation theory and spline functions'' , Reidel  (1984)  pp. 65–87</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Schempp (ed.)  K. Zeller (ed.) , ''Multivariate approximation theory'' , Birkhäuser  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Schempp (ed.)  K. Zeller (ed.) , ''Multivariate approximation theory'' , '''II''' , Birkhäuser  (1982)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W. Schempp (ed.)  K. Zeller (ed.) , ''Multivariate approximation theory'' , '''III''' , Birkhäuser  (1985)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Schempp (ed.)  K. Zeller (ed.) , ''Constructive theory of functions of several variables'' , Springer  (1977)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G.G. Lorentz,  "Approximation of functions" , Holt, Rinehart &amp; Winston  (1966)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Golomb,  "Approximation by functions of fewer variables"  R.E. Langer (ed.) , ''On numerical approximation'' , Univ. of Wisconsin Press  (1959)  pp. 275–327</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  L.L. Schumaker,  "Spline functions, basic theory" , Wiley  (1981)</TD></TR></table>

Latest revision as of 20:29, 10 January 2021


The case when the function $ f $ to be approximated depends on two or more variables:

$$ f (t) = f ( t _ {1} \dots t _ {m} ) ,\ \ m \geq 2 . $$

(See Approximation of functions.) In comparison with the one-dimensional case, the study of problems on the approximation of functions of $ m $( $ m \geq 2 $) variables is highly complicated because of essentially new circumstances that arise related to dimension. First of all this applies to the domain on which the approximation is carried out. Simply-connected compact sets (in the one-dimensional case intervals) in $ \mathbf R ^ {m} $( even in the plane) can have a large variety of shapes; therefore it becomes necessary to classify the domains according to, for instance, the smoothness properties of their boundaries. Complications also arise in the description of difference-differential properties of functions of $ m $ variables. Generally speaking, these properties can be different in different directions, and their characterization ought to take into account both the geometry of the domain on which the function is defined and the behaviour of the function while approaching the boundary, so that the study of boundary properties of the function is of great importance. If one attempts to simplify the solution of an approximation problem by passing to a domain with a simpler structure, the problem that arises is to extend a function $ f $ from a domain $ Q \subset \mathbf R ^ {m} $ into a canonical domain $ Q _ {1} $( for instance a parallelepiped or the whole space $ \mathbf R ^ {m} $) containing $ Q $ while preserving certain smoothness properties (see Extension theorems). This kind of problems is closely connected with imbedding theorems, and also with questions arising in the solution of the boundary value problems of mathematical physics.

An increase in the number of independent variables complicates, of course, also the tools of approximation, because when the dimension is increased, with it, e.g., the degrees of the polynomials increase. An algebraic polynomial of degrees $ n _ {1} \dots n _ {m} $ in the variables $ t _ {1} \dots t _ {m} $, respectively, has the form

$$ \tag{1 } P _ {n _ {1} \dots n _ {m} } ( t _ {1} \dots t _ {m} ) = \ \sum a _ {k _ {1} \dots k _ {m} } t _ {1} ^ {k _ {1} } \dots t _ {m} ^ {k _ {m} } , $$

where $ a _ {k _ {1} \dots k _ {m} } $ are real coefficients. The summation is carried out with respect to $ k _ \nu $, $ \nu = 1 \dots m $, from 0 to $ n _ \nu $, so that, e.g., the subspace of polynomials of degree 3 in each of the $ m $ variables has dimension $ 4 ^ {m} $. Sometimes the overall or total degree $ n $ of the polynomial is fixed; then the summation in (1) is carried out over all indices satisfying the inequality $ 0 \leq k _ {1} + \dots + k _ {m} \leq n $. A real trigonometric polynomial of degrees $ n _ {1} \dots n _ {m} $ in the variables $ t _ {1} \dots t _ {m} $ can be written in the form

$$ T _ {n _ {1} \dots n _ {m} } ( t _ {1} \dots t _ {m} ) = \ \sum a _ {k _ {1} \dots k _ \nu } \mathop{\rm exp} \left ( i \sum _ {\nu = 1 } ^ { {m } } k _ \nu t _ \nu \right ) , $$

