Symmetric approximation of frames and bases in Hilbert spaces

Abstract

We consider existence and uniqueness of symmetric approximation of frames by normalized tight frames and of symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces H . More precisely, we determine whether a given frame or basis possesses a normalized tight frame or orthonormal basis that is quadratically closest to it, if there exists such frames or bases at all. A crucial role is played by the Hilbert-Schmidt property of the operator (P-|F|), where F is the adjoint operator of the frame transform F*: H --> l_2 of the initial frame or basis and (1-P) is the projection onto the kernel of F. The result is useful in wavelet theory.