Construction of Orthonormal Multivariate Wavelets
by
Brian Lindmark
A thesis submitted to the Graduate Faculty of
Auburn University
in partial ful llment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
August 6, 2011
Copyright 2011 by Brian Lindmark
Approved by
Richard Zalik, Chair, Professor of Mathematics and Statistics
Bertram Zinner, Associate Professor of Mathematics and Statistics
Stephen Stuckwisch, Assistant Professor of Mathematics and Statistics
Abstract
The purpose of this paper is to explain the construction of orthonormal multivariate
wavelets associated with a multiresolution analysis. This paper primarily uses the work of R.
A. Zalik [10], where he outlines a method of constructing orthonormal multivariate wavelets
given an existing orthonormal multivariate wavelet associated with an MRA, and attempts
to clarify it for a wider audience. In the last section, I use the result in constructing some
orthonormal multivariate wavelets in various examples.
ii
Acknowledgments
I?d like to thank my wife Kirsten for her encouragement throughout this process. I?d
also like to thank my adviser, Dr. Zalik, for introducing me to the subject of wavelets and
helping to guide my studies for the past two years, as well as for his help in my editing
process. Finally, I?d like to thank Dr.  Angel San Antol  n Gil for helping to focus my e orts
and being available for additional help when needed.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
iv
Chapter 1
Introduction
In what follows, d > 1 will be an integer, arbitrary but  xed; Z will denote the set of
integers and R the set of real numbers; boldface lowcase letters will always denote elements
of Rd; x y will stand for the standard dot product of the vectors x and y; i will be reserved
for the imaginary number p 1. The inner product of two functions f;g2L2(Rd) will be
denoted by hf;gi, their bracket product by [f;g], and the norm of f by jjfjj; thus,
hf;gi:=
Z
Rd
f(t)g(t)dt;
[f;g](t) :=
X
k2Zd
f(t + k)g(t + k);
and
jjfjj:=
p
hf;fi:
The Fourier transform of a function f will be denoted by bf. If f2L(Rd),
bf(x) :=
Z
Rd
e 2 it xf(t)dt
For every j2Z and k2Zd, the operators Dj and Tk are de ned in L2(Rd) by
Djf(t) := 2dj=2f(2jt)
and
Tkf(t) := f(t k)
1
A set of functions f 1;:::; mg L2(Rd) is called an orthonormal multivariate wavelet, if
the sequence
fDjTk l;j2Z;k2Zd;1 l mg
it generates is an orthonormal basis of L2(Rd).
A multiresolution analysis (MRA) in L2(Rd) is a sequence fVj;j2Zg of closed linear
subspaces of L2(Rd) such that:
Vj Vj+1 for every j2Z (i)
For every j2Z;f(t)2Vj if and only if f(2t)2Vj+1 (ii)
[
j2Z
Vj is dense in L2(Rd): (iii)
There is a function u such that fTku;k2Zdg is an orthonormal basis of V0: (iv)
Let T := [0;1], and let Td denote the d-dimensional torus. A function f will be called
Zd-periodic if it is de ned in Rd, and for every k2Zd and x2Rd we have f(x + k) = f(x).
Claim: It follows from the de nition of MRA that there is a Zd-periodic function p2L2(Td)
such that
bu(2x) = p(x)bu(x) a.e.
Proof. Let fTku;k2Zdg be an orthonormal basis of V0. In particular, u(t)2V0. Thus, by
(ii), u(t2)2V 1. By (i) and (iv), we can write
u
 t
2
 
=
X
k2Zd
akTku(t)
2
Where ak =
 
u(t2);Tku(t)
 
