  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.
