Difference between revisions of "Complete system of functions"
From Encyclopedia of Mathematics
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| − | An [[Orthonormal system|orthonormal system]] of functions | + | {{TEX|done}} |
| + | 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 | ||
| − | + | $$\left\lbrace\sqrt\frac2\pi\cos nx\right\rbrace,\quad n=0,1,\ldots,$$ | |
| − | forms a complete system of functions in the space | + | 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
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