e continue the construction of the section
(
Piecewise linear MRA
). It
remains to satisfy the requirement
(
Multiresolution analysis
)5. We use
the proposition (
OST property 3
).
In context of the proposition (
OST property 3
),
let


(Starting scaling function)

Proof
Direct verification.
It remains to verify that
of the formula (
Starting scaling
function
) satisfies the formula (
Riesz
basis condition
).
We could evaluate
and then estimate as in the formula (
Riesz
basis condition
). However there is a better way for
with finite support.
Proposition
(Shifted Fourier transform
equality) If
then
For the function
of the form (
Starting scaling
function
) only three terms
are
nonzero:
Hence
The property (
Riesz basis condition
) is
now evident and existence of the function
of the definition (
Multiresolution
analysis
)5 follows from the proposition (
OST
property 3
).
