Pairs of oblique duals in spaces of periodic functions
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Pairs of oblique duals in spaces of periodic functions Ole Christensen · Say Song Goh
Received: 24 February 2008 / Accepted: 7 January 2009 / Published online: 3 March 2009 © Springer Science + Business Media, LLC 2009
Abstract We construct non-tight frames in finite-dimensional spaces consisting of periodic functions. In order for these frames to be useful in practice one needs to calculate a dual frame; while the canonical dual frame might be cumbersome to work with, the setup presented here enables us to obtain explicit constructions of some particularly convenient oblique duals. We also provide explicit oblique duals belonging to prescribed spaces different from the space where we obtain the expansion. In particular this leads to oblique duals that are trigonometric polynomials. Keywords Periodic frames · Oblique dual generators · Polyphase splines · Mixed Gramians Mathematics Subject Classification (2000) 41A58 · 42A16 · 42A05 1 Introduction During the last few years, frames in finite-dimensional Hilbert spaces have become increasingly popular as well in mathematics as in engineering. The
Communicated by Qiyu Sun. O. Christensen (B) Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark e-mail: [email protected] S. S. Goh Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Republic of Singapore e-mail: [email protected]
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O. Christensen, S.S. Goh
focus has been on frames in Cn , especially the construction of tight frames; see, e.g., [1, 2]. The present paper is a contribution to the theory for frames in finitedimensional spaces, but from a different angle. We construct non-tight frames in finite-dimensional spaces consisting of periodic functions, i.e., in subspaces of L2 (0, 2π ). The fact that the frames are non-tight makes it cumbersome to find the canonical dual frame, but we show that by allowing generators outside the space where we have the frame expansion, we have considerable freedom in the choice of oblique duals. We use this freedom to provide explicit constructions of some particularly convenient oblique duals, as well as oblique duals belonging to prescribed spaces different from the space where we obtain the expansion. In the rest of this introduction we give a more technical description of our aims, and provide a short presentation of some basic facts about frames and polyphase splines. Let N ∈ N be given. Define the translation operator 2π T : L2 (0, 2π ) → L2 (0, 2π ), (T f )(·) := f · − , N and let R := {0, 1, . . . , N − 1}. Given functions φ1 , . . . , φr ∈ L2 (0, 2π ), we consider the vector space V := span T φm : m = 1, . . . , r; ∈ R ; (1.1) here T = TT · · · T with factors. The functions φ1 , . . . , φr are known as generators of V. The multiresolution and wavelet subspaces studied in periodic wavelet analysis are typical examples of vector spaces of the form (1.1). In periodic wavelet analysis, the number N takes consecutive nonnegative powers of 2, or more gene
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