On the Search for Tight Frames of Low Coherence
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(2021) 27:2
On the Search for Tight Frames of Low Coherence Xuemei Chen1 · Douglas P. Hardin2 · Edward B. Saff2 Received: 15 February 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We introduce a projective Riesz s-kernel for the unit sphere Sd−1 and investigate properties of N -point energy minimizing configurations for such a kernel. We show that these configurations, for s and N sufficiently large, form frames that are well-separated (have low coherence) and are nearly tight. Our results suggest an algorithm for computing well-separated tight frames which is illustrated with numerical examples. Keywords Frame · Energy · Tight · Coherence · Separation Mathematics Subject Classification Primary 42C15 · 31C20 · Secondary 42C40 · 74G65
Communicated by Pete Casazza. X. Chen: The research of this author was supported by the U. S. National Science Foundation under Grant DMS-1908880. D. P. Hardin and E. B. Saff: The research of these authors was supported, in part, by the U. S. National Science Foundation under Grant DMS-1516400.
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Xuemei Chen [email protected] Douglas P. Hardin [email protected] Edward B. Saff [email protected]
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Department of Mathematics and Statistics, University of North Carolina Wilmington, Wilmington, USA
2
Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA 0123456789().: V,-vol
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Page 2 of 27
Journal of Fourier Analysis and Applications
(2021) 27:2
1 Introduction A set of vectors X = {xi }i∈I is a frame1 for a separable Hilbert space H if there exist A, B > 0 such that for every x ∈ H , Ax2 ≤
|x, xi |2 ≤ Bx2 .
i∈I
The constant A (B, resp.) is called the lower (upper, resp.) frame bound. When A = B, X is called a tight frame, which generalizes the concept of an orthonormal basis in the sense that the recovery formula x = A1 i∈I x, xi xi holds for every x ∈ H . For the N is a frame of Hd finite dimensional space H = Hd , where H = R or C, X = {xi }i=1 N spans Hd . We shall also use X to denote the matrix whose ith if and only if {xi }i=1 column is xi and therefore we have X is tight with frame bound A ⇐⇒ X X ∗ = AId , where Id is the d × d identity matrix. N ⊂ Hd : x = 1} be the collection of all N -point Let S(d, N ) := {X = {xi }i=1 i d−1 configurations on S , the unit sphere of Hd , where · denotes the 2 norm. If we have a unit norm tight frame X ∈ S(d, N ), then it is well known that the frame bound has to be N /d since N X X ∗ = Id . (1.1) d Benedetto and Fickus show in [2] that frames that attain min
X ∈S (d,N )
|xi , x j |2
(1.2)
i = j
are precisely the unit norm tight frames. We will call the function |x, y|2 the frame potential kernel. Ehler and Okoudjou [23] generalized this result to the p-frame potential kernel |x, y| p , see also [5] for recent results on p-frames. Separation is a desirable property of a frame. It is quantified by the coherence ξ(X ) of a frame defined for X ∈ S(d, N ) by ξ(X ) := max |xi , x j |.
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