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Frames, bases and group representations. (English) Zbl 0971.42023
A set of vectors $\left\{{x}_{n}\right\}$ in a Hilbert space $H$ is a frame if there exist constants $A,B>0$ such that ${A\parallel x\parallel }^{2}\le \sum |〈x,{x}_{n}〉{|}^{2}\le B{\parallel x\parallel }^{2}$ 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 $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ are disjoint frames for $H$, then $\left\{{x}_{n}+{y}_{n}\right\}$ is a frame for $H$, and if $\left\{{x}_{n}\right\}$ and $\left\{{y}_{n}\right\}$ are strongly disjoint normalized tight frames for $H$, then $\left\{A{x}_{n}+B{y}_{n}\right\}$ is a normalized tight frame for all bounded operators $A,B$ on $H$ for which $A{A}^{*}+B{B}^{*}=I$. Frame vectors for unitary systems are introduced, and the special case of a Gabor type unitary system is discussed. A group representation $\left(\pi ,𝒢,H\right)$ is called a frame representation if there eixsts a vector $\eta \in H$ such that $\pi \left(𝒢\right)\eta$ is a normalized tight frame for $H$. The set of all $\eta$ such that $\pi \left(𝒢\right)\eta$ 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 C. E. Heil and D. F. Walnu, “The art of frame theory,” [Taiwanese J. Math. 4, No. 2, 129-201 (2000; Zbl 0966.42022)] by 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 O. Christensen.

##### MSC:
 42C40 Wavelets and other special systems 47N40 Applications of operator theory in numerical analysis 47C05 Operators in topological algebras 46B28 Spaces of operators; tensor products; approximation properties 46N99 Miscellaneous applications functional analysis