Tight frames, partial isometries, and signal reconstruction

This article gives a procedure to convert a frame which is not a tight frame into a Parseval frame for the same space, with the requirement that each element in the resulting Parseval frame can be explicitly written as a linear combination of the elements in the original frame. Several examples are considered, such as a Fourier frame on a spiral. The procedure can be applied to the construction of Parseval frames for L^2(B(0,R)), the space of square integrable functions whose domain is the ball of radius R. When a finite number of measurements are used to reconstruct a signal in L^2(B(0,R)), error estimates arising from such approximation are discussed.