where complex coefficients $ a _ {k _ {1} \dots k _ {n} } $ with indices of opposite sign are complex conjugate, and the summation is carried out with respect to $ k _ \nu $, $ \nu = 1 \dots m $, from $ - n _ \nu $ to $ n _ \nu $. Such a polynomial can be represented in the form of a linear combination of all possible products of the form $ \phi _ {k _ {1} } ( t _ {1} ) \dots \phi _ {k _ {m} } ( t _ {m} ) $, where $ \phi _ {k _ \nu } ( t _ \nu ) $ is either $ \sin k _ \nu t _ \nu $( $ 0 \leq k _ \nu \leq n _ \nu $) or $ \cos k _ \nu t _ \nu $( $ 0 \leq k _ \nu \leq n _ \nu $). Multi-dimensional splines have a wide field of applications; they consist of "pieces" of algebraic polynomials in $ m $ variables "tied" together in order to obtain specified smoothness properties. In the case $ m = 2 $, splines of the simplest form are formed by polynomials "tied" together along straight lines parallel to the coordinate axes. As a tool of approximation functions $ g ( t _ {1} \dots t _ {m} ) $ are used which are polynomials or splines only in one of the variables. For the approximation of non-periodic functions given on the whole space $ \mathbf R ^ {m} $( or on an unbounded subset of $ \mathbf R ^ {m} $) one may use entire functions of exponential type. These can be represented in the form of sums of absolutely convergent power series

$$ \tag{2 } G _ {n _ {1} \dots n _ {m} } ( t _ {i} \dots t _ {m} ) = \ \sum _ {\begin{array}{c} k _ \nu \geq 0 , \\ \nu = 1 \dots m \end{array} } a _ {k _ {1} \dots k _ {m} } t _ {1} ^ {k _ {1} } \dots t _ {m} ^ {k _ {m} } $$

under the condition that for any $ \epsilon > 0 $ and all complex $ t _ {1} \dots t _ {m} $ one has

$$ | G _ {n _ {1} \dots n _ {m} } ( t _ {1} \dots t _ {m} ) | \leq M _ \epsilon \ \mathop{\rm exp} \ \sum _ {\nu = 1 } ^ { m } ( n _ \nu + \epsilon ) | t _ \nu | , $$

where the constant $ M _ \epsilon $ depends only on $ \epsilon $( see [1]). It should be noted that, unlike polynomials, the function (2) is determined by an infinite number of parameters.

In the multi-dimensional case one has Weierstrass' theorem on the possibility of approximating a function $ f \in C (Q) $, i.e. continuous on a bounded and closed set $ Q \subset \mathbf R ^ {m} $, or a function $ f \in \widetilde{C} ( \mathbf R ^ {m} ) $, continuous on the whole space $ \mathbf R ^ {m} $ with period $ 2 \pi $ in each variable, with arbitrary accuracy by means of algebraic (respectively, trigonometric) polynomials. A similar result holds in the spaces $ L _ {p} (Q) $ and $ \widetilde{L} _ {p} ( \mathbf R ^ {m} ) $( in the periodic case) ( $ 1 \leq p < \infty $). General results and theorems concerning properties of the best approximation, the existence and uniqueness, the characteristic properties of functions of best approximation, and general relations of duality when approximating by (means of) a convex set (of functions) and, in particular, by a subspace, can be extended to normed linear spaces of functions of $ m $ variables (see [3] and [4]). However, to obtain explicit formulations of these theorems, taking into account a specific metric and specific properties of the approximating subspace in the multi-dimensional case, involves great difficulties.

Quite thorough investigations were carried out on questions concerning the connection between the smoothness properties of a function of several variables and the rate of decrease of its best approximation by algebraic and trigonometric polynomials, as well as by entire functions.

Let $ Q $ be an arbitrary open set in $ \mathbf R ^ {m} $( in particular, $ Q = \mathbf R ^ {m} $), let $ \mathbf e $ be a unit vector in $ \mathbf R ^ {m} $, let $ h > 0 $ and let $ Q _ {h \mathbf e } $ be the set of points $ t \in Q $ such that the segment $ [ t , t + h \mathbf e ] \in Q $. If $ f \in L _ {p} (Q) $ and $ 1 \leq p \leq \infty $, then the quantity

$$ \omega _ {\mathbf e } ( f ; \delta ) _ {L _ {p} (Q) } = \ \sup _ {h \leq \delta } \ \| f ( t + h \mathbf e ) - f (t) \| _ {L _ {p} ( Q _ {h \mathbf e } ) } $$

is called the modulus of continuity of the function $ f ( t _ {1} \dots t _ {m} ) $ in the direction $ \mathbf e $ with respect to the metric of $ L _ {p} (Q) $. The quantity

$$ \omega ( f ; \delta ) _ {L _ {p} (Q) } = \ \sup _ {\mathbf e } \ \omega _ {\mathbf e } ( f ; \delta ) _ {L _ {p} (Q) } $$

is called the modulus of continuity of $ f $ in $ L _ {p} (Q) $.

