(Stability of wavelet splitting
1) Assume the condition (
Sparse tensor
product setup
). Then for any
,
and a decomposition
we
have
for
.
The first inequality holds for
.

Remark

The above formulation is suboptimal. The result actually takes place for
.
The original sources contain a generic result. However, such sources are out
of print and out of stock in online book stores. The below proof is a quick
fix for the present special case.

Proof

We
estimate
and apply the proposition
(
Bernstein inequality for
wavelets
), recalling
Therefore, we obtain the first part of inequality
.
To obtain the second part of
we apply the proposition
(
Jackson inequality for wavelets
2
):
We would like to
conclude
To prove such inequality we assume the alternative and find a contradiction.
The contrary statement is the following: there exists a sequence
s.t.
and
Note that
where the
has compact support. Hence, even in
only a finite number of terms
is not orthogonal and such finite number is independent of
.
Thus, it suffices to
disprove
We take a subsequence of
converging to a weak limit
(see the proposition (
Weak
compactness of bounded set
)) and arrive to the task of disproving
existence of
such that the decomposition
has divergent
series
The sum
is finite and has
terms.