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The inequality
 
The inequality
  
<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/b/b015/b015850/b0158501.png" /></td> </tr></table>
+
$$
 +
\| f \|  ^ {2}  = (f, f)  \geq  \
 +
\sum _ {\alpha \in A }
  
<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/b/b015/b015850/b0158502.png" /></td> </tr></table>
+
\frac{| (f, \phi _  \alpha  ) |  ^ {2} }{( \phi _  \alpha  , \phi _  \alpha  ) }
 +
=
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b0158503.png" /> is an element of a (pre-) Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b0158504.png" /> with scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b0158505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b0158506.png" /> is an orthogonal system of non-zero elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b0158507.png" />. The right-hand side of Bessel's inequality never contains more than a countable set of non-zero components, whatever the cardinality of the index set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b0158508.png" />. Bessel's inequality follows from the Bessel identity
+
$$
 +
= \
 +
\sum _ {\alpha \in A } \left | \left ( f,
 +
 +
\frac{\phi _  \alpha  }{\| \phi _  \alpha  \| }
 +
\right ) \right |  ^ {2} ,
 +
$$
  
<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/b/b015/b015850/b0158509.png" /></td> </tr></table>
+
where  $  f $
 +
is an element of a (pre-) Hilbert space  $  H $
 +
with scalar product  $  (f, \phi ) $
 +
and  $  \{ {\phi _  \alpha  } : {\alpha \in A } \} $
 +
is an orthogonal system of non-zero elements of  $  H $.  
 +
The right-hand side of Bessel's inequality never contains more than a countable set of non-zero components, whatever the cardinality of the index set  $  A $.  
 +
Bessel's inequality follows from the Bessel identity
  
which is valid for any finite system of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585010.png" />. In this formula the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585011.png" /> are the Fourier coefficients of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585012.png" /> with respect to the orthogonal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585013.png" />, i.e.
+
$$
 +
\left \|
 +
f - \sum _ {i = 1 } ^ { n }
 +
x ^ {\alpha _ {i} }
 +
\phi _ {\alpha _ {i}  } \
 +
\right \|  ^ {2}  \equiv \
 +
| f |  ^ {2} -
 +
\sum _ {i = 1 } ^ { n }
 +
\lambda _ {\alpha _ {i}  }
 +
| x ^ {\alpha _ {i} } |  ^ {2} ,
 +
$$
  
<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/b/b015/b015850/b01585014.png" /></td> </tr></table>
+
which is valid for any finite system of elements  $  \{ {\phi _ {\alpha _ {i}  } } : {i = 1 \dots n } \} $.  
 +
In this formula the  $  x ^ {\alpha _ {i} } $
 +
are the Fourier coefficients of the vector  $  f $
 +
with respect to the orthogonal system  $  \{ \phi _ {\alpha _ {1}  } \dots \phi _ {\alpha _ {n}  } \} $,
 +
i.e.
  
The geometric meaning of Bessel's inequality is that the orthogonal projection of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585015.png" /> on the linear span of the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585017.png" />, has a norm which does not exceed the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585018.png" /> (i.e. the hypothenuse in a right-angled triangle is not shorter than one of the other sides). For a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585019.png" /> to belong to the closed linear span of the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585021.png" />, it is necessary and sufficient that Bessel's inequality becomes an equality. If this occurs for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585022.png" />, one says that the [[Parseval equality|Parseval equality]] holds for the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585024.png" />.
+
$$
 +
x ^ {\alpha _ {i} }  = \
 +
\frac{1} {\lambda  _ {\alpha _ {i}  } }
 +
(f, \phi _ {\alpha _ {i}  } ),\ \
 +
\lambda _ {\alpha _ {i}  }  = \
 +
( \phi _ {\alpha _ {i}  } , \phi _ {\alpha _ {i}  } ).
 +
$$
  
For a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585025.png" /> of linearly independent (not necessarily orthogonal) elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585026.png" /> Bessel's identity and Bessel's inequality assume the form
+
The geometric meaning of Bessel's inequality is that the orthogonal projection of an element  $  f $
 +
on the [[linear span]] of the elements $  \phi _  \alpha  $,
 +
$  \alpha \in A $,
 +
has a norm which does not exceed the norm of $  f $(
 +
i.e. the hypothenuse in a right-angled triangle is not shorter than one of the other sides). For a vector  $  f $
 +
to belong to the closed linear span of the vectors  $  \phi _  \alpha  $,
 +
$  \alpha \in A $,
 +
it is necessary and sufficient that Bessel's inequality becomes an equality. If this occurs for any  $  f \in H $,
 +
one says that the [[Parseval equality|Parseval equality]] holds for the system  $  \{ {\phi _  \alpha  } : {\alpha \in A } \} $
 +
in  $  H $.
  
