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An orthogonal system of vectors is a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o0703801.png" /> of non-zero vectors of a Euclidean (Hilbert) space with a scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o0703802.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o0703803.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o0703804.png" />. If under these conditions the norm of each vector is equal to one, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o0703805.png" /> is said to be an [[orthonormal system]]. A complete orthogonal (orthonormal) system of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o0703806.png" /> is called an orthogonal (orthonormal) basis.
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An orthogonal system of vectors is a set  $  \{ x _  \alpha  \} $
 +
of non-zero vectors of a Euclidean (Hilbert) space with a scalar product $  ( \cdot , \cdot ) $
 +
such that $  ( x _  \alpha  , x _  \beta  ) = 0 $
 +
when $  \alpha \neq \beta $.  
 +
If under these conditions the norm of each vector is equal to one, then $  \{ x _  \alpha  \} $
 +
is said to be an [[orthonormal system]]. A complete orthogonal (orthonormal) system of vectors $  \{ x _  \alpha  \} $
 +
is called an orthogonal (orthonormal) basis.
  
 
''M.I. Voitsekhovskii''
 
''M.I. Voitsekhovskii''
  
An orthogonal coordinate system is a coordinate system in which the coordinate lines (or surfaces) intersect at right angles. Orthogonal coordinate systems exist in any Euclidean space, but, generally speaking, do not exist in an arbitrary space. In a two-dimensional smooth affine space, orthogonal systems can always be introduced at least in a sufficiently small neighbourhood of every point. It is sometimes possible to introduce orthogonal coordinate systems in the large. In an orthogonal system, the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o0703807.png" /> is diagonal; the diagonal components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o0703808.png" /> are called Lamé coefficients. The [[Lamé coefficients|Lamé coefficients]] of an orthogonal system in space are expressed by the formulas
+
An orthogonal coordinate system is a coordinate system in which the coordinate lines (or surfaces) intersect at right angles. Orthogonal coordinate systems exist in any Euclidean space, but, generally speaking, do not exist in an arbitrary space. In a two-dimensional smooth affine space, orthogonal systems can always be introduced at least in a sufficiently small neighbourhood of every point. It is sometimes possible to introduce orthogonal coordinate systems in the large. In an orthogonal system, the metric tensor $  g _ {ij} $
 +
is diagonal; the diagonal components $  g _ {ij} $
 +
are called Lamé coefficients. The [[Lamé coefficients|Lamé coefficients]] of an orthogonal system in space are expressed by the formulas
  
<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/o/o070/o070380/o0703809.png" /></td> </tr></table>
+
$$
 +
L _ {u}  = \sqrt {\left (
 +
\frac{\partial  x }{\partial  u }
 +
\right )  ^ {2} + \left (
 +
\frac{\partial  y }{\partial  u }
 +
\right )  ^ {2}
 +
+ \left (
 +
\frac{\partial  z }{\partial  u }
 +
\right )  ^ {2} } ,
 +
$$
  
<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/o/o070/o070380/o07038010.png" /></td> </tr></table>
+
$$
 +
L _ {v}  = \sqrt {\left (
 +
\frac{\partial  x }{\partial  v }
 +
\right )
 +
^ {2} + \left (
 +
\frac{\partial  y }{\partial  v }
 +
\right )  ^ {2}
 +
+ \left (
 +
\frac{\partial  z }{\partial  v }
 +
\right )  ^ {2} } ,
 +
$$
  
<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/o/o070/o070380/o07038011.png" /></td> </tr></table>
+
$$
 +
L _ {w}  = \sqrt {\left (
 +
\frac{\partial  x }{\partial  w }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038014.png" /> are Cartesian coordinates. The Lamé coefficients are also used to express the line element:
+
\right )  ^ {2} + \left (
 +
\frac{\partial  y }{\partial  w }
 +
\right )  ^ {2} + \left (
 +
\frac{\partial  z }{\partial  w }
 +
\right )  ^ {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/o/o070/o070380/o07038015.png" /></td> </tr></table>
+
where  $  x $,
 +
$  y $
 +
and  $  z $
 +
are Cartesian coordinates. The Lamé coefficients are also used to express the line element:
 +
 
 +
$$
 +
ds  = \sqrt {L _ {u}  ^ {2}  du  ^ {2} + L _ {v}  ^ {2}  dv  ^ {2} + L _ {w}  ^ {2}
 +
dw  ^ {2} } ,
 +
$$
  
 
the element of surface area:
 
the element of surface area:
  
