Frames, bases and group representations

Geometric aspects of frame representations of abelian groups

“…Finally, by the same computation, the projections P A and P B commute, whence by Corollary 2.15 in [18], the samples commute.…”

“…If a group representation has a frame vector as above, we say that the representation is a frame representation. A frame representation of a group must be unitarily equivalent to a subrepresentation of the (left) regular representation [18,Theorem 3.11] and [25]. This, in turn, implies that the measure µ is equivalent to the restriction of Haar measure λ to the set F 1 .…”

“…providedx j andỹ j are the standard duals of x j and y j , respectively. Such frames are called strongly disjoint (see [18]) and this procedure is what engineers call multiplexing.…”

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“…Hilbert 空间(向量)框架首先是由 R. Duffin 和 A. Schaeffer 为研究非调和 Fourier 级数及其一些重要的应用而 引入的 [3] ；近 20 年来，它在理论研究和应用方面都得到迅速发展，文献 [4]对其作了比较详细的介绍。…”

Analysis of Fractals, Image Compression, Entropy Encoding, Karhunen-Loève Transforms

“…Investigation of this relation would have theoretical and practical potential and thus is becoming a subject of growing interest both in and outside mathematics. It has been recently shown (by Palle Jorgensen [Jor06], Ola Bratteli [BJ02], David Larson, X. Dai [DL98], [DLS98], [HL00], [LSST06], [LTW06], [Lar07] and others) that a unifying approach to wavelets, dynamical systems, iterated function systems, self-similarity and fractals may be based on the systematic use of operator analysis and representation theory. See Figure 1 below.…”