. By taking the Fourier transform of both sides, we get
Z
Rd
e 2 it xu(t=2)dt =
Z
Rd
e 2 it x
X
k2Zd
aku(t k)dt:
By changing variables, we get
Z
Rd
e 2 i(2s) xu(s)2dds =
Z
Rd
e 2 i(s+k) x
X
k2Zd
aku(s)ds:
Which yields,
2dbu(2x) =
X
k2Zd
ake 2 ik xbu(x)
If we let p(x) = 2 d
X
k2Zd
ake 2 ik x, then the result follows.
The functionuis called ascalingfunctionfor the MRA, andpis called thelowpassfilter
associated with u.
We will denote the orthogonal complement of Vj in Vj+1 by Wj. Thus, Vj+1 = Vj Wj.
Let f 1;:::; mg be an orthonormal multivariate wavelet in L2(Rd); for j2Z, let Pj denote
the closure of the linear span of
fDjTk l;k2Zd;1 l mg
and let Vj :=
X
r<j
Pr. Note that  1;:::; m 2 V1. We say that f 1;:::; mg is associated
with an MRA if M := fVj;j2Zg is a multiresolution analysis. If this is the case, we also
say that f 1;:::; mg is associated with M. The de nition implies that f 1;:::; mg is an
orthonormal multivariate wavelet associated with M if and only iffTk l;k2Zd;1 l mg
is an orthonormal basis of W0.
Given fu1;:::;umg L2(Rd), we will adopt the following notation:
T(u1;:::;um) :=fTkul;k2Zd;1 l mg;
3
and
S(u1;:::;um) := span T(u1;:::;um):
The following is a special case of a theorem by Guo, Lebate et al. [4, Proposition 2.1]
Theorem 1. Assume that T(u1;:::;um) and T(h1;:::;hn) are orthonormal sequences in
L2(Rd) such that S(u1;:::;um) = S(h1;:::;hm). Then m = n.
Proof. Since S(u1;:::;um) = S(h1;:::;hm),
ul(x) =
X
k2Zd
nX
j=1
hul;TkhjiTkhj(x) l = 1;:::;m: and hj(x) =
X
k2Zd
mX
l=1
hhj;TkuliTkul(x)
This implies that
1 =jjuljj2 =
X
k2Zd
nX
j=1
jhul;Tkhjij2 and 1 =jjhjjj2 =
X
k2Zd
mX
l=1
jhhj;Tkulij2
Also note that hul;Tkhji=hT kul;hji Thus we can show,
m =
mX
l=1
jjuljj2 =
mX
l=1
X
k2Zd
nX
j=1
jhul;Tkhjij2 =
nX
j=1
X
k2Zd
mX
l=1
jhT kul;hjij2 =
nX
j=1
jjhjjj2 = n
In [7] Wilson and Weiss showed that iff 1;:::; lgis an orthonormal multivariate wavelet
in L2(Rd) associated with a multiresolution analysis, then m = 2d 1. Hence, when combined
with Theorem 1, we have that in the case of orthonormal multivariate wavelets associated
with the same MRA, m = n = 2d 1.
The following theorem is from [5, p. 57], which we adapt to suit the above de nition of
the Fourier transform.
Theorem 2. If  is a scaling function for an MRA fVj;j2Zg and p is the associated low
pass  lter, then h2L2(R) is an orthonormal wavelet associated with this MRA if and only
4
if there is a measurable unimodular and Z-periodic function v(x), such that
bh(2x) = e2 ixv(2x)p(x+ 1=2)b (x) a:e:
The main results of this paper will be generalizing the following corollary to wavelets in
L2(Rd).
Corollary 1. If h is an orthonormal wavelet associated with an MRA, then  is an orthonor-
mal wavelet associated with the same MRA if and only if there is a measurable unimodular
and Z-periodic function q(x) such that
b (x) = q(x)bh(x) a:e:
The following theorems will be referenced multiple times in the paper and will be in-
cluded here as a reference.
Theorem 3. (Parseval?s Identity) Let f2L2(Td), and ck := bf(k) be the Fourier coe cients
of f. Then
X
k2Zd
jckj2 =jjfjj2L2(Td)
Theorem 4. (Plancherel?s Theorem) Let f;g2L2(Rd). Then,
Z
Rd
jf(t)j2dt =
Z
Rd
jbf(x)j2dx
Corollary 2. Z
Rd
f(t)g(t)dt =
Z
Rd
bf(x)bg(x)dx
Theorem 5. (Fubini?s Theorem) Let X, Y be measure spaces. If
Z
X Y
jf(x;y)jd(x;y) <1:
5
Then Z
X
Z
Y
jf(x;y)jdydx =
Z
Y
Z
X
jf(x;y)jdxdy =
Z
X Y
jf(x;y)jd(x;y)
Corollary 3. If
X
n
Z
A
jf(n;x)jdx<1;
then,
X
n
Z
A
f(n;x)dx =
Z
A
X
n
f(n;x)dx
Theorem 6. (Gram-Schmidt Orthogonalization) Let fu1;:::;ung2S linearly independent,
where S is an inner product space. Then we can  nd a set f~u1;:::; ~ung2S of orthonormal
vectors that span the same space.
6
Chapter 2
Main Results
Lemma 1. (a) T(u1:::um) is an orthogonal sequence in L2(Rd) if and only if
[bul;buj] (x) = 0 a.e., l;j = 1;:::;m l6= j