<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/o/o070/o070380/o07038016.png" /></td> </tr></table>
+
$$
 +
d \sigma  = \sqrt {( L _ {u} L _ {v}  du  dv)  ^ {2} + ( L _ {u} L _ {w}  du  dw)
 +
^ {2} + ( L _ {v} L _ {w}  dv  dw)  ^ {2} } ,
 +
$$
  
 
the volume element:
 
the volume element:
  
<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/o/o070/o070380/o07038017.png" /></td> </tr></table>
+
$$
 +
dV  = L _ {u} L _ {v} L _ {w}  du  dv  dw,
 +
$$
  
 
and the operations of vector analysis:
 
and the operations of vector analysis:
  
<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/o/o070/o070380/o07038018.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm grad} _ {u}  \phi  =
 +
\frac{1}{L _ {u} }
 +
 +
\frac{\partial  \phi }{\partial  u }
 +
,\ \
 +
\mathop{\rm grad} _ {v}  \phi  =
 +
\frac{1}{L _ {v} }
 +
 +
\frac{\partial  \phi }{\partial  v }
 +
,
 +
$$
  
<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/o/o070/o070380/o07038019.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm grad} _ {w}  \phi  =
 +
\frac{1}{L _ {w} }
 +
 +
\frac{\partial  \phi }{\partial  w }
 +
,
 +
$$
  
<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/o/o070/o070380/o07038020.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm div}  \mathbf a  =
 +
\frac{1}{L _ {u} L _ {v} L _ {w} }
 +
\left [
 +
\frac \partial {\partial  u }
 +
( a _ {u} L _ {v} L _ {w} ) +
 +
\frac \partial {\partial  v }
 +
( a _ {v} L _ {u} L _ {w} ) \right . +
 +
$$
  
<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/o/o070/o070380/o07038021.png" /></td> </tr></table>
+
$$
 +
+ \left .
  
<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/o/o070/o070380/o07038022.png" /></td> </tr></table>
+
\frac \partial {\partial  w }
 +
( a _ {w} L _ {u} L _ {v} ) \right ] ;
 +
$$
  
<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/o/o070/o070380/o07038023.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm rot} _ {u}  \mathbf a  =
 +
\frac{1}{L _ {v} L _ {w} }
 +
\left [
 +
\frac \partial {\partial  v }
 +
( a _ {w} L _ {w} ) -  
 +
\frac \partial {\partial  w }
 +
( a _ {v} L _ {v} ) \right ] ,
 +
$$
  
<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/o/o070/o070380/o07038024.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm rot} _ {v}  \mathbf a  =
 +
\frac{1}{L _ {u} L _ {w} }
 +
\left [
 +
\frac \partial {\partial  w }
 +
( a _ {u} L _ {u} ) -  
 +
\frac \partial {\partial  u }
 +
( a _ {w} L _ {w} ) \right ] ,
 +
$$
  
<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/o/o070/o070380/o07038025.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm rot} _ {w}  \mathbf a  =
 +
\frac{1}{L _ {u} L _ {v} }
 +
\left [
 +
\frac \partial {\partial  u }
 +
( a _ {v} L _ {v} ) -  
 +
\frac \partial {\partial  v }
 +
( a _ {u} L _ {u} ) \right ] ,
 +
$$
  
<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/o/o070/o070380/o07038026.png" /></td> </tr></table>
+
$$
 +
\Delta \phi  =
 +
\frac{1}{L _ {u} L _ {v} L _ {w} }
 +
\left [
 +
\frac \partial {
 +
\partial  u }
 +
\left (
 +
\frac{L _ {v} L _ {w} }{L _ {u} }
 +
 +
\frac{\partial  \phi }{\partial  u }
 +
\right ) +
 +
\frac \partial {\partial  v }
 +
\left (
 +
\frac{L _ {u} L _ {w} }{L _ {v} }
 +
 +
\frac{\partial  \phi }{\partial  v }
 +
\right ) \right . +
 +
$$
 +
 
 +
$$
 +
+ \left .
 +
 
 +
\frac \partial {\partial  w }
 +
\left (
 +
\frac{L _ {u} L _ {v} }{L _ {w} }
 +
 +
\frac{\partial  \phi }{\partial  w }
 +
\right ) \right ] .
 +
$$
  
