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Every frame is a sum of three (but not two) orthonormal bases – and other frame representations. (English) Zbl 0935.46022

Let \(H\) be a Hilbert space. A sequence \((x_n)\) in \(H\) is called a frame for \(H\) if there are constants \(0<A\leq B\) such that \( A\|x\|^2 \leq \sum_n |\langle x, x_n \rangle |^2 \leq B \|x\|^2, \) for all \(x\in H \). A frame is called tight if \(A=B\). In this paper the author shows that every frame for a Hilbert space \(H\) can be written as a (multiple of a) sum of three orthonormal bases for \(H\). It is also obtained that a frame can be represented as a linear combination of two orthonormal bases if and only if it is a Riesz basis. Furthermore, he shows that every frame can be written as a (multiple of a) sum of two tight frames with frame bounds one or a sum of an orthonormal basis and a Riesz basis.

MSC:

46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47B65 Positive linear operators and order-bounded operators
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References:

[1] Berberian, S.K. (1961).Introduction to Hilbert Space, Oxford University Press. · Zbl 0121.09302
[2] Conway, J.B. (1985).A Course in Functional Analysis, Springer Verlag, New York. · Zbl 0558.46001
[3] Halmos, P. (1967).A Hilbert Space Problem Book, Van Norstrand. · Zbl 0144.38704
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