Harmonic analysis on Euclidean spaces Definition

Tight frame

Definition

Let $H$ be a Hilbert space with inner product $\ip{\cdot}{\cdot}$ , and let $(f_i)_{i \in I}$ be a family of vectors of $H$ . The family is a frame if there are constants $0 < A \le B < \infty$ with $A \norm{f}^2 \le \sum_{i \in I} \abs{\ip{f}{f_i}}^2 \le B \norm{f}^2 \quad \text{for all } f \in H.$ The numbers $A$ and $B$ are frame bounds. The frame is tight if one can take $A = B$ , that is, if $\sum_{i \in I} \abs{\ip{f}{f_i}}^2 = A \norm{f}^2$ for all $f \in H$ . A tight frame with $A = B = 1$ is a Parseval frame.

Remarks

For a tight frame with bound $A$ , every $f \in H$ satisfies $f = A^{-1} \sum_{i \in I} \ip{f}{f_i} f_i$ , with convergence in $H$ . Every orthonormal basis is a Parseval frame, and the union of two orthonormal bases is a tight frame with $A = 2$ .

Example: in $\R^2$ , let $u_1 = (0, 1)$ , $u_2 = (-\tfrac{\sqrt3}{2}, -\tfrac12)$ , $u_3 = (\tfrac{\sqrt3}{2}, -\tfrac12)$ . For $f = (x, y)$ , $\sum_k \ip{f}{u_k}^2 = y^2 + \tfrac14(\sqrt3 x + y)^2 + \tfrac14(\sqrt3 x - y)^2 = \tfrac32 (x^2 + y^2)$ . So $(u_1, u_2, u_3)$ is a tight frame with bound $\tfrac32$ .

Non-example: in $\R^2$ , the family $(e_1, e_1, e_2)$ gives $\sum \ip{f}{f_i}^2 = 2x^2 + y^2$ . It is a frame with optimal bounds $A = 1$ and $B = 2$ , but it is not tight.

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Hilbert space concept
Inner product concept
Tight frame definition

Tag	`ST8B3EV6YA`
URI	`m2://ST8B3EV6YA`
Kind	definition
Writing	W0 stub
MSC	42C15
Defines	Tight frame
Uses	Hilbert space, Inner product

Tight frame

Definition

Remarks

Reading path

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