 
The most frequently used orthogonal coordinate systems are: on a plane — [[Cartesian coordinates|Cartesian coordinates]]; [[Elliptic coordinates|elliptic coordinates]]; [[Parabolic coordinates|parabolic coordinates]]; and [[Polar coordinates|polar coordinates]]; in space — [[Cylinder coordinates|cylinder coordinates]]; [[Bicylindrical coordinates|bicylindrical coordinates]]; [[Bipolar coordinates|bipolar coordinates]]; [[Paraboloidal coordinates|paraboloidal coordinates]]; and [[Spherical coordinates|spherical coordinates]].
 
The most frequently used orthogonal coordinate systems are: on a plane — [[Cartesian coordinates|Cartesian coordinates]]; [[Elliptic coordinates|elliptic coordinates]]; [[Parabolic coordinates|parabolic coordinates]]; and [[Polar coordinates|polar coordinates]]; in space — [[Cylinder coordinates|cylinder coordinates]]; [[Bicylindrical coordinates|bicylindrical coordinates]]; [[Bipolar coordinates|bipolar coordinates]]; [[Paraboloidal coordinates|paraboloidal coordinates]]; and [[Spherical coordinates|spherical coordinates]].
Line 47: Line 186:
 
''D.D. Sokolov''
 
''D.D. Sokolov''
  
An orthogonal system of functions is a finite or countable system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038027.png" /> belonging to a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038028.png" /> and satisfying the condition
+
An orthogonal system of functions is a finite or countable system of functions $  \{ \phi _ {i} \} $
 +
belonging to a space $  L _ {2} ( X, S, \mu ) $
 +
and satisfying the condition
 +
 
 +
$$
 +
\int\limits _ { X } \phi _ {i} ( x) \overline \phi _ {j} ( x)  d \mu ( x)  = \
 +
\left \{ \begin{array}{l}
 +
0 \\
 +
\lambda _ {i}  >  0  
 +
\end{array}
 +
\  \begin{array}{l}
 +
\textrm{ if }  i \neq j, \\
 +
 
 +
\textrm{ if }  i = j .  
 +
\end{array}
 +
 
 +
\right .$$
 +
 
 +
If  $  \lambda _ {i} = 1 $
 +
for all  $  i $,
 +
then the system is orthonormal. It is supposed that the measure  $  \mu $
 +
defined on the  $  \sigma $-algebra  $  S $
 +
of subsets of the set  $  X $
 +
is countably additive, complete, and has a countable base. This definition encompasses all orthogonal systems studied in analysis. Such systems are obtained for various concrete realizations of the measure space  $  ( X, S, \mu ) $.
  
<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/o/o070/o070380/o07038029.png" /></td> </tr></table>
+
The greatest interest is in complete orthonormal systems  $  \{ \phi _ {n} \} $,
 +
which possess the property that for any function  $  f \in L _ {2} ( X, S, \mu ) $
 +
there is a unique series  $  \sum c _ {n} \phi _ {n} $
 +
which converges to  $  f $
 +
in the metric of the space  $  L _ {2} ( X, S, \mu ) $.
 +
The coefficients  $  c _ {n} $
 +
are defined by the Fourier formula:
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038030.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038031.png" />, then the system is orthonormal. It is supposed that the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038032.png" /> defined on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038033.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038034.png" /> of subsets of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038035.png" /> is countably additive, complete, and has a countable base. This definition encompasses all orthogonal systems studied in analysis. Such systems are obtained for various concrete realizations of the measure space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038036.png" />.
+
$$
 +
c _ {n}  = \int\limits _ { X } f \overline \phi _ {n}  d \mu .
 +
$$
  
The greatest interest is in complete orthonormal systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038037.png" />, which possess the property that for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038038.png" /> there is a unique series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038039.png" /> which converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038040.png" /> in the metric of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038041.png" />. The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038042.png" /> are defined by the Fourier formula:
+
These systems exist by virtue of the separability of the space  $  L _ {2} ( X, S, \mu ) $.
 +
A universal method of constructing complete orthonormal systems is given by the Gram–Schmidt [[Orthogonalization method|orthogonalization method]]. This method can be applied to any complete linearly independent sequence  $  \{ f _ {n} \} $
 +
of functions in $  L _ {2} ( X, S, \mu ) $.
  
