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Introduction to Finite normalized tight frames

Introduction to Finite normalized tight frames -...

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Introduction to “Finite normalized tight frames” Jiahao Chen and Inmi Kim December 3, 2007 A finite normalized tight frame (FNTF) in a real or complex Euclidean space K d ( K = R or C ) is a Bessel sequence { x n } N n = 1 that generalizes orthonormal bases that maintains the decomposition property y = 1 A N X n = 1 y , x n fi x n y K d (1) while giving up the basis property. 1 Properties of FNTFs A FNTF is a Bessel sequence { x n } N n = 1 in a finite dimensional Hilbert space H of dimension d that is • A tight frame (Theorem 2.1): (a) { x n } N n = 1 is a A -tight frame with frame operator S ⇐⇒ S = AI where A is a positive constant and I is the identity map. (I.e. Eq. 1 holds.) (b) { x n } N n = 1 is a normalized A -tight frame if A 1, with A = 1 if and only if { x n } N n = 1 is an orthonormal basis for H . • Normalized: each element has k x n k = 1. • Finite (Theorem 2.3): A normalized Bessel sequence in a finite dimensional Hilbert space H must be a se- quence of finite length. A FNTF is a generalization of an orthonormal basis in the sense that the frame operator S = L * L is proportional to the identity and the Grammian operator G = LL * , G ( m , n ) = ⟨ x m , x n has diagonal entries equal to one. The frame constant A is a measure of the degeneracy of the FNTF. Theorem 3.1 (value of frame constant). If { x n } N n = 1 is a FNTF for a Hilbert space of dimension d , then it has frame constant A = N d .
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