verywhere in this
section we assume that the following condition holds. See the proposition
(
Existence of
smooth compactly supported wavelets
) for context.
Therefore, according to the proposition
(
Reproduction of polynomials
4
),
where the
is the collection of all polynomials with degree up to and including
.
We introduce the
notations
so
that
Condition
(Sufficiently fine scale) We assume that
the parameter
is sufficiently large so
that
For a polynomial
we introduce the
notation
In
particular,
and, according to the proposition
(
Reproduction of polynomials
4
),
We would like to find an
orthogonal
basis for the space
.
The functions
are already
orthogonal.
Before we perform GramSchmidt orthogonalization of
and
we need to establish that both collections are linearly independent.
Proposition
(Restricted linear independence)
The collection
is linearly independent on
.
Proof
The proof may be found in
[Lemarie]
or
[Meyer]
. For a particular
this may be verified directly.
Proposition
(Maximal dimension 1) The collection
is linearly independent. The collection
is linearly independent.
We proceed with orthogonalization. We seek the functions
such
that
We substitute the first relationship into the
second:
The
is a square symmetric positivedefinite matrix. Hence, there exists a Choleski
decomposition:
and the
choice
is sufficient to produce
orthogonal
collection
.
The collection
is
orthogonal
to
because it is a linear combination of
.
Proposition
(Resolution
structure for adapted scale functions) The spaces
have the
structure
