TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 360, Number 6, June 2008, Pages 3307–3326 S 0002-9947(08)04435-8 Article electronically published on January 30, 2008 FRAME REPRESENTATIONS AND PARSEVAL DUALS WITH APPLICATIONS TO GABOR FRAMES DEGUANG HAN Abstract. Let {x n } be a frame for a Hilbert space H. We investigate the conditions under which there exists a dual frame for {x n } which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether {x n } can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is obtained in terms of the frame excess. For a frame {π(g)ξ : g ∈ G} induced by a projective unitary representation π of a group G, it is possible that {π(g)ξ : g ∈ G} can have a Parseval dual, but does not have a Parseval dual of the same type. The primary aim of this paper is to present a complete characterization for all the projective unitary representations π such that every frame {π(g)ξ : g ∈ G} (with a necessary lower frame bound condition) has a Parseval dual of the same type. As an application of this characterization together with a result about lattice tiling, we prove that every Gabor frame G(g, L, K) (again with the same necessary lower frame bound condition) has a Parseval dual of the same type if and only if the volume of the fundamental domain of L×K is less than or equal to 1 2 . 1. Introduction A frame for a Hilbert space H is a sequence of vectors {x n }⊂ H for which there exist constants 0 <A ≤ B< ∞ such that, for every x ∈ H, (1.1) A‖x‖ 2 ≤  |〈 x,x n 〉| 2 ≤ B‖x‖ 2 . The optimal constants (maximal for A and minimal for B) are known respectively as the upper and lower frame bounds. A frame is called a tight frame if A = B, and is called a Parseval frame if A = B = 1. If a sequence {x n } satisfies the upper bound condition in (1.1), then {x n } is also called a Bessel sequence. Frames are generalizations of Riesz bases and were first introduced by Duffin and Schaeffer [9] to deal with some difficult problems in nonharmonic Fourier analysis. The study of frame theory has drawn a lot of attention in recent years, partially because of its applications in signal processing as well as its close connections with other mathematical fields such as wavelet theory, time-frequency analysis, operator and operator algebra theory (cf. [8, 14, 15, 12, 13, 16, 17, 18, 20, 21, 27, 25, 29]). Received by the editors February 22, 2005 and, in revised form, October 3, 2006. 2000 Mathematics Subject Classification. Primary 42C15, 46C05, 47B10. Key words and phrases. Frames, Parseval duals, frame representations, Gabor frames, lattice tiling. c 2008 American Mathematical Society Reverts to public domain 28 years from publication 3307