(b) T(u1:::um) is an orthonormal sequence in L2(Rd) if and only if
[bul;buj] (x) =  l;j a.e., l;j = 1;:::;m
Proof. It su ces to prove (b).
Let a;b2Zd and k = b a. Then
hTaul;Tbuji=hul;Tkuji
=
Z
Rd
ul(t)uj(t k)dt
=
Z
Rd
bul(x)buj(x)e2 ik xdx (Plancherel?s Theorem)
=
X
n2Zd
Z
Td
bul(y + n)buj(y + n)e2 ik (y+n)dy (\periodize" the integral)
=
Z
Td
X
n2Zd
bul(y + n)buj(y + n)e2 ik ydy (Fubini?s Theorem)
=
Z
Td
[bul;buj] (y)e2 ik ydy (1)
Thus hTaul;Tbuji are the Fourier coe cients of [bul;buj], and Parseval?s Identity implies that
jj[bul;buj]jj2L2(Td) =
X
k2Zd
jhul;Tkujij2 (2)
7
Assume T(u1;:::;um) is an orthonormal sequence in L2(Rd). Then for l6= j, we have
that the right hand side of (2) is equal to 0, which implies that [bul;buj](x) = 0 a.e.. When
l = j, we have that the hul;Tkuli are the Fourier coe cients of the function 1, and by the
uniqueness of Fourier coe cients (since [bul;bul](x) is Zd-periodic and in L2(Td)), we have
that [bul;bul](x) = 1 a.e., and thus [bul;buj](x) =  l;j a.e..
Conversely, assume [bul;buj] =  l;j a.e.. Then when l6= j, (1) implies that hTaul;Tbuji=
R
Td 0 dy = 0. When l = j, (1) implies that hTaul;Tbuli =
R
Td e
2 ik ydy =  a;b. Thus
T(u1;:::;um) is an orthonormal sequence in L2(Rd).
Lemma 2. If T(h1;:::;hm) is an orthonormal sequence in L2(Rd) and S(u1;:::;um)  
S(h1:::hm), then there are Zd-periodic functions pl;j(x)2L2(Td), uniquely de ned a.e., such
that
bul(x) =
mX
r=1
pl;r(x)bhr(x) a.e., l = 1;:::;m (3)
Proof. Since T(h1:::hm) is an orthonormal sequence in L2(Rd) we can write the orthogonal
projection of ul onto S(hj), which we will denote by ul;r, as
ul;r(t) =
X
k2Zd
al;r;khr(t k):
Where al;r;k =hul;Tkhri, and since S(u1:::um) S(h1:::hm), we can write
ul(t) =
mX
r=1
ul;r(t) =
mX
r=1
X
k2Zd
al;r;khr(t k):
If we take the Fourier Transform of both sides, we get
bul(x) =
mX
r=1
X
k2Zd
al;r;kbhr(x)e 2 ik x
=
mX
r=1
bhr(x) X
k2Zd
al;r;ke 2 ik x
8
If we let pl;r(x) =
X
k2Zd
al;r;ke 2 ik x, then the result follows.
Lemma 3. Assume that T(h1;:::;hm) is an orthonormal sequence in L2(Rd) and that S(u1;:::;um) 
S(h1;:::;hm), and assume there are Zd-periodic functions pl;j(x) 2L2(Td) such that (3) is
satis ed. Then T(u1;:::;um) is an orthonormal sequence if and only if
mX
r=1
pl;r(x)pj;r(x) =  l;j a.e., l;j = 1;:::;m (4)
Proof. Let ul;r denote the orthogonal projection of ul onto S(hr). Then
cul;r(x) = pl;r(x)bhr(x) a.e. l;r = 1;:::;m:
Note that bul(x) =
mX
r=1
cul;r(x) and that since T(h1;:::;hm) is an orthonormal sequence in
L2(Rd);ul;r is orthogonal to uj;s for any r6= s.
Hence,
hul;Tkuji=
* mX
r=1
ul;r(t);
mX
s=1
Tkuj;s(t)
+
=
Z
Rd
mX
r=1
ul;r(t)uj;r(t k)dt (by orthogonality)
=
Z
Rd
mX
r=1
cul;r(x)cuj;r(x)e2 ik xdx (Plancherel?s Theorem)
=
X
n2Zd
Z
Td
mX
r=1
cul;r(y + n)cuj;r(y + n)e2 ik (y+n)dy
=
Z
Td
 mX
r=1
[cul;r; cuj;r](y)
!
e2 ik ydy (Fubini?s Theorem)
Thus, we have that hul;Tkuji are Fourier coe cients of
mX
r=1
[cul;r; cuj;r](x). But these are
the same Fourier coe cients as [bul;buj] (x), found in our proof of Lemma 1. Hence, by the
uniqueness of Fourier coe cients, [bul;buj] (x) =
mX
r=1
[cul;r; cuj;r](x) a.e. and thus, by Lemma 1,
9
T(u1;:::;um) is an orthonormal sequence if and only if
mX
r=1
[cul;r; cuj;r](x) = [bul;buj](x) =  l;j a:e: l;j = 1;:::;m
But,
[cul;r; cuj;r](x) =
X
k2Zd
cul;r(x + k)cuj;r(x + k)
=
X
k2Zd
pl;r(x + k)bhr(x + k)pj;r(x + k)bhr(x + k)
=
X
k2Zd
pl;r(x)pj;r(x)bhr(x + k)bhr(x + k) (since p is Zd-periodic)
= pl;r(x)pj;r(x)
hb
hr; bhr
i
(x)
= pl;r(x)pj;r(x) (by Lemma 1)
Hence, T(u1;:::;um) is an orthonormal sequence if and only if
 l;j = [bul;buj](x) a.e. l;j = 1;:::;m (Lemma 1)
=
mX
r=1
[cul;r; cuj;r](x)
=
mX
r=1
pl;r(x)pj;r(x)
Lemma 4. Assume that T(u1;:::;um) and T(h1;:::;hm) are orthonormal sequences in L2(Rd).
Then S(u1;:::;um) = S(h1;:::;hm) if and only if there are Zd-periodic functions pl;r(x) 2
L2(Td) that satisfy (3) and the matrix
P(x) :=
 
