Vol. 11, No. 2, 2018

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What is odd about binary Parseval frames?

Zachery J. Baker, Bernhard G. Bodmann, Micah G. Bullock, Samantha N. Branum and Jacob E. McLaney

Vol. 11 (2018), No. 2, 219–233
Abstract

This paper examines the construction and properties of binary Parseval frames. We address two questions: When does a binary Parseval frame have a complementary Parseval frame? Which binary symmetric idempotent matrices are Gram matrices of binary Parseval frames? In contrast to the case of real or complex Parseval frames, the answer to these questions is not always affirmative. The key to our understanding comes from an algorithm that constructs binary orthonormal sequences that span a given subspace, whenever possible. Special regard is given to binary frames whose Gram matrices are circulants.

Keywords
frames, Parseval frames, binary Parseval frame, binary cyclic frame, finite-dimensional vector spaces, binary numbers, orthogonal extension principle, switching equivalence, Naimark complement, Gram matrices, Gram–Schmidt orthogonalization
Mathematical Subject Classification 2010
Primary: 42C15
Secondary: 15A33
Milestones
Received: 31 August 2015
Revised: 7 March 2016
Accepted: 23 March 2017
Published: 17 September 2017

Communicated by David Royal Larson
Authors
Zachery J. Baker
Department of Mathematics
University of Houston
Houston, TX
United States
Bernhard G. Bodmann
Department of Mathematics
University of Houston
Houston, TX
United States
Micah G. Bullock
Department of Mathematics
University of Houston
Houston, TX
United States
Samantha N. Branum
Department of Mathematics
University of Houston
Houston, TX
United States
Jacob E. McLaney
Department of Mathematics
University of Houston
Houston, TX
United States