Frame of a vector space

A frame is an object from the mathematical branch of functional analysis, in particular in the field of Hilbert space theory. There is a special system of generators of a Hilbert space.

Definition

It should be a separable Hilbert space with inner product and induced norm of which is called A family of frame, if there is such that for all the inequality

Applies. This means that the norm of the result of the Fourier coefficients is directly related to the norm of the function.

Can it be selected, then referred to the frame as tight or tight.

If the above inequality specifically met, it is called the frame also Parsevalframe. In this case, for all the Parseval equation

Example

• The vectors are a tight frame for the

Properties

• Each frame is a system of generators of the following ( topological ) sense: It is.
• Each orthonormal basis is a Parsevalframe.
• In particular Parsevalframes behave similarly benign as orthonormal bases, as shall the development for this. However, in contrast to orthonormal bases, this decomposition is not unique, ie, there may be other coefficients indicate with