I would like to have an idea what is the field Frame Theory. Can anyone describe what this field is and what kind of problems will be considered in this topic. Thanks.
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Maybe if you told us where you saw it, we may have a better chance of knowing and answering. – user99680 Feb 05 '14 at 03:10
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I only want to know what is this subfield of analysis, I didn't encounter it. – AmirHosein Sadeghimanesh Feb 05 '14 at 03:24
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The basic idea of frames is as follows:
Let $H$ be a Hilbert Space. A sequence of functions/vectors $\{x_n\}$ in $H$ is a frame of $H$ if $\exists$ constants $0 < A \leq B < \infty$ s.t. $\forall$ $f \in H$, $A||f||^2 \leq \sum_n |\langle f, x_n \rangle |^2 \leq B||f||^2$.
Note that if $\{x_n\}$ is an ortho-normal basis of $H$, then $\sum_n |\langle f, x_n \rangle |^2 = ||f||^2$. Thus the ortho-normal basis is a special case of frames. Essentially frames are a "relaxed" version of bases, where you have built-in redundancy. From there things develop into some very interesting maths/applications.
As a simple example let $u_1 = (0,1)$, $u_2 = \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$, $u_3 = \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$, then it is straight-forward to show (through the definition above) that this is a frame of $\textbf{R}^2$ with bounds $A = B = \frac{3}{2}$.
daniel
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