<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/o/o070/o070380/o07038043.png" /></td> </tr></table>
+
Important examples of [[orthogonal series]] are obtained by considering the space  $  L _ {2} [ a, b] $ (in this case,  $  X = [ a, b] $,
 +
$  S $
 +
is the system of Lebesgue-measurable sets and  $  \mu $
 +
is the Lebesgue measure). Many theorems on the convergence or summability of a series  $  \sum a _ {n} \phi _ {n} $
 +
with respect to a general orthogonal system  $  \{ \phi _ {n} \} $
 +
in the space  $  L _ {2} [ a, b] $
 +
are also valid for series with respect to orthonormal systems in the space  $  L _ {2} ( X, S, \mu ) $.  
 +
Moreover, in this particular case, interesting concrete orthogonal systems have been constructed which possess some nice properties. These systems include the Haar, Rademacher, Walsh–Paley, and Franklin systems.
  
These systems exist by virtue of the separability of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038044.png" />. A universal method of constructing complete orthonormal systems is given by the Gram–Schmidt [[Orthogonalization method|orthogonalization method]]. This method can be applied to any complete linearly independent sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038045.png" /> of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038046.png" />.
+
a) The [[Haar system]] $  \{ \chi _ {n} \} _ {n=1}  ^  \infty  $:
 +
$  \chi _ {1} ( x) = 1 $,
 +
$  x \in [ 0, 1] $,
  
Important examples of [[Orthogonal series|orthogonal series]] are obtained by considering the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038047.png" /> (in this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038049.png" /> is the system of Lebesgue-measurable sets and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038050.png" /> is the Lebesgue measure). Many theorems on the convergence or summability of a series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038051.png" /> with respect to a general orthogonal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038052.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038053.png" /> are also valid for series with respect to orthonormal systems in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038054.png" />. Moreover, in this particular case, interesting concrete orthogonal systems have been constructed which possess some nice properties. These systems include the Haar, Rademacher, Walsh–Paley, and Franklin systems.
+
$$
 +
\chi _ {m} ( x) = \left \{
  
a) The Haar system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038055.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038057.png" />,
+
\begin{array}{ll}
 +
\sqrt {2  ^ {n} }  & \textrm{ if }  x \in \left ( 2k-
 +
\frac{2}{2  ^ {n+1} }
 +
, 2k-
 +
\frac{1}{2  ^ {n+1} }
 +
\right ) , \\
 +
- \sqrt {2  ^ {n} }  & \textrm{ if }  x \in \left ( 2k-
 +
\frac{1}{2  ^ {n+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/o/o070/o070380/o07038058.png" /></td> </tr></table>
+
\frac{2k}{2  ^ {n+1} }
 +
\right ) ,  \\
 +
0 & \textrm{ at  the  remaining  points  of  } \
 +
[ 0, 1],  \\
 +
\end{array}
 +
\right.
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038061.png" />. Series with respect to the Haar system are typical examples of martingales (cf. [[Martingale|Martingale]]) and thus the general theorems of martingale theory are also correct for them. Moreover, the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038062.png" /> is a [[Basis|basis]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038064.png" />, and the Fourier series with respect to the Haar system of any integrable function converges almost-everywhere.
+
where $  m = 2  ^ {n} + k $,
 +
$  1 \leq  k \leq  2  ^ {n} $,
 +
$  m = 2, 3 ,\dots $.  
 +
Series with respect to the Haar system are typical examples of martingales (cf. [[Martingale]]) and thus the general theorems of martingale theory are also correct for them. Moreover, the system $  \{ \chi _ {n} \} _ {n=1} ^  \infty  $
 +
is a [[basis]] in $  L _ {p} [ 0, 1] $,
 +
$  p \geq  1 $,  
 +
and the Fourier series with respect to the Haar system of any integrable function converges almost-everywhere.
  
b) The Rademacher system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038065.png" />:
+
b) The [[Rademacher system]]  $  \{ r _ {n} \} _ {n=0}  ^  \infty  $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038066.png" /></td> </tr></table>
+
$$
 +
r _ {n} ( x)  =   \mathop{\rm sign}  \sin  2  ^ {n+1} \pi x,\  x \in [ 0, 1],
 +
$$
  
 
is an important example of a stochastically-independent orthogonal system of functions and is used both in probability theory and in the theory of orthogonal and general series of functions.
 
