Symmetric approximation of frames and bases in Hilbert spaces
M. Frank, V. I. Paulsen, T. R. Tiballi
TL;DR
The paper investigates how to achieve symmetric, order-independent approximations of frames by normalized tight frames and of bases by orthonormal bases in Hilbert spaces. It centers on the frame transform $F$ and its polar decomposition $F=W|F|$, establishing precise existence and uniqueness conditions for symmetric approximations in finite and infinite settings via Hilbert–Schmidt properties of operators like $(P-|F|)$ (or $(I-|F|)$). In finite dimensions the symmetric approximation always exists and is unique, given by the normalized tight frame $\{W(e_i)\}$; in the separable infinite case, existence and uniqueness hinge on the Hilbert–Schmidt condition, with the minimizer again tied to $W(e_i)$. For arbitrary linearly independent infinite sets, symmetric orthogonalizations exist exactly under a dimension condition and have a canonical form $\nu_i=(V+W)(e_i)$, with uniqueness occurring when $\ker F=\{0\}$. These results have implications for wavelet theory and stable, canonical representations in signal processing and numerical computations.
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.