### Abstract

A frame is a family $\{f_i \}_{i=1}^{\infty}$ of elements in
aHilbert space $\cal H $with the property that every element in
$\cal H $ can be written as a(infinite) linear combination of the
frame elements. Frame theorydescribes how one can choose the
corresponding coefficients, which arecalled frame coefficients.
From the mathematical point of view this is gratifying, but for
applications it is a problem that the calculationrequires
inversion of an operator on $\cal H $. \The projection method is
introduced in order to avoid thisproblem. The basic idea is
toconsider finite subfamilies $\{f_i \}_{i=1}^{n}$ of the frame
and theorthogonal projection $P_n$ onto its span. For $f \in \h
,P_nf$ has a representation as a linear combination of $f_i ,
i=1,2,..n,$and the corresponding coefficients can be calculated
using finite dimensionalmethods. We find conditions implying that
those coefficients convergeto the correct frame coefficients as $n
\rightarrow \infty$, in which casewe have avoided the inversion
problem. In the same spirit we approximatethe solution to a moment
problem. It turns out, that the class of``well-behaving frames''
are identical for the two problems we consider.

Original language | English |
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Journal | J. Elec. Imaging |

Volume | 6 (1997) |

Pages (from-to) | 479-483 |

Publication status | Published - 1997 |

## Cite this

Christensen, O., & Casazza, P. (1997). Approximation of the Frame Coefficients using Finite
Dimensional Methods.

*J. Elec. Imaging*,*6 (1997)*, 479-483.