Introduction to Finite normalized tight frames

Introduction to Finite normalized tight frames -...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 ....
View Full Document

Page1 / 3

Introduction to Finite normalized tight frames -...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online