In the periodic case, the best approximation $ \widetilde{E} _ {n _ {1} \dots n _ {m} } ( f _ {\widetilde{L} _ {p} } ( \mathbf R ^ {m} ) ) $ of a function $ f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} ) $ having (Sobolev generalized) partial derivatives

$$ D _ \nu ^ {r _ \nu } f = \ \frac{\partial ^ {r _ \nu } }{\partial t _ \nu ^ {r _ \nu } } f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} ) $$

(where the $ r _ \nu \geq 0 $ are integers and $ D ^ {0} f = f $, $ \nu = 1 \dots m $) by trigonometric polynomials $ T _ {n _ {1} \dots n _ {m} } $, satisfies the inequality

$$ \tag{3 } \widetilde{E} _ {n _ {1} \dots n _ {m} } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq M \sum _ {\nu = 1 } ^ { m } n _ \nu ^ {- r _ \nu } \omega _ {\mathbf e _ \nu } ( D _ \nu ^ {r _ \nu } f ; n _ \nu ^ {-1} ) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } , $$

where $ \mathbf e _ \nu $ is the unit vector directed along $ t _ \nu $, and where the constant $ M $ depends neither on $ f $ nor on $ n _ \nu $. For a function $ f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} ) $ having generalized partial derivatives

$$ D ^ {\mathbf r } f = \ \frac{\partial ^ {r} }{\partial t _ {1} ^ {r _ {1} } \dots \partial t _ {m} ^ {r _ {m} } } f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} ) $$

of order $ r = r _ {1} + \dots + r _ {m} $( where $ \mathbf r = ( r _ {1} \dots r _ {m} ) $), the following inequalities hold:

$$ \tag{4 } \widetilde{E} _ {n \dots n } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq \ \frac{M}{n ^ {r} } \sum _ {r _ {1} + \dots + r _ {m} =r } \omega ( D ^ {\mathbf r } f ; \ n ^ {-1} ) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } . $$

If

$$ \omega ( D ^ {\mathbf r } f ; \delta ) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq K \delta ^ \alpha ,\ \ 0 < \alpha < 1 , $$

i.e. if the function $ D ^ {\mathbf r } f $ satisfies a Hölder condition, then

$$ \tag{5 } \widetilde{E} _ {n \dots n } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq \ \frac{M}{n ^ {r + \alpha } } ,\ \ r = 0 , 1 ,\dots; \ \ 0 < \alpha \leq 1 . $$

In the last case the inverse theorem asserts that if for a function $ f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} ) $ inequality (5) holds for all $ n = 1 , 2 \dots $ then the derivatives $ D ^ {\mathbf r } f \in \widetilde{L} _ {p} ( \mathbf R ^ {m} ) $ exist and satisfy for any $ \mathbf h \in \mathbf R ^ {m} $ the inequalities

$$ \tag{6 } \| D ^ {\mathbf r } f ( \mathbf t + \mathbf h ) - D ^ {\mathbf r } f ( \mathbf t ) \| _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq \ K | \mathbf h | ^ \alpha $$

for $ 0 < \alpha < 1 $; whereas

$$ \tag{7 } \| D ^ {\mathbf r } f ( \mathbf t + \mathbf h ) - 2 D ^ {\mathbf r} f ( \mathbf t) - D ^ {\mathbf r } f ( \mathbf t - \mathbf h ) \| _ {\widetilde{L} _ {p} ( \mathbf R ^ {m} ) } \leq K | \mathbf h | $$

for $ \alpha = 1 $. Here $ K $ does not depend on the length $ | \mathbf h | = ( h _ {1} ^ {2} + \dots + h _ {m} ) ^ {1/2} $ of the vector $ \mathbf h = ( h _ {1} \dots h _ {m} ) $.

Theorems analogous to those given above also hold for non-periodic functions $ f \in L _ {p} ( \mathbf R ^ {m} ) $ if entire functions of exponential type are used as tools of approximation. The above-mentioned results can be extended also to classes of functions whose smoothness is described in terms of moduli of continuity (moduli of smoothness) of a higher order (see [1]).

