IJPAM: Volume 63, No. 2 (2010)

EXISTENCE AND CONSTRUCTION OF
FINITE FRAMES WITH A GIVEN FRAME OPERATOR

Peter G. Casazza$^1$, Manuel T. Leon$^2$
$^{1,2}$Department of Mathematics
University of Missouri
Columbia, MO 65211-4100, USA
$^1$e-mail: [email protected]


Abstract.Let $ S $ be a positive self-adjoint invertible operator on an $N$-dimensional Hilbert space $ H_N$ and let $ M \ge N $. We give necessary and sufficient conditions on real sequences $ a_1 \ge a_2 \ge \cdots \ge a_M \ge 0 $ so that there is a frame $ \{ \varphi_n \}_{n=1}^{M} $ for $ H_N$ with frame operator $ S $ and $ \norm{\varphi_n}=a_n$, for all $n=1,2,\dots M$. As a consequence, given any frame operator $ S $ as above, there is a set of equal norm vectors in $H_{N}$ which have precisely $ S $ as their frame operator.

Received: June 9, 2010

AMS Subject Classification: 47A05, 47A10

Key Words and Phrases: finite tight frame, orthogonal matrix

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2010
Volume: 63
Issue: 2