The norms are defined on the space by
Since all practical desired frequency responses arising in digital filter design problems are bounded on the unit circle, it follows that forms a Banach space under any norm.
The weighted norms are defined by
The case gives the popular root mean square norm, and can be interpreted as the total energy of in many physical contexts.
An advantage of working in is that the norm is provided by an inner product,
As approaches infinity in Eq. (1), the error measure is dominated by the largest values of . Accordingly, it is customary to define
Suppose the norm of is finite, and let
The norms for impulse response sequences are defined in a manner exactly analogous with the frequency response norms , viz.,
The and norms are strictly concave functionals for (see below).
By Parseval's theorem, we have , i.e., the and norms are the same for .
The Frobenious norm of an matrix is defined as
Theorem. The unique rank matrix which minimizes is given by , where is a singular value decomposition of , and is formed from by setting to zero all but the largest singular values.
Proof. See Golub and Kahan .
The induced norm of a matrix is defined in terms of the norm defined for the vectors on which it operates,
The Hankel matrix corresponding to a time series is defined by , i.e.,
The Hankel norm of a filter frequency response is defined as the spectral norm of the Hankel matrix of its impulse response,
If is strictly stable, then is finite for all , and all norms defined thus far are finite. Also, the Hankel matrix is a bounded linear operator in this case.
The Hankel norm is bounded below by the norm, and bounded above by the norm ,