is an important example of a stochastically-independent orthogonal system of functions and is used both in probability theory and in the theory of orthogonal and general series of functions.
  
c) The Walsh–Paley system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038067.png" /> is defined using the Rademacher functions:
+
c) The [[Walsh–Paley system]]  $  \{ W _ {n} \} _ {n=0}  ^  \infty  $
 +
is defined using the Rademacher functions:
  
<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/o/o070/o070380/o07038068.png" /></td> </tr></table>
+
$$
 +
W _ {0} ( x)  = 1,\ \
 +
W _ {n} ( x)  = \prod _ { k= 0} ^ { m }  [ r _ {k} ( x)] ^ {q _ {k} } ,\ \
 +
x \in [ 0, 1],
 +
$$
  
where the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038070.png" /> are defined using the binary expansion of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038071.png" />:
+
where the numbers $  m $
 +
and $  q _ {k} $
 +
are defined using the binary expansion of the number $  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/o/o070/o070380/o07038072.png" /></td> </tr></table>
+
$$
 +
= \sum _ { k= 0} ^ { m }  q _ {k} 2  ^ {k} .
 +
$$
  
d) The Franklin system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038073.png" /> is obtained by Gram–Schmidt orthogonalization of the sequence of functions
+
d) The [[Franklin system]]  $  \{ \Phi _ {n} ( x) \} _ {n=1}  ^  \infty  $
 +
is obtained by Gram–Schmidt orthogonalization of the sequence of functions
  
<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/o/o070/o070380/o07038074.png" /></td> </tr></table>
+
$$
 +
u _ {1} ( x)  = x,\ \
 +
u _ {2} ( x)  = 1- x,
 +
$$
  
<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/o/o070/o070380/o07038075.png" /></td> </tr></table>
+
$$
 +
u _ {n} ( x)  = \int\limits _ { 0 } ^ { x }  \chi _ {n-1} ( t)  dt,\  n \geq  3 ,\  x \in [ 0, 1].
 +
$$
  
It is an example of an orthogonal basis of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038076.png" /> of continuous functions.
+
It is an example of an orthogonal basis of the space $  C[ 0, 1] $
 +
of continuous functions.
  
 
In the theory of multiple orthogonal series, function systems of the form
 
In the theory of multiple orthogonal series, function systems of 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/o/o070/o070380/o07038077.png" /></td> </tr></table>
+
$$
 +
\phi _ {n _ {1}  } ( x _ {1} ), \dots, \phi _ {n _ {m}  } ( x _ {m} ) ,
 +
$$
  
<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/o/o070/o070380/o07038078.png" /></td> </tr></table>
+
$$
 +
x _ {i}  \in  [ a, b],\; n _ {i}  = 1, 2, \dots \; 1  \leq  i  \leq  m,
 +
$$
  
are examined, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038079.png" /> is an orthonormal system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038080.png" />. These systems are orthonormal on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038081.png" />-dimensional cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038082.png" />, and are complete if the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038083.png" /> is complete.
+
are examined, where $  \{ \phi _ {n} \} _ {n=1}  ^  \infty  $
 +
is an orthonormal system in $  L _ {2} [ a, b] $.  
 +
These systems are orthonormal on the $  m $-dimensional cube $  J _ {m} = [ a, b] \times \dots \times [ a, b] $,  
 +
and are complete if the system $  \{ \phi _ {n} \} $
 +
is complete.
  
 
====References====
 
====References====
Line 103: Line 331:
  
 
====Comments====
 
====Comments====
A complete system of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038084.png" /> in a Hilbert space, or, more generally, an inner product space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038085.png" />, is a set of elements such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038086.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038087.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038088.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038089.png" />.
+
A complete system of elements $  \{ \phi _  \alpha  \} $
 +
in a Hilbert space, or, more generally, an inner product space $  V $,  
 +
is a set of elements such that for any $  \phi \in V $,  
 +
if $  \langle  \phi , \phi _  \alpha  \rangle = 0 $
 +
for all $  \alpha $,  
 +
then $  \phi = 0 $.
  
Cf. also [[Complete system of functions|Complete system of functions]]. The Walsh–Paley system is a complete orthonormal system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070380/o07038090.png" />.
+
Cf. also [[Complete system of functions|Complete system of functions]]. The Walsh–Paley system is a complete orthonormal system in $  L _ {2} ( 0, 1) $.

