## Spherical two-distance sets and related topics in harmonic analysis

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##### Date

2014##### Author

Yu, Wei-Hsuan

##### Advisor

Barg, Alexander

Benedetto, John Joseph

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Show full item record##### Abstract

This dissertation is devoted to the study of applications of
harmonic analysis. The maximum size of spherical few-distance sets
had been studied by Delsarte at al. in the 1970s. In particular,
the maximum size of spherical two-distance sets in $\mathbb{R}^n$
had been known for $n \leq 39$ except $n=23$ by linear programming
methods in 2008. Our contribution is to extend the known results
of the maximum size of spherical two-distance sets in
$\mathbb{R}^n$ when $n=23$, $40 \leq n \leq 93$ and $n \neq 46,
78$. The maximum size of equiangular lines in $\mathbb{R}^n$ had
been known for all $n \leq 23$ except $n=14, 16, 17, 18, 19$ and
$20$ since 1973. We use the semidefinite programming method to
find the maximum size for equiangular line sets in $\mathbb{R}^n$
when $24 \leq n \leq 41$ and $n=43$.
We suggest a method of constructing spherical two-distance sets
that also form tight frames. We derive new structural properties
of the Gram matrix of a two-distance set that also forms a tight
frame for $\mathbb{R}^n$. One of the main results in this part is
a new correspondence between two-distance tight frames and certain
strongly regular graphs. This allows us to use spectral properties
of strongly regular graphs to construct two-distance tight
frames. Several new examples are obtained using this
characterization.
Bannai, Okuda, and Tagami proved that a tight spherical designs of
harmonic index 4 exists if and only if there exists an equiangular
line set with the angle $\arccos (1/(2k-1))$ in the Euclidean
space of dimension $3(2k-1)^2-4$ for each integer $k \geq 2$. We
show nonexistence of tight spherical designs of harmonic index $4$
on $S^{n-1}$ with $n\geq 3$ by a modification of the semidefinite
programming method. We also derive new relative bounds for
equiangular line sets. These new relative bounds are usually
tighter than previous relative bounds by Lemmens and Seidel.

University of Maryland, College Park, MD 20742-7011 (301)314-1328.

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