Normalized tight frames in finite dimensions. (English) Zbl 0990.42019
Haussmann, Werner (ed.) et al., Recent progress in multivariate approximation. Proceedings of the 4th international conference, Witten-Bommerholz, Germany, September 24-29, 2000. Basel: Birkhäuser. ISNM, Int. Ser. Numer. Math. 137, 249-252 (2001).
A frame in a Hilbert space \(H\) is a collection of vectors \(\{v_k\}\) such that for any vector \(x\in H\),
\[
C_1\|x\|^2 \leq \sum_k |\langle x, v_k \rangle|^2 \leq C_2 \|x\|^2.
\]
\(0<C_1 \leq C_2<\infty\) are the frame bounds. If \(C_1 = C_2\) we have a tight frame. And we say that a frame is normalized if for each \(k\), \(\|v_k\|=1\).
The author considers finite dimensional Hilbert spaces, i.e., \({\mathbb R}^d\) and \({\mathbb C}^d\), and in each case, given \(n \geq d\), presents an example of a normalized tight frame consisting of \(n\) elements.
For the entire collection see [Zbl 0972.00049].
The author considers finite dimensional Hilbert spaces, i.e., \({\mathbb R}^d\) and \({\mathbb C}^d\), and in each case, given \(n \geq d\), presents an example of a normalized tight frame consisting of \(n\) elements.
For the entire collection see [Zbl 0972.00049].
Reviewer: Wojciech Czaja (College Park)
MSC:
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