Latest revision as of 08:58, 4 March 2022


An orthogonal system of vectors is a set $ \{ x _ \alpha \} $ of non-zero vectors of a Euclidean (Hilbert) space with a scalar product $ ( \cdot , \cdot ) $ such that $ ( x _ \alpha , x _ \beta ) = 0 $ when $ \alpha \neq \beta $. If under these conditions the norm of each vector is equal to one, then $ \{ x _ \alpha \} $ is said to be an orthonormal system. A complete orthogonal (orthonormal) system of vectors $ \{ x _ \alpha \} $ is called an orthogonal (orthonormal) basis.

M.I. Voitsekhovskii

An orthogonal coordinate system is a coordinate system in which the coordinate lines (or surfaces) intersect at right angles. Orthogonal coordinate systems exist in any Euclidean space, but, generally speaking, do not exist in an arbitrary space. In a two-dimensional smooth affine space, orthogonal systems can always be introduced at least in a sufficiently small neighbourhood of every point. It is sometimes possible to introduce orthogonal coordinate systems in the large. In an orthogonal system, the metric tensor $ g _ {ij} $ is diagonal; the diagonal components $ g _ {ij} $ are called Lamé coefficients. The Lamé coefficients of an orthogonal system in space are expressed by the formulas

$$ L _ {u} = \sqrt {\left ( \frac{\partial x }{\partial u } \right ) ^ {2} + \left ( \frac{\partial y }{\partial u } \right ) ^ {2} + \left ( \frac{\partial z }{\partial u } \right ) ^ {2} } , $$

$$ L _ {v} = \sqrt {\left ( \frac{\partial x }{\partial v } \right ) ^ {2} + \left ( \frac{\partial y }{\partial v } \right ) ^ {2} + \left ( \frac{\partial z }{\partial v } \right ) ^ {2} } , $$

$$ L _ {w} = \sqrt {\left ( \frac{\partial x }{\partial w } \right ) ^ {2} + \left ( \frac{\partial y }{\partial w } \right ) ^ {2} + \left ( \frac{\partial z }{\partial w } \right ) ^ {2} } , $$

where $ x $, $ y $ and $ z $ are Cartesian coordinates. The Lamé coefficients are also used to express the line element:

$$ ds = \sqrt {L _ {u} ^ {2} du ^ {2} + L _ {v} ^ {2} dv ^ {2} + L _ {w} ^ {2} dw ^ {2} } , $$

the element of surface area:

$$ d \sigma = \sqrt {( L _ {u} L _ {v} du dv) ^ {2} + ( L _ {u} L _ {w} du dw) ^ {2} + ( L _ {v} L _ {w} dv dw) ^ {2} } , $$

the volume element:

$$ dV = L _ {u} L _ {v} L _ {w} du dv dw, $$

and the operations of vector analysis:

$$ \mathop{\rm grad} _ {u} \phi = \frac{1}{L _ {u} } \frac{\partial \phi }{\partial u } ,\ \ \mathop{\rm grad} _ {v} \phi = \frac{1}{L _ {v} } \frac{\partial \phi }{\partial v } , $$

$$ \mathop{\rm grad} _ {w} \phi = \frac{1}{L _ {w} } \frac{\partial \phi }{\partial w } , $$

$$ \mathop{\rm div} \mathbf a = \frac{1}{L _ {u} L _ {v} L _ {w} } \left [ \frac \partial {\partial u } ( a _ {u} L _ {v} L _ {w} ) + \frac \partial {\partial v } ( a _ {v} L _ {u} L _ {w} ) \right . + $$

$$ + \left . \frac \partial {\partial w } ( a _ {w} L _ {u} L _ {v} ) \right ] ; $$

$$ \mathop{\rm rot} _ {u} \mathbf a = \frac{1}{L _ {v} L _ {w} } \left [ \frac \partial {\partial v } ( a _ {w} L _ {w} ) - \frac \partial {\partial w } ( a _ {v} L _ {v} ) \right ] , $$

$$ \mathop{\rm rot} _ {v} \mathbf a = \frac{1}{L _ {u} L _ {w} } \left [ \frac \partial {\partial w } ( a _ {u} L _ {u} ) - \frac \partial {\partial u } ( a _ {w} L _ {w} ) \right ] , $$

