TY - RPRT

T1 - Riesz Frames and Approximation of the Frame Coefficients

AU - Christensen, Ole

PY - 1996

Y1 - 1996

N2 - A frame is a familyof elements in a Hilbert space with the
propertythat every element in the Hilbert space can be written as
a (infinite)linear combination of the frame elements. Frame theory
describes howone can choose the corresponding coefficients, which
are calledframe coefficients. From the mathematical point of view
this isgratifying, but for applications it is a problem that the
calculationrequires inversion of an operator on the Hilbert
space.The projection method is introduced in order to avoid this
problem.The basic idea is to consider finite subfamiliesof the
frame and the orthogonal projection onto its span. Forfin QTR
H,P_nf has a representation as a linear combinationof
f_i,i=1,2,..,n, and the corresponding coefficients can be
calculatedusing finite dimensional methods. We find conditions
implying that thosecoefficients converge to the correct frame
coefficients as n goes to infinityin which case we have avoided
the inversion problem. It turns out, thatthe class of
''well-behaving frames'' are identical for the two problemswe
consider.

AB - A frame is a familyof elements in a Hilbert space with the
propertythat every element in the Hilbert space can be written as
a (infinite)linear combination of the frame elements. Frame theory
describes howone can choose the corresponding coefficients, which
are calledframe coefficients. From the mathematical point of view
this isgratifying, but for applications it is a problem that the
calculationrequires inversion of an operator on the Hilbert
space.The projection method is introduced in order to avoid this
problem.The basic idea is to consider finite subfamiliesof the
frame and the orthogonal projection onto its span. Forfin QTR
H,P_nf has a representation as a linear combinationof
f_i,i=1,2,..,n, and the corresponding coefficients can be
calculatedusing finite dimensional methods. We find conditions
implying that thosecoefficients converge to the correct frame
coefficients as n goes to infinityin which case we have avoided
the inversion problem. It turns out, thatthe class of
''well-behaving frames'' are identical for the two problemswe
consider.

M3 - Report

BT - Riesz Frames and Approximation of the Frame Coefficients

ER -