e proceed to extend
results of the sections
(
Vanishing
moments for biorthogonal wavelets
) and
(
Vanishing moments of
wavelet
) to Sobolev spaces
,
see the chapter (
Sobolev spaces
). The
section (
Construction
of approximation spaces
) is important prerequisite.
Note
that


(Derivative vs scale)

Proof
First, we prove the result for
on
:
The method of the prove is assume that
does not hold and arrive to contradiction using the proposition
(
Vanishing moments vs
approximation 3
). If
does not hold then there exists an increasing sequence
,
such
that
Note that the scale operation does not alter the
norm,
see the formula (
Property of
scale and transport 2
), hence we only need to estimate the numerator of
to show that, in fact, LHS of
cannot blow up.
In context of the proposition
(
Vanishing moments vs
approximation 3
) we take the sequence
,
then
Note
that
Thus, for
to
approximate
the
the
maxnorm has to grow like
:
We also use the formula (
Derivative vs
scale
):
or
This estimate is in contradiction with
,
thus,
is proven.
We extend the estimate to
as follows.
Let
then
We apply
.
The
is a
constant:
.
We use the proposition (
Frame property
2
).
Next, we extend the result to
.
Let
,
then
We have proven the estimate in case of
.
To extend the result to
we note that the procedure in the section
(
Construction
of MRA and wavelets on half line or an interval
) is a finite linear
combination taken within
.
