rth-orderΣ∆quantization withF by:
ef =F({qn}).
And we also know that,
f =FTW,w∗ (f) =F({πWn(f)}) =F({xn}).
Hence, the error can be written as:
kf−efk2=kF({xn} −F({qn})k2=kF({xn−qn})k2=kF(∆rn)k2=kF(D∗)r(u)k2.
Here,u= [u1,u2, . . . ,un]∗and
D=
I −πW2 0 · · · 0 0 I −πW3 · · · 0
. .. . ..
0 · · · 0 I −πWn 0 · · · 0 0 I
. (4.5.1)
Note that D is invertible and
D−1=
I πW2 πW2πW3 · · · πW2πW3· · ·πWn 0 I πW3 · · · πW3πW4· · ·πWn
. .. . ..
0 · · · 0 I πWn
0 · · · 0 0 I
.
Hence we get kF(D∗)r(u)k2≤ kF(D∗)rkOPkuk2 ≤C√
NkF(D∗)rkOP, where C is a positive constant by Theorem 4.2.7 and Theorem 4.3.4. Thus by Theorem 4.5.2 we know
(a) (b)
Figure 4.2: Numerical experiment
Sobolev left inverse leads the minimal squared error.
For now we can not give a accurate estimation for the error boundkF(D∗)rkOP. Instead, we give the following numerical experiment for the error performance of rth order Σ∆
quantization with Sobolev left inverse. The figure(a)and(b)shows the result whenr=2 andr=3. In each figure, the x-coordinate is always the size N of the fusion frame. In the top graph, the y-coordinate is the operator norm ofF(D∗)rwhich is the upper bound of the error. In the middle graph, the y-coordinate is the operator norm ofF(D∗)rtimesN2r, and in the bottom graph, y-coordinate is ln((F(D∗)r))divided by ln(N), which shows the order of decay speed. We can observe the error is almost equal toO(N−r).
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