A recursive construction of a class of finite normalized tight frames

Abstract Finite normalized tight frames are interesting because they provide decompositions in applications and some physical interpretations. In this article, we give a recursive method for constructing them. c (2014) Wavelets and Linear Algebr

Modular frames for Hilbert C*-modules and symmetric approximation of frames

We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related modular operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modulesover unital C*-algebras that admit an orthonormal Riesz basis. Interrelations and applications to classical linear frame theory are indicated. As an application we describe the nature of families of operators {S_i} such that SUM_i S*_iS_i=id_H, where H is a Hilbert space. Resorting to frames in Hilbert spaces we discuss some measures for pairs of frames to be close to one another. Most of the measures are expressed in terms of norm-distances of different kinds of frame operators. In particular, the existence and uniqueness of the closest (normalized) tight frame to a given frame is investigated. For Riesz bases with certain restrictions the set of closetst tight frames often contains a multiple of its symmetric orthogonalization (i.e. L\"owdin orthogonalization).Comment: SPIE's Annual Meeting, Session 4119: Wavelets in Signal and Image Processing; San Diego, CA, U.S.A., July 30 - August 4, 2000. to appear in: Proceedings of SPIE v. 4119(2000), 12 p

Characterization and computation of canonical tight windows for Gabor frames

Let $(g_{nm})_{n,m\in Z}$ be a Gabor frame for $L_2(R)$ for given window $g$. We show that the window $h^0=S^{-1/2} g$ that generates the canonically associated tight Gabor frame minimizes $\|g-h\|$ among all windows $h$ generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical $h^0$ is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames. Finally, a Newton-type method for a fast numerical calculation of \ho is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples