efinition
(Shift and sampling operators)
1. For any
we define "shift operator"
acting on
sequences:
2. We define "downsampling operator"
as follows:
3. We define "upsampling
operator":
4. We define
"convolution":
5. We define the "tilde
operation":
Compare with the proposition
(
Recursive
relationships for wavelet transform
)a,b for motivation.
Proof
Direct verification.
Proposition
(Interaction of
upsampling with Fourier transform) For a
we
have
Proposition
(Interaction of tilde
with Fourier transform) For a
we
have
Proposition
(Interaction of approximation and detail operators with Fourier
transform) For a
we
have
