Ellipsoidal tight frames and projection decompositions of operators

By K. Dykema, Daniel Freeman, Keri Kornelson, D. Larson, Marc S. Ordower, E. Weber. 2003

73 citations recorded by the source corpus.

We prove the existence of tight frames whose elements lie on an arbitrary ellipsoidal surface within a real or complex separable Hilbert space H, and we analyze the set of attainable frame bounds. In the case where H is real and has finite dimension, we give an algorithmic proof. Our main tool in the infinite-dimensional case is a result we have proven which concerns the decomposition of a positive invertible operator into a strongly converging sum of (not necessarily mutually orthogonal) self-adjoint projections. This decomposition result implies the existence of tight frames in the ellipsoidal surface determined by the positive operator. In the real or complex finite dimensional case, this provides an alternate (but not algorithmic) proof that every such surface contains tight frames with every prescribed length at least as large as dim H. A corollary in both finite and infinite dimensions is that every positive invertible operator is the frame operator for a spherical frame.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/06a86af19d4720c077b9c0fd5760d0741ebff6ca

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In Applied Mathematics

1. V. Kaftal
2. Shuang Zhang
3. Mariano A. Ruiz
4. D. Larson
5. Jorge Antezana
6. Demetrio Stojanoff
7. Pedro G. Massey

In Mathematical Physics

1. V. Kaftal
2. Shuang Zhang
3. D. Larson

Citations in the corpus

• Operator-valued frames (2007, background): “te frames are the Gabor (Weyl-Heisenberg) frames of time-frequency analysis and the wavelet frames [7]. Some papers dealing with infinite frames which relate directly or indirectly to this article are [15, 13, 9, 2, 10, 20, 19, 21]. Work on this article began in January 1999, when the first-named auth”
• THE SCHUR-HORN THEOREM FOR OPERATORS AND FRAMES WITH PRESCRIBED NORMS AND FRAME OPERATOR. (2005, background): “Nevertheless, if k > n, c = (1, . . . , 1) ∈ Ck and S = knI ∈ Mn(C), then F (S, c) is the set of spherical tight frames of k elements in Cn. Dykema, Freeman, Korleson, Larson, Ordower and Weber [10] have shown that, in this case, F (S, c) has a rich geometrical structure, with several orbits of qual”