IJPAM: Volume 75, No. 4 (2012)

FRAME BOUNDS FOR SUBSPACES OF $L_2(\R)$

Yeon Hyang Kim$^1$, Matthew J. Petro$^2$,
Andrew Dugowson$^3$, Robert Fraser$^4$
$^1$Department of Mathematics
Central Michigan University
Mount Pleasant, Michigan, 48859, USA
$^2$Computer Aided Engineering
University of Wisconsin-Madison
1410, Engineering Drive, Madison, WI 53706, USA
$^3$Department of Mathematics
Pomona College
333, N. College Way, Claremont, CA 91711, USA
$^4$Department of Mathematics
Case Western Reserve University
Cleveland, Ohio, 44106, USA


Abstract. Let $H$ be a separable Hilbert space with inner product $\inpro{\cdot, \cdot}$. We say a set $\{f_k : k \in \Z\}$ is a (fundamental) frame for $H$ if there exist $A, B > 0$ such that for each $f \in H$,

\begin{displaymath}
A \norm{f}_H^2 \leq \sum_{n \in \Z}{\abs{\left\langle f, f_k \right\rangle}^2} \leq B \norm{f}_H^2.
\end{displaymath} (1)

In case $\{f_k : k \in \Z\}$ is a frame for the subspace $\overline{span}\{f_k : k \in \Z\} $, we say that $\{f_k : k \in \Z\}$ is a frame sequence.

A Weyl-Heisenberg frame sequence is a frame sequence which is generated by translated and modulated versions of $L_2$-functions.

In this paper, we characterize Weyl-Heisenberg frame sequences using infinite Hermitian matrices and obtain the optimal frame bounds in terms of the operator norms of these matrices. This work is inspired by a paper by Casazza and Christensen, where sufficient conditions for a Weyl-Heisenberg system to be a frame sequence are studied.

Received: August 24, 2010

AMS Subject Classification: 42C15

Key Words and Phrases: frame sequences, Hermitian matrix, Weyl-Heisenberg systems, modulation frames

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 75
Issue: 4