<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/b/b015/b015850/b01585027.png" /></td> </tr></table>
+
For a system  $  \{ {\phi _  \alpha  } : {\alpha = 1, 2 , . . . } \} $
 +
of linearly independent (not necessarily orthogonal) elements of  $  H $
 +
Bessel's identity and Bessel's inequality assume 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/b/b015/b015850/b01585028.png" /></td> </tr></table>
+
$$
 +
\left \| f -
 +
\sum _ {\alpha , \beta = 1 } ^ { n }
 +
b _ {n} ^ {\alpha \beta }
 +
(f, \phi _  \beta  )
 +
\phi _  \alpha  \right \|  ^ {2\ } \equiv
 +
$$
  
<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/b/b015/b015850/b01585029.png" /></td> </tr></table>
+
$$
 +
\equiv \
 +
\| f \|  ^ {2} -
 +
\sum _ {\alpha , \beta = 1 } ^ { n }  b _ {n} ^ {\alpha
 +
\beta } (f, \phi _  \alpha  ) (f, \phi _  \beta  ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585030.png" /> are the elements of the matrix inverse to the Gram matrix (cf. [[Gram determinant|Gram determinant]]) of the first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585031.png" /> vectors of the initial system.
+
$$
 +
\| f \|  ^ {2}  \geq  \sum _ {\alpha , \beta = 1 } ^ { n }  b _ {n} ^ {\alpha \beta } (f, \phi _  \alpha  ) (f, \phi _  \beta  ),
 +
$$
 +
 
 +
where b _ {n} ^ {\alpha \beta } $
 +
are the elements of the matrix inverse to the Gram matrix (cf. [[Gram determinant|Gram determinant]]) of the first $  n $
 +
vectors of the initial system.
  
 
The inequality was derived by F.W. Bessel in 1828 for the trigonometric system.
 
The inequality was derived by F.W. Bessel in 1828 for the trigonometric system.
Line 29: Line 105:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.D. Kudryavtsev,  "Mathematical analysis" , '''2''' , Moscow  (1973)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.D. Kudryavtsev,  "Mathematical analysis" , '''2''' , Moscow  (1973)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Usually, the orthogonal system of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585032.png" /> is orthonormalized, i.e. one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015850/b01585033.png" />. Bessel's inequality then takes the form
+
Usually, the orthogonal system of elements $  \{ \phi _  \alpha  \} $
 +
is orthonormalized, i.e. one sets $  \psi _  \alpha  = \phi _  \alpha  / \| \phi _  \alpha  \| $.  
 +
Bessel's inequality then takes 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/b/b015/b015850/b01585034.png" /></td> </tr></table>
+
$$
 +
\sum _ {\alpha \in A }
 +
| (f, \psi _  \alpha  ) |  \leq  \
 +
\| f \|  ^ {2} ,
 +
$$
  
 
which is easier to remember. In this form it is used in approximation theory, Fourier analysis, the theory of orthogonal polynomials, etc.
 
which is easier to remember. In this form it is used in approximation theory, Fourier analysis, the theory of orthogonal polynomials, etc.

