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Zbl 0971.42023
Han, Deguang; Larson, David R.
Frames, bases and group representations.
(English)
[J] Mem. Am. Math. Soc. 697, 94 p. (2000). ISSN 0065-9266

A set of vectors $\{x_n\}$ in a Hilbert space $H$ is a frame if there exist constants $A,B>0$ such that $A\|x\|^2 \le \sum |\langle x,x_n\rangle|^2 \le B\|x\|^2$ for all $x\in H$. A special role is played by normalized tight frames, i.e., frames for which we can use $A=B=1$ in the definition. Frames play a role in signal processing and other applied areas, but they are also mathematically very interesting. In this paper an operator-theoretic approach to frames is given. It is proved that frames are inner direct summands of Riesz bases and that normalized tight frames are inner direct summands of orthonormal bases. Different concepts for disjointness of frames are defined; it is proved that if $\{x_n\}$ and $\{y_n\}$ are disjoint frames for $H$, then $\{x_n+y_n\}$ is a frame for $H$, and if $\{x_n\}$ and $\{y_n\}$ are strongly disjoint normalized tight frames for $H$, then $\{Ax_n+By_n\}$ is a normalized tight frame for all bounded operators $A,B$ on $H$ for which $AA^*+BB^*=I$. Frame vectors for unitary systems are introduced, and the special case of a Gabor type unitary system is discussed. A group representation $(\pi, {\cal G},H)$ is called a frame representation if there eixsts a vector $\eta \in H$ such that $\pi({\cal G})\eta$ is a normalized tight frame for $H$. The set of all $\eta$ such that $\pi({\cal G})\eta$ is a normalized tight frame for $H$ is characterized. As consequences of this approach to frame theory, new proofs of several known results are given. Besides the references given in the paper, the reader can find discussions of frames from a more classical point of view in the survey papers Continuous and discrete wavelet transforms'' [SIAM Review 31, No. 4, 628-666 (1989; Zbl 0683.42031)] by {\it C. E. Heil} and {\it D. F. Walnu}, The art of frame theory,'' [Taiwanese J. Math. 4, No. 2, 129-201 (2000; Zbl 0966.42022)] by {\it P. G. Casazza}, and Frames, bases, and discrete Gabor/wavelet expansions'', Bull. Am. Math. Soc. 38, No. 3, 273-291 (2001; Zbl 0982.42018)] by {\it O. Christensen}.
[Ole Christensen (Lyngby)]
MSC 2000:
*42C40 Wavelets
47N40 Appl. of operator theory in numerical analysis
47C05 Operators in topological algebras
46B28 Normed linear spaces of linear operators, etc.
46N99 Appl. of functional analysis

Keywords: frames; wavelet; super frame; super wavelet; Gabor frame; Riesz bases; normalized tight frames

Citations: Zbl 0683.42031; Zbl 0966.42022; Zbl 0982.42018

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