In the case of approximating functions $ f \in L _ {p} (Q) $ by algebraic polynomials $ P _ {n _ {1} \dots n _ {m} } $ on a bounded parallelepiped (as well on some other bounded sets) direct theorems similar to (3), (4) and (5) have been proved. Inverses of these theorems, like in the case of functions defined on a bounded interval, are possible only on a proper subset $ Q _ {1} $ of $ Q $. Inverse theorems are known assuming a better order of approximation in the neighbourhood of boundary of $ Q $( see [13]); there are also direct theorems asserting the possibility of achieving such an improvement in a neighbourhood of angular points (see [14]). Necessary and sufficient conditions for a function $ f $ to belong to a class $ H _ {C} ^ {r + \alpha } (Q) $( defined in the metric of $ C (Q) $) by conditions similar to the conditions (6) and (7)) at the cost of raising the order of approximation in a neighbourhood of the boundary (like in the one-dimensional case) are unknown (1983). However, one has the following results of a negative character (see [13]). Let $ Q = \{ {\mathbf t } : {\mathbf t \in \mathbf R ^ {2} , | \mathbf t | \leq 1 } \} $. There does not exist a sequence $ \lambda _ {n} ^ {( \alpha ) } ( | \mathbf t | ) $, $ n = 1 , 2 ,\dots $; $ 0 < \alpha < 1 $, of functions in $ Q $ having the two following properties:

1) For any function $ f \in H _ {C} ^ \alpha (Q) $ there are a constant $ M $ and a sequence of polynomials

$$ P _ {n} ( \mathbf t ) = \ \sum _ {0 \leq k _ {1} + k _ {2} \leq n } a _ {k _ {1} , k _ {2} } ^ {(n)} t _ {1} ^ {k _ {1} } t _ {2} ^ {k _ {2} } ,\ \ n = 1 , 2 \dots $$

such that

$$ \tag{8 } | f ( \mathbf t ) - P _ {n} ( \mathbf t ) | \leq M \lambda _ {n} ^ {( \alpha ) } ( | \mathbf t | ) ,\ \ \mathbf t \in Q . $$

2) The existence of a constant $ M > 0 $ and of a sequence of polynomials $ P _ {n} ( \mathbf t ) $ satisfying (8) implies that $ f \in H _ {C} ^ \alpha (Q) $ for any function $ f $ defined on $ Q $.

To illustrate the specific nature of approximation of functions of several variables, one can mention the following results.

Let $ \widetilde{E} _ {n _ {1} , n _ {2} } (f) _ {\widetilde{X} } $ be the best approximation of a $ 2 \pi $- periodic function $ f $ of two variables by trigonometric polynomials $ T _ {n _ {1} , n _ {2} } $ in the metric of $ \widetilde{X} $( $ \widetilde{X} = \widetilde{C} ( \mathbf R ^ {2} ) $ or $ \widetilde{X} = \widetilde{L} _ {p} ( \mathbf R ^ {2} ) $), and let $ E _ {n _ {1} , \infty } (f) _ {\widetilde{X} } $ be the best approximation of $ f $ in $ \widetilde{X} $ by functions $ T _ {n _ {1} , \infty } $, i.e. by trigonometric polynomials of degree at most $ n _ {1} $ in the variable $ t _ {1} $ and with coefficients that are functions of $ t _ {2} $. The quantity $ \widetilde{E} _ {\infty , n _ {2} } (f) _ {\widetilde{X} } $ is defined similarly.

If $ 1 < p < \infty $, one has the inequalities

$$ \widetilde{E} _ {n _ {1} , n _ {2} } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {2} ) } \leq \ A _ {p} \{ \widetilde{E} _ {n _ {1} , \infty } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {2} ) } + E _ {\infty , n _ {2} } (f) _ {\widetilde{L} _ {p} ( \mathbf R ^ {2} ) } \} , $$

where $ A _ {p} $ depends only on $ p $.

If $ \widetilde{X} = \widetilde{L} _ {1} ( \mathbf R ^ {2} ) $ or $ \widetilde{X} = \widetilde{C} ( \mathbf R ^ {2} ) $, then

$$ \tag{9 } \widetilde{E} _ {n _ {1} , n _ {2} } (f) _ {\widetilde{X} } \leq $$

$$ \leq \ A \mathop{\rm ln} ( 2 + \mathop{\rm min} \{ n _ {1} , n _ {2} \} ) ( \widetilde{E} _ {n _ {1} , \infty } (f) _ {\widetilde{X} } + \widetilde{E} _ {\infty , n _ {2} } (f) _ {\widetilde{X} } ) , $$

where $ A $ is an absolute constant, and the factor $ \mathop{\rm ln} ( 2 + \mathop{\rm min} \{ n _ {1} , n _ {2} \} ) $ in (9) cannot be replaced by another factor tending slower to infinity as $ \mathop{\rm min} \{ n _ {1} , n _ {2} \} \rightarrow \infty $( see [15]).