Latest revision as of 04:37, 9 May 2022


The inequality

$$ \| f \| ^ {2} = (f, f) \geq \ \sum _ {\alpha \in A } \frac{| (f, \phi _ \alpha ) | ^ {2} }{( \phi _ \alpha , \phi _ \alpha ) } = $$

$$ = \ \sum _ {\alpha \in A } \left | \left ( f, \frac{\phi _ \alpha }{\| \phi _ \alpha \| } \right ) \right | ^ {2} , $$

where $ f $ is an element of a (pre-) Hilbert space $ H $ with scalar product $ (f, \phi ) $ and $ \{ {\phi _ \alpha } : {\alpha \in A } \} $ is an orthogonal system of non-zero elements of $ H $. The right-hand side of Bessel's inequality never contains more than a countable set of non-zero components, whatever the cardinality of the index set $ A $. Bessel's inequality follows from the Bessel identity

$$ \left \| f - \sum _ {i = 1 } ^ { n } x ^ {\alpha _ {i} } \phi _ {\alpha _ {i} } \ \right \| ^ {2} \equiv \ | f | ^ {2} - \sum _ {i = 1 } ^ { n } \lambda _ {\alpha _ {i} } | x ^ {\alpha _ {i} } | ^ {2} , $$

which is valid for any finite system of elements $ \{ {\phi _ {\alpha _ {i} } } : {i = 1 \dots n } \} $. In this formula the $ x ^ {\alpha _ {i} } $ are the Fourier coefficients of the vector $ f $ with respect to the orthogonal system $ \{ \phi _ {\alpha _ {1} } \dots \phi _ {\alpha _ {n} } \} $, i.e.

$$ x ^ {\alpha _ {i} } = \ \frac{1} {\lambda _ {\alpha _ {i} } } (f, \phi _ {\alpha _ {i} } ),\ \ \lambda _ {\alpha _ {i} } = \ ( \phi _ {\alpha _ {i} } , \phi _ {\alpha _ {i} } ). $$

The geometric meaning of Bessel's inequality is that the orthogonal projection of an element $ f $ on the linear span of the elements $ \phi _ \alpha $, $ \alpha \in A $, has a norm which does not exceed the norm of $ f $( i.e. the hypothenuse in a right-angled triangle is not shorter than one of the other sides). For a vector $ f $ to belong to the closed linear span of the vectors $ \phi _ \alpha $, $ \alpha \in A $, it is necessary and sufficient that Bessel's inequality becomes an equality. If this occurs for any $ f \in H $, one says that the Parseval equality holds for the system $ \{ {\phi _ \alpha } : {\alpha \in A } \} $ in $ H $.

For a system $ \{ {\phi _ \alpha } : {\alpha = 1, 2 , . . . } \} $ of linearly independent (not necessarily orthogonal) elements of $ H $ Bessel's identity and Bessel's inequality assume the form

$$ \left \| f - \sum _ {\alpha , \beta = 1 } ^ { n } b _ {n} ^ {\alpha \beta } (f, \phi _ \beta ) \phi _ \alpha \right \| ^ {2\ } \equiv $$

$$ \equiv \ \| f \| ^ {2} - \sum _ {\alpha , \beta = 1 } ^ { n } b _ {n} ^ {\alpha \beta } (f, \phi _ \alpha ) (f, \phi _ \beta ), $$

$$ \| f \| ^ {2} \geq \sum _ {\alpha , \beta = 1 } ^ { n } b _ {n} ^ {\alpha \beta } (f, \phi _ \alpha ) (f, \phi _ \beta ), $$

where $ b _ {n} ^ {\alpha \beta } $ are the elements of the matrix inverse to the Gram matrix (cf. Gram determinant) of the first $ n $ vectors of the initial system.

The inequality was derived by F.W. Bessel in 1828 for the trigonometric system.

References

[1] L.D. Kudryavtsev, "Mathematical analysis" , 2 , Moscow (1973) (In Russian)

Comments

Usually, the orthogonal system of elements $ \{ \phi _ \alpha \} $ is orthonormalized, i.e. one sets $ \psi _ \alpha = \phi _ \alpha / \| \phi _ \alpha \| $. Bessel's inequality then takes the form

$$ \sum _ {\alpha \in A } | (f, \psi _ \alpha ) | \leq \ \| f \| ^ {2} , $$

which is easier to remember. In this form it is used in approximation theory, Fourier analysis, the theory of orthogonal polynomials, etc.

References

[a1] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
[a2] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982) pp. 203ff
[a3] P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126
[a4] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)
How to Cite This Entry:
Bessel inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bessel_inequality&oldid=15247
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article