pl;r(x)
 m
l;r=1
(5)
is nonsingular almost everywhere.
10
Proof. First, assume there are Zd-periodic functions pl;j(x)2L2(Td) that satisfy (1) and the
matrix (5) is nonsingular almost everywhere. Let
U(x) :=
0
BB
BB
@
bu1(x)
...
cum(x)
1
CC
CC
A
and H(x) :=
0
BB
BB
@
bh1(x)
...
chm(x)
1
CC
CC
A
:
Then
U(x) = P(x)H(x) a.e.
If P(x) is nonsingular almost everywhere, setting
Q(x) :=
8
><
>:
[P(x)] 1 if P(x) is nonsingular
0 if P(x) is singular
;
yields that Q(x) is Zd-periodic and
H(x) = Q(x)U(x) a.e.
If we let
Q(x) :=
 
ql;r(x)
 m
l;r=1
;
then
bhl(x) =
mX
r=1
ql;r(x)bur(x):
11
We then have
1 =jjbhljj2 =jj
mX
r=1
ql;rburjj2 =
* mX
r=1
ql;r(x)bur(x);
mX
s=1
ql;s(x)bus(x)
+
=
Z
Rd
mX
r=1
mX
s=1
ql;r(x)bur(x)ql;s(x)bus(x)dx
=
X
k2Zd
Z
Td
mX
r=1
mX
s=1
ql;r(y + k)bur(y + k)ql;s(y + k)bus(y + k)dy
=
Z
Td
mX
r=1
mX
s=1
ql;r(y)ql;s(y)
X
k2Zd
bur(y + k)bus(y + k)dy (since q is Zd periodic)
=
Z
Td
mX
r=1
mX
s=1
ql;r(y)ql;s(y)[bur;bus](y)dy
=
Z
Td
mX
r=1
jql;r(y)j2dy (by Lemma 1)
 
