Symmetric approximation of frames and bases in Hilbert spaces

Abstract

Abstract. We introduce the symmetric approximation of frames by normalized tight frames extending the concept of the symmetric orthogonalization of bases by orthonormal bases in Hilbert spaces. We prove existence and uniqueness results for the symmetric approximation of frames by normalized tight frames. Even in the case of the symmetric orthogonalization of bases, our techniques and results are new. A crucial role is played by whether or not a certain operator related to the initial frame or basis is Hilbert-Schmidt.

On a conjecture by Mbekhta about best approximation by polar factors

“…These facts are essential to obtain information about the minimizers. For instance, this implication of the conjecture holds true when the operator norm is replaced by the Hilbert-Schmidt norm for arbitrary Hilbert spaces [6,8], or by any strictly convex unitarily invariant norm for finite dimensional Hilbert spaces [2]. In these works, best approximation by partial isometries with the Hilbert-Schmidt norm has deserved special attention due its importance for frame theory.…”

Tight frames, partial isometries, and signal reconstruction

“…In [9], Frank, Paulsen, and Tiballi obtain a Parseval frame from a given frame that spans the same subspace as the original frame and is closest to it in some sense, which they call symmetric approximation. The approach in [9] is to use the polar decomposition of the synthesis operator of the original frame. This idea inspires the method developed in the present work to obtain Parseval frames for the spiral sampling case.…”

“…If for two frames {f i } i∈N and {g i } i∈N of two Hilbert subspaces K and L of H, respectively, there exists an invertible bounded linear operator T : K → L such that T (f i ) = g i for every index i, then these two frames are said to be weakly similar [9]. A Parseval frame {ν i } n i=1 in a finite dimensional Hilbert subspace L ⊆ H is said to be a symmetric approximation of a finite frame {f i } n i=1 in a Hilbert subspace K ⊆ H if the frames {f i } n i=1 and {ν i } n i=1 are weakly similar and the inequality…”

“…is valid for all Parseval frames {µ i } n i=1 in Hilbert subspaces of H that are weakly similar to {f i } n i=1 [9]. If K = L, the frames are called similar.…”

Compactly supported wavelets and representations of the Cuntz relations II

“…. , ψ (N −1) in L 2 (R) such that the functions in (5) have the basis property, either in the strict sense, or in the sense of frames. In that case, the system…”