Frame (Hilbert space)

A frame is an object from the mathematical sub-area of functional analysis, in particular from the area of ​​Hilbert space theory. It is a special generating system of a Hilbert space .

definition

Let it be a separable Hilbert space with a scalar product and the norm induced by it. A family is called a frame of if there is such that for all the inequality ${\displaystyle H}$ ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle \|\cdot \|={\sqrt {\langle \cdot ,\cdot \rangle }}.}$ ${\displaystyle \left\{f_{j}\right\}_{j\in \mathbb {Z} }\subset H}$ ${\displaystyle H}$ ${\displaystyle 0<m\leq M}$ ${\displaystyle f\in H}$

${\displaystyle m\left|\left|f\right|\right|^{2}\leq \sum _{j\in \mathbb {Z} }\left|\langle f,f_{j}\rangle \right|^{2}\leq M\left|\left|f\right|\right|^{2}}$

applies. This means that the norm of the sequence of Fourier coefficients is directly related to the norm of the function . ${\displaystyle \ell ^{2}}$ ${\displaystyle (\langle f,f_{j}\rangle )_{j\in \mathbb {Z} }}$ ${\displaystyle f}$

If there is a choice, then the frame is referred to as tight or tight . ${\displaystyle m=M}$

If the above inequality is especially true for , the frame is also called a Parseval frame . In this case, applies to all the Parseval's identity ${\displaystyle m=M=1}$ ${\displaystyle f\in H}$

${\displaystyle \left|\left|f\right|\right|^{2}=\sum _{j\in \mathbb {Z} }\left|\langle f,f_{j}\rangle \right|^{2}}$.

example

• The vectors are a tight frame for the ${\displaystyle (1/2,1/2),\,(1/2,i/2),\,(1/2,-1/2),\,(1/2,-i/2)}$ ${\displaystyle \mathbb {C} ^{2}.}$

properties

• Every frame is a generating system of in the following (topological) sense: It applies . ${\displaystyle \left\{f_{j}\right\}_{j\in \mathbb {Z} }}$ ${\displaystyle H}$ ${\displaystyle {\overline {\operatorname {span} \{f_{j}\}}}=H}$
• Every orthonormal basis is a parseval frame.
• Parseval frames in particular behave in a similarly benign manner to orthonormal bases, since the development applies to them. In contrast to orthonormal bases, however, this decomposition is not unique, that is, there can also be other coefficients with ${\displaystyle \textstyle f=\sum _{j\in \mathbb {Z} }\langle f,f_{j}\rangle f_{j}}$ ${\displaystyle \{c_{j}\}_{j\in \mathbb {Z} }}$ ${\displaystyle \textstyle f=\sum _{j\in \mathbb {Z} }c_{j}f_{j}\,.}$

literature

• Ole Christensen: An Introduction to Frames and Riesz Bases . Birkhäuser 2002, ISBN 0-8176-4295-1 .

Web links

• Frame at PlanetMath