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On the value of the fifth maximal projection constant. (English) Zbl 1519.41010

Summary: Let \(\lambda(m)\) denote the maximal absolute projection constant over real \(m\)-dimensional subspaces. This quantity is extremely hard to determine exactly, as testified by the fact that the only known value of \(\lambda(m)\) for \(m > 1\) is \(\lambda(2) = 4 / 3\). There is also numerical evidence indicating that \(\lambda(3) = (1 + \sqrt{5}) / 2\). In this paper, relying on a new construction of certain mutually unbiased equiangular tight frames, we show that \(\lambda(5) \geq 5(11 + 6 \sqrt{5}) / 59 \approx 2.06919\). This value coincides with the numerical estimation of \(\lambda(5)\) obtained by B. L. Chalmers, thus reinforcing the belief that this is the exact value of \(\lambda(5)\).

MSC:

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
15A42 Inequalities involving eigenvalues and eigenvectors
41A44 Best constants in approximation theory
42C15 General harmonic expansions, frames
46B20 Geometry and structure of normed linear spaces
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References:

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