Fundamental peculiarities arise in problems of interpolation for functions of several variables. For instance, contrary to the one-dimensional case, the existence of an algebraic interpolation polynomial depends substantially on the position of the interpolation nodes. Nevertheless, effective methods of constructing polynomials and splines interpolating a function $ f ( t _ {1} \dots t _ {m} ) $ on a distribution of nodes chosen in a definite way, have been established (see Interpolation). For multi-dimensional interpolation by splines, in a number of cases order estimates of the approximation errors, for the function $ f $ as well as for its partial derivatives, have been found; in this direction, two-dimensional splines of low degree, as well as local (Hermitian) splines of arbitrary degree, have been studied in more detail (see [7], [10][12]). Among the other linear methods for approximating functions of several variables, multiple Fourier sums and their various means have been studied most extensively. Here, order estimates of the approximation with respect to classes of functions are known, and in some case asymptotically exact results were obtained (see [5], [6] and [8]).

References

[1] S.M. Nikol'skii, "Approximation of functions of several variables and imbedding theorems" , Springer (1975) (Translated from Russian)
[2] R.S. Guter, L.D. Kudryavtsev, B.M. Levitan, "Elements of the theory of functions" , Moscow (1963) pp. 106–198 (In Russian)
[3] N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian)
[4] V.M. Tikhomirov, "Some problems in approximation theory" , Moscow (1976) (In Russian)
[5] V.K. Dzyadyk, "Introduction to the theory of uniform approximation of functions by polynomials" , Moscow (1977) (In Russian)
[6] A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)
[7] S.B. Stechkin, Yu.N. Subbotin, "Splines in numerical mathematics" , Moscow (1976) (In Russian)
[8] A.I. Stepanets, "Uniform approximation by trigonometric polynomials. Linear methods" , Kiev (1980) (In Russian)
[9] P.J. Laurent, "Approximation et optimisation" , Hermann (1972)
[10] J.H. Ahlberg, E.N. Nilson, J.F. Walsh, "Theory of splines and their applications" , Acad. Press (1967)
[11] V.L. Miroshichenko, "Methods of spline functions" , Moscow (1980)
[12] R.S. Varga, "Functional analysis and approximation theory in numerical analysis" , Reg. Conf. Ser. Appl. Math. , 3 , SIAM (1971)
[13] S.M. Nikol'skii, "Approximation of functions of several variables by polynomials" Siberian Math. J. , 10 : 4 (1969) pp. 792–799 Sibirsk. Mat. Zh. , 10 (1969) pp. 1075–1083
[14] Yu.A. Brudnyi, "Approximation of functions defined in a convex polyhedron" Soviet Math. Doklady , 11 : 6 (1970) pp. 1587–1590 Dokl. Akad. Nauk SSSR , 195 (1970) pp. 1007–1009
[15] V.N. Teml'yakov, "Best approximations for functions of two variables" Soviet Math. Doklady , 16 : 4 (1975) pp. 1051–1055 Dokl. Akad. Nauk SSSR , 223 (1975) pp. 1079–1082

Comments

References

[a1] E.W. Cheney, "Four lectures on multivariate approximation" S.P. Singh (ed.) J.H.W. Burry (ed.) B. Watson (ed.) , Approximation theory and spline functions , Reidel (1984) pp. 65–87
[a2] W. Schempp (ed.) K. Zeller (ed.) , Multivariate approximation theory , Birkhäuser (1979)
[a3] W. Schempp (ed.) K. Zeller (ed.) , Multivariate approximation theory , II , Birkhäuser (1982)
[a4] W. Schempp (ed.) K. Zeller (ed.) , Multivariate approximation theory , III , Birkhäuser (1985)
[a5] W. Schempp (ed.) K. Zeller (ed.) , Constructive theory of functions of several variables , Springer (1977)
[a6] G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966)
[a7] M. Golomb, "Approximation by functions of fewer variables" R.E. Langer (ed.) , On numerical approximation , Univ. of Wisconsin Press (1959) pp. 275–327
[a8] L.L. Schumaker, "Spline functions, basic theory" , Wiley (1981)
How to Cite This Entry:
Approximation of functions of several real variables. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximation_of_functions_of_several_real_variables&oldid=12943
This article was adapted from an original article by V.N. KonovalovN.P. Korneichuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article