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A concept in the theory of orthogonal systems. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b0158801.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b0158802.png" /> be two complete systems of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b0158803.png" /> (i.e. measurable functions that are square-integrable on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b0158804.png" />), forming a [[Biorthogonal system|biorthogonal system]] of functions. The system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b0158805.png" /> is said to be a Bessel system if, for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b0158806.png" />, the series
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A concept in the theory of orthogonal systems. Let $\left\{{\psi_n}\right\}$ and $\left\{{g_n}\right\}$ be two complete systems of functions in $L_2 \! \left({a, b}\right) = L_2$ (i.e. measurable functions that are square-integrable on the segment $a, b$), forming a [[Biorthogonal system|biorthogonal system]] of functions. The system $\left\{{\psi_n}\right\}$ is said to be a Bessel system if, for any function $f \in L_2$, the series
 +
$$
 +
\sum_{n \, = \, 1}^{\infty} c_{n}^{2}
 +
$$
  
<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/b015880/b0158807.png" /></td> </tr></table>
+
is convergent; here, $c_n = \left({f, g_n}\right)$ are the coefficients of the expansion
 +
$$
 +
f \sim \sum_{n \, = \, 1}^{\infty} c_n \psi_n
 +
$$
  
is convergent; here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b0158808.png" /> are the coefficients of the expansion
+
of the function $f$ with respect to the system $\left\{{\psi_n}\right\}$. For a system $\left\{{\psi_n}\right\}$ to be a Bessel system it is necessary and sufficient that it be possible to define a bounded linear operator $A$ on the space $L_2$ such that the system $\left\{{\phi_n}\right\}$ defined by the equation $A \psi_n = \phi_n$ ($n = 1, 2, \dots$) is a complete orthonormal system. If the system $\left\{{\psi_n}\right\}$ is a Bessel system, there exists a constant $M$ such that for any $f \in L_2$
 +
$$
 +
\sum_{n \, = \, 1}^{\infty} \left({f, g_n}\right)^2 \leq M \left\|{f}\right\|_{L_2}^2.
 +
$$
  
<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/b015880/b0158809.png" /></td> </tr></table>
 
 
of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b01588010.png" /> with respect to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b01588011.png" />. For a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b01588012.png" /> to be a Bessel system it is necessary and sufficient that it be possible to define a bounded linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b01588013.png" /> on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b01588014.png" /> such that the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b01588015.png" /> defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b01588016.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b01588017.png" />) is a complete orthonormal system. If the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b01588018.png" /> is a Bessel system, there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b01588019.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b01588020.png" />
 
 
<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/b015880/b01588021.png" /></td> </tr></table>
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,  H. Steinhaus,  "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR></table>
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 +
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Latest revision as of 05:56, 14 January 2017

A concept in the theory of orthogonal systems. Let $\left\{{\psi_n}\right\}$ and $\left\{{g_n}\right\}$ be two complete systems of functions in $L_2 \! \left({a, b}\right) = L_2$ (i.e. measurable functions that are square-integrable on the segment $a, b$), forming a biorthogonal system of functions. The system $\left\{{\psi_n}\right\}$ is said to be a Bessel system if, for any function $f \in L_2$, the series $$ \sum_{n \, = \, 1}^{\infty} c_{n}^{2} $$

is convergent; here, $c_n = \left({f, g_n}\right)$ are the coefficients of the expansion $$ f \sim \sum_{n \, = \, 1}^{\infty} c_n \psi_n $$

of the function $f$ with respect to the system $\left\{{\psi_n}\right\}$. For a system $\left\{{\psi_n}\right\}$ to be a Bessel system it is necessary and sufficient that it be possible to define a bounded linear operator $A$ on the space $L_2$ such that the system $\left\{{\phi_n}\right\}$ defined by the equation $A \psi_n = \phi_n$ ($n = 1, 2, \dots$) is a complete orthonormal system. If the system $\left\{{\psi_n}\right\}$ is a Bessel system, there exists a constant $M$ such that for any $f \in L_2$ $$ \sum_{n \, = \, 1}^{\infty} \left({f, g_n}\right)^2 \leq M \left\|{f}\right\|_{L_2}^2. $$


References

[1] S. Kaczmarz, H. Steinhaus, "Theorie der Orthogonalreihen" , Chelsea, reprint (1951)
How to Cite This Entry:
Bessel system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bessel_system&oldid=40179
This article was adapted from an original article by P.I. Lizorkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article