Approximation of the Frame Coefficients using Finite Dimensional Methods

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.