Normalized Tight Frames

# Normalized Tight Frames - Recent Progress in Multivariate...

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Normalized Tight Frames in Finite Dimensions Georg Zimmermann Abstract For any dimension d and for any n d , we construct normalized tight frames with n elements for C I d and for IR d . The Problem Recent Progress in Multivariate Approximation W. Haussmann, K. Jetter, and M. Reimer (eds.) International Series of Numerical Mathematics Vol. 17, pp. 249–252 c 2001 Birkh¨ auser Verlag Basel/Switzerland In one of the problem sessions, the following question was raised. Consider frames for IR 3 with n 3 elements and the property that every frame vector has Euclidean norm 1 . What is the minimal quotient of the frame bounds for each n ? We answer this question in full generality by showing that for any IR d and any n d , there exist tight frames with the desired property, so for any d , the minimal quotient of the frame bounds equals 1 for each n d . 1 Definitions and Preliminaries Definition. A frame for a Hilbert space H is a family of vectors { v k } k I in H with the property C 1 k x k 2 H X k I |h x, v k i| 2 C 2 k x k 2 H for all x H , where the frame bounds C 1 , C 2 satisfy 0 < C 1 C 2 < .

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