$$ \mathop{\rm rot} _ {w} \mathbf a = \frac{1}{L _ {u} L _ {v} } \left [ \frac \partial {\partial u } ( a _ {v} L _ {v} ) - \frac \partial {\partial v } ( a _ {u} L _ {u} ) \right ] , $$

$$ \Delta \phi = \frac{1}{L _ {u} L _ {v} L _ {w} } \left [ \frac \partial { \partial u } \left ( \frac{L _ {v} L _ {w} }{L _ {u} } \frac{\partial \phi }{\partial u } \right ) + \frac \partial {\partial v } \left ( \frac{L _ {u} L _ {w} }{L _ {v} } \frac{\partial \phi }{\partial v } \right ) \right . + $$

$$ + \left . \frac \partial {\partial w } \left ( \frac{L _ {u} L _ {v} }{L _ {w} } \frac{\partial \phi }{\partial w } \right ) \right ] . $$

The most frequently used orthogonal coordinate systems are: on a plane — Cartesian coordinates; elliptic coordinates; parabolic coordinates; and polar coordinates; in space — cylinder coordinates; bicylindrical coordinates; bipolar coordinates; paraboloidal coordinates; and spherical coordinates.

D.D. Sokolov

An orthogonal system of functions is a finite or countable system of functions $ \{ \phi _ {i} \} $ belonging to a space $ L _ {2} ( X, S, \mu ) $ and satisfying the condition

$$ \int\limits _ { X } \phi _ {i} ( x) \overline \phi _ {j} ( x) d \mu ( x) = \ \left \{ \begin{array}{l} 0 \\ \lambda _ {i} > 0 \end{array} \ \begin{array}{l} \textrm{ if } i \neq j, \\ \textrm{ if } i = j . \end{array} \right .$$

If $ \lambda _ {i} = 1 $ for all $ i $, then the system is orthonormal. It is supposed that the measure $ \mu $ defined on the $ \sigma $-algebra $ S $ of subsets of the set $ X $ is countably additive, complete, and has a countable base. This definition encompasses all orthogonal systems studied in analysis. Such systems are obtained for various concrete realizations of the measure space $ ( X, S, \mu ) $.

The greatest interest is in complete orthonormal systems $ \{ \phi _ {n} \} $, which possess the property that for any function $ f \in L _ {2} ( X, S, \mu ) $ there is a unique series $ \sum c _ {n} \phi _ {n} $ which converges to $ f $ in the metric of the space $ L _ {2} ( X, S, \mu ) $. The coefficients $ c _ {n} $ are defined by the Fourier formula:

$$ c _ {n} = \int\limits _ { X } f \overline \phi _ {n} d \mu . $$

These systems exist by virtue of the separability of the space $ L _ {2} ( X, S, \mu ) $. A universal method of constructing complete orthonormal systems is given by the Gram–Schmidt orthogonalization method. This method can be applied to any complete linearly independent sequence $ \{ f _ {n} \} $ of functions in $ L _ {2} ( X, S, \mu ) $.

Important examples of orthogonal series are obtained by considering the space $ L _ {2} [ a, b] $ (in this case, $ X = [ a, b] $, $ S $ is the system of Lebesgue-measurable sets and $ \mu $ is the Lebesgue measure). Many theorems on the convergence or summability of a series $ \sum a _ {n} \phi _ {n} $ with respect to a general orthogonal system $ \{ \phi _ {n} \} $ in the space $ L _ {2} [ a, b] $ are also valid for series with respect to orthonormal systems in the space $ L _ {2} ( X, S, \mu ) $. Moreover, in this particular case, interesting concrete orthogonal systems have been constructed which possess some nice properties. These systems include the Haar, Rademacher, Walsh–Paley, and Franklin systems.

