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Difference between revisions of "Complete system of functions"

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An [[Orthonormal system|orthonormal system]] of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023900/c0239001.png" /> in some Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023900/c0239002.png" /> such that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023900/c0239003.png" /> there is no function orthogonal to all the functions in that family. A system of functions that is complete in one space may be incomplete in another. For example, the system
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An [[Orthonormal system|orthonormal system]] of functions $\{\phi(x)\}$ in some Hilbert space $H$ such that in $H$ there is no function orthogonal to all the functions in that family. A system of functions that is complete in one space may be incomplete in another. For example, the system
  
<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/c/c023/c023900/c0239004.png" /></td> </tr></table>
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$$\left\lbrace\sqrt\frac2\pi\cos nx\right\rbrace,\quad n=0,1,\ldots,$$
  
forms a complete system of functions in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023900/c0239005.png" /> but does not form a complete system in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023900/c0239006.png" />.
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forms a complete system of functions in the space $L[0,\pi]$ but does not form a complete system in the space $L[-\pi,\pi]$.

Latest revision as of 18:23, 14 August 2014

An orthonormal system of functions $\{\phi(x)\}$ in some Hilbert space $H$ such that in $H$ there is no function orthogonal to all the functions in that family. A system of functions that is complete in one space may be incomplete in another. For example, the system

$$\left\lbrace\sqrt\frac2\pi\cos nx\right\rbrace,\quad n=0,1,\ldots,$$

forms a complete system of functions in the space $L[0,\pi]$ but does not form a complete system in the space $L[-\pi,\pi]$.

How to Cite This Entry:
Complete system of functions. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_system_of_functions&oldid=12534
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article