The * norms* are defined on the space by

norms are technically pseudo-norms; if functions in are replaced by equivalence classes containing all functions equal almost everywhere, then a norm is obtained.

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

where is real, positive, and integrable. Typically, . If for a set of nonzero measure, then a pseudo-norm results.

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

and this is often called the

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 [3].

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 [1],

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