Z
Td
jql;n(y)j2dy for any n2[1;:::;m]
=jjql;njj2L2(Td)
Therefore ql;n2L2(Td) for l;n = 1;:::;m and thus S(u1;:::;um) = S(h1;:::;hm).
Conversely, assume that S(u1;:::;um) = S(h1;:::;hm). Then (3) implies that there are
Zd-periodic matrices
P(x) =
 
pl;r(x)
 m
l;r=1
and Q(x) =
 
ql;r(x)
 m
l;r=1
such that
pl;r;ql;r2L2(Td); l;r = 1;:::;m
U(x) = P(x)H(x) a.e.
H(x) = Q(x)U(x) a.e.
Thus
U(x) = P(x)Q(x)U(x) a.e.
12
Which implies that
P(x)Q(x) = I a.e.
and thus P(x) is nonsingular almost everywhere.
Theorem 7. Assume that T(h1;:::;hm) is an orthonormal sequence in L2(Rd), and let
fu1;:::;ung be a set of functions de ned on Rd. Then T(u1;:::;un) is an orthonormal se-
quence and
S(h1;:::;hm) = S(u1;:::;un)
if and only if m = n, there are Zd-periodic functions pl;r(x)2L2(Td) such that (3) is satis ed
and the matrix (5) is orthogonal almost everywhere.
Proof. Assume thatT(h1;:::;hm) andT(u1;:::;un) are orthonormal and such thatS(h1;:::;hm) =
S(u1;:::;un). Then m = n by Theorem 1. Lemma 2 implies that (3) is satis ed. Since (3)
is satis ed and T(h1;:::;hm) is orthonormal, Lemma 3 implies that (4) is satis ed. If we
de ne Pl(x) as the l-th row of P(x), we see that the left hand side of (4) is equivalent to
Pl(x) Pj(x) , which tells us that (5) is orthogonal.
Now assume m = n, there are Zd-periodic functions pl;r(x) 2L2(Td) such that (3) is
satis ed and (5) is orthogonal a.e.. Since (3) is satis ed,
S(u1;:::;um) S(h1;:::;hm):
Since (5) is orthogonal a.e., (4) is satis ed. We can then use Lemma 3 to show that
T(u1;:::;um) is an orthonormal sequence. Since (5) is orthogonal a.e., it is also nonsingular
a.e., and we can then use Lemma 4 to conclude that
S(h1;:::;hm) = S(u1;:::;um):
13
As we remarked above, iff 1;:::; mgis an orthonormal multivariate wavelet in L2(Rd)
associated with an MRA, then m = 2d 1. Thus, an immediate consequence of Theorem 7
is
Theorem 8. Assume that f 1;:::; mg is an orthonormal multivariate wavelet in L2(Rd)
associated with an MRA, and let f 1;:::; ng be a set of functions de ned in L2(Rd). Then
f 1;:::; ng is an orthonormal multivariate wavelet associated with the same MRA as
f 1;:::; mg, if and only if m = n = 2d 1, and there are Zd-periodic functions pl;r(x) 2
L2(Td) such that
b l(x) =
mX
r=1
pl;r(x)b r(x) a.e., l = 1;:::;m
and the matrix (5) is orthogonal a.e.
14
Chapter 3
Examples
We?ll start with some basic constructions of the matrix P(x), and then move to some
more complicated formulations. The following de nitions will hold for all of the examples
section.
 (x) =  [0;1)(x)
 (x) =
8
>>>>
<
>>>
>:
1 if x2[0; 12)
 1 if x2[12;1)
0 elsewhere
Note that  (x) is known as the Haar Wavelet. For more information on its construction and
properties, see [5, 6, 7]
Let
 1(x;y) =  (x) (y)
 2(x;y) =  (x) (y)
 3(x;y) =  (x) (y)
The construction of f 1; 1; 3g is outlined in [6, p.82], and is shown to be orthonormal
wavelet in L2(R2) generated by the scaling function  (x;y) =  (x) (y). For information
regarding the construction of multivariate wavelets from a scaling function, see [9, Theorem
15
9].
Since each  j is separable, the Fourier transforms are easily found and equal to,
b 1(u;v) = b (u)b (v)
b 2(u;v) = b (u)b (v)
b 3(u;v) = b (u)b (v)
Example 1. Let
P(u;v) =
0
BB
BB
@
cos(2 u)  sin(2 u) 0
sin(2 u) cos(2 u) 0
0 0 1
1
CC
CC
A
This is a basic rotation matrix, which has the property of being Z2-periodic and or-
thogonal, hence ful lling the conditions for Theorem 8. If we apply this to our equation, we
get, 0
BB
BB
@
b~ 
1(u;v)
b~ 
2(u;v)
b~ 
3(u;v)
1
CC
CC
A
=
0
BB
BB
@
cos(2 u)  sin(2 u) 0
sin(2 u) cos(2 u) 0
0 0 1
1
CC
CC
A
0
BB
BB
@
b 1(u;v)
b 2(u;v)
b 3(u;v)
1
CC
CC
A
Recall that cos(2 u) = 12(e2 iu +e 2 iu) and sin(2 u) = 12i(e2 iu e 2 iu). Hence,
0
BB
BB
@
~ 1(x;y)
~ 2(x;y)
~ 3(x;y)
1
CC
CC
A
=
0
BB
BB
B@
 
