View source for Bessel system

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

<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, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015880/b0158808.png" /> are the coefficients of the expansion

<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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kaczmarz,   H. Steinhaus,   "Theorie der Orthogonalreihen" , Chelsea, reprint  (1951)</TD></TR></table>