a) The Haar system $ \{ \chi _ {n} \} _ {n=1} ^ \infty $: $ \chi _ {1} ( x) = 1 $, $ x \in [ 0, 1] $,

$$ \chi _ {m} ( x) = \left \{ \begin{array}{ll} \sqrt {2 ^ {n} } & \textrm{ if } x \in \left ( 2k- \frac{2}{2 ^ {n+1} } , 2k- \frac{1}{2 ^ {n+1} } \right ) , \\ - \sqrt {2 ^ {n} } & \textrm{ if } x \in \left ( 2k- \frac{1}{2 ^ {n+1} } ,\ \frac{2k}{2 ^ {n+1} } \right ) , \\ 0 & \textrm{ at the remaining points of } \ [ 0, 1], \\ \end{array} \right. $$

where $ m = 2 ^ {n} + k $, $ 1 \leq k \leq 2 ^ {n} $, $ m = 2, 3 ,\dots $. Series with respect to the Haar system are typical examples of martingales (cf. Martingale) and thus the general theorems of martingale theory are also correct for them. Moreover, the system $ \{ \chi _ {n} \} _ {n=1} ^ \infty $ is a basis in $ L _ {p} [ 0, 1] $, $ p \geq 1 $, and the Fourier series with respect to the Haar system of any integrable function converges almost-everywhere.

b) The Rademacher system $ \{ r _ {n} \} _ {n=0} ^ \infty $:

$$ r _ {n} ( x) = \mathop{\rm sign} \sin 2 ^ {n+1} \pi x,\ x \in [ 0, 1], $$

is an important example of a stochastically-independent orthogonal system of functions and is used both in probability theory and in the theory of orthogonal and general series of functions.

c) The Walsh–Paley system $ \{ W _ {n} \} _ {n=0} ^ \infty $ is defined using the Rademacher functions:

$$ W _ {0} ( x) = 1,\ \ W _ {n} ( x) = \prod _ { k= 0} ^ { m } [ r _ {k} ( x)] ^ {q _ {k} } ,\ \ x \in [ 0, 1], $$

where the numbers $ m $ and $ q _ {k} $ are defined using the binary expansion of the number $ n $:

$$ n = \sum _ { k= 0} ^ { m } q _ {k} 2 ^ {k} . $$

d) The Franklin system $ \{ \Phi _ {n} ( x) \} _ {n=1} ^ \infty $ is obtained by Gram–Schmidt orthogonalization of the sequence of functions

$$ u _ {1} ( x) = x,\ \ u _ {2} ( x) = 1- x, $$

$$ u _ {n} ( x) = \int\limits _ { 0 } ^ { x } \chi _ {n-1} ( t) dt,\ n \geq 3 ,\ x \in [ 0, 1]. $$

It is an example of an orthogonal basis of the space $ C[ 0, 1] $ of continuous functions.

In the theory of multiple orthogonal series, function systems of the form

$$ \phi _ {n _ {1} } ( x _ {1} ), \dots, \phi _ {n _ {m} } ( x _ {m} ) , $$

$$ x _ {i} \in [ a, b],\; n _ {i} = 1, 2, \dots \; 1 \leq i \leq m, $$

are examined, where $ \{ \phi _ {n} \} _ {n=1} ^ \infty $ is an orthonormal system in $ L _ {2} [ a, b] $. These systems are orthonormal on the $ m $-dimensional cube $ J _ {m} = [ a, b] \times \dots \times [ a, b] $, and are complete if the system $ \{ \phi _ {n} \} $ is complete.

References

[1] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
[2] B.I. Golubov, "Series with respect to the Haar system" J. Soviet Math. , 1 : 6 (1973) pp. 704–726 Itogi Nauk. Mat. Anal. 1970 (1971) pp. 109–146
[3] L.A. Balashov, A.I. Rubenshtein, "Series with respect to the Walsh system and their generalizations" J. Soviet Math. , 1 : 6 (1973) pp. 727–763 Itogi Nauk. Mat. Anal. 1970 (1971) pp. 147–202
[4] J.L. Doob, "Stochastic processes" , Chapman & Hall (1953)
[5] M. Loève, "Probability theory" , Springer (1977)
[6] A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)

A.A. Talalyan

Comments

A complete system of elements $ \{ \phi _ \alpha \} $ in a Hilbert space, or, more generally, an inner product space $ V $, is a set of elements such that for any $ \phi \in V $, if $ \langle \phi , \phi _ \alpha \rangle = 0 $ for all $ \alpha $, then $ \phi = 0 $.

Cf. also Complete system of functions. The Walsh–Paley system is a complete orthonormal system in $ L _ {2} ( 0, 1) $.

How to Cite This Entry:
Orthogonal system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_system&oldid=49349
This article was adapted from an original article by M.I. Voitsekhovskii, D.D. Sokolov, A.A. Talalyan (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article