摘要
A frame is a fmaily {f_i}_(i=1)~∞ of elements in a Hilbert space with the property that every element in can be written as a (infinite) linear combination of the frame elements. Frame theory describes how one can choose the corresponding coefficients, which are called frame coef- ficients. From the mathematical point of view this is gratifying, but for applications it is a problem that the calculation requires inversion of an operator on The projection method is introduced in order to avoid this problem. The basic idea is to con- sider finite subfamilies {f_i}_(i=1)~n of the frame and the orthogonal projection P_n onto its span. For f∈P_n f has a representation as a linear combination of f_i,i=1,2,…,n and the correspond- ing coefficients, can be calculated using finite dimensional methods. We find conditions implying that those coefficients converge to the correct frame coefficients as n→∞, in which case we have avoided the inversion problem. In the same spirit we approximate the solution to a moment prob- lem. It turns out, that the class of 'well-behaving frames' are identical for the two problems we consider.
A frame is a fmaily {f_i}_(i=1)~∞ of elements in a Hilbert space with the property that every element in can be written as a (infinite) linear combination of the frame elements. Frame theory describes how one can choose the corresponding coefficients, which are called frame coef- ficients. From the mathematical point of view this is gratifying, but for applications it is a problem that the calculation requires inversion of an operator on The projection method is introduced in order to avoid this problem. The basic idea is to con- sider finite subfamilies {f_i}_(i=1)~n of the frame and the orthogonal projection P_n onto its span. For f∈P_n f has a representation as a linear combination of f_i,i=1,2,…,n and the correspond- ing coefficients, can be calculated using finite dimensional methods. We find conditions implying that those coefficients converge to the correct frame coefficients as n→∞, in which case we have avoided the inversion problem. In the same spirit we approximate the solution to a moment prob- lem. It turns out, that the class of 'well-behaving frames' are identical for the two problems we consider.
基金
The first named author is partially supported by NSF DMS 9201357
Danish NSRC grant 9401958
Missouri Research Board grant C-3-41743
a Missouri Research Council Summer Fellowship