1
2
 
 (x+ 1) + (x 1)
 
 12i
 
 (x+ 1)  (x 1)
  
 (y)
 
1
2i
 
 (x+ 1)  (x 1)
 
+ 12
 
 (x+ 1) + (x 1)
  
 (y)
 (x) (y)
1
CC
CC
CA
Note that in the above example, the new wavelets are linear combinations of shifted
versions of our original  ( ) and  ( ) in each variable. Thus, it is clear that the new wavelets
have bounded support since  ( ) and  ( ) have bounded support. This will hold true for all
of the future examples by the same reasoning.
16
Example 2. For a more general case, we can look back to our introduction of pl;r(x), where
we de ned it as
X
k2Zd
al;r;ke2 ik x. Hence, we can choose each pl;r(u;v) to be linear com-
binations of two-dimensional Zd-periodic complex exponentials. We will maintain the or-
thogonality condition by only using the main diagonal and single terms, rather than linear
combinations.
Let
P(u;v) =
0
BB
BB
@
e4 iue12 iv 0 0
0  e2 iue 6 iv 0
0 0 e 2 iu
1
CC
CC
A
Once again,  nding the inverse Fourier transform is simple since the wavelet and each
pl;r(u;v) are separable. Thus,
0
BB
BB
@
~ 1(x;y)
~ 2(x;y)
~ 3(x;y)
1
CC
CC
A
=
0
BB
BB
@
 (x+ 2) (y + 6)
  (x+ 1) (y 3)
 (x 1) (y)
1
CC
CC
A
Example 3. For the most general case, we begin with three rows which are linearly inde-
pendent, and use the Gram-Schmidt Process to orthogonalize the rows, and thus create an
orthogonal matrix.
Let
~P(u;v) =
0
BB
BB
@
3e4 iu 0 4
e4 iu 4e 2 iue 2 iv 0
0 0 5e10 iv
1
CC
CC
A
Note that all these rows are linearly independent. If we de ne our inner product
hPl(x);Pj(x)i:=
3X
r=1
Z
T2
pl;r(u;v)pj;r(u;v)dudv;
17
then by using Gram-Schmidt orthogonalization on our example ~P(u;v), we get that
P(u;v) =
0
BB
BB
@
3
5e
4 iu 0 4
5
4
25p26e
4 iu 1p
26e
 2 i(u+v)  3
25p26
0 0 e10 iv
1
CC
CC
A
Which yields our new wavelet,
0
BB
BB
@
~ 1(x;y)
~ 2(x;y)
~ 3(x;y)
1
CC
CC
A
=
0
BB
BB
@
3
5 (x+ 2) (y) +
4
5 (x) (y)
4
25p26 (x+ 2) (y) +
1p
26 (x 1) (y 1) 
3
25p26 (x) (y)
 (x) (y + 5)
1
CC
CC
A
18
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