Wavelets presentation

Outline
Introduction
Compactly Supported Wavelets
Multivariable Wavelets
References
Compactly Supported Wavelets Method for
Solving Solving PDEs
Asaph Keikara Muhumuza
July 6, 2017
Asaph Keikara Muhumuza Compactly Supported Wavelets Method for Solving Solving PDEs

Outline
Introduction
References
Introduction
Background
From Fourier Analysis to Wavelet Analysis
Time Frequency Localization
Wavelet Transform
Wavelet Function Admissibility
Multiresolution Analysis
Deﬁnition
Riesz Sequence and Basis
Compact Support
Smooth Wavelets
General Representation
Multi-dimension Multiresoultion Analysis
Regularity of Multivariate WaveletsAsaph Keikara Muhumuza Compactly Supported Wavelets Method for Solving Solving PDEs

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
In early 70’s, sums functions, called atomic decomposition
by Coifman [8], were widely used speciﬁclly in Hardy space
theory.
One method used to establish that a general function f has
such a decomposition is to start with ’Calderon formula’:for
a fonction f, which holds that
f(x) =
+∞
0
+∞
−∞
(ψt ∗ f) (y) ˆψt (x − y)dy
dt
t
(1)
. where the denotes convolution.
Here ψt (x) = t−1ψ(x/t) and similarly ˆφt (x), for appropriate
functions ψ and ˆψ.
This is typically an example of a continuous wavelet
function.

Outline
Introduction
References
Background
Wavelet Transform
Definition
Background
The first orthogonal wavelets were discovered by
Strömberg [14].
A discrete version the Calderón formula had also been
used for similar purposes by Jawerth [12] and long before
this there were results by Haar [10], Franklin [11],
Ciesielski [16], and many others.
Lamerié and Meyer [13], independently of Strömberg,
constructed new orthogonal wavelet expansions.
Mallat and Meyer [13], introduced the notion of
multiresolution a systematic framework for understanding
these orthogonal expansions.
Soon after, Daubenchie [1, 2], gave the construction of
wavelets, non-zero only on finite interval.Asaph Keikara Muhumuza Compactly Supported Wavelets Method for Solving Solving PDEs

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Existence and Some Applications of Wavelets
Figure: Existence and Applications of Wavelets

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Basic Types of Wavelets
Figure: Illustration of Common type of Wavelets

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Basic Illustrations and Notations
Figure: Connection between L2(R) to Fourier and then Wavelets

Outline
Introduction
References
Background
Wavelet Transform
Definition
Basic Notations and Definitions
For a complex-valued (Lebesgue) measurable function f
on R, let
||f||p = |f(x)|p
dx
1/p
1 ≤ p < ∞
By Lp we denote a Banach space of all measurable
functions f on R such that ||f||p < ∞.
In Particular, L2(R) is a Hilbert space with with an inner
product defined by
f, g =
R
f(x)g(x)dx, f, g ∈ L2(R)
As a result ||f|| = f, f 1/2

Outline
Introduction
References
Background
Wavelet Transform
Definition
Fourier Transform
Definition
Periodic Function is function f such that f(x + T) = f(x) for all
x ∈ R and T a positive constant called a period. e.g., sin x is
periodic since sin(x + 2x), sin(x + 4x), · · · are all equal to sin x.
Definition
Fourier Transform Let f ∈ L1(R). The Fourier transform of f is
the function ˆf : R → C defined by
ˆf(t) =
1
√
2π
+∞
−∞
f(x)e−itx
dx

Outline
Introduction
References
Background
Wavelet Transform
Definition
Convolution
Definition
Convolution Let f, g ∈ L1(R). We define the convolution of
f g by
(f g)(x) =
+∞
−∞
f(x − y)g(y)dy
such that the integral exists for all x ∈ R (and (f g)(x) = 0
otherwise).
Example Given H(x) = e−|x| then,
ˆH =
1
√
2π
∞
−∞
e−|x|−itx
dx =
2
π
1
1 + t2
.

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Windowed Fourier Transform
Given a signal f(t) (t cts time variable), one is interested in
frequency content locally.
The Fourier transform
(Ff)(ω, t) =
1
√
2π I∈R
f(t)e−iωt
dt (2)
Time localization can be achieved by ﬁrst windowing the
signal f so as to cut off a well localized slice of f and taking
the transform.
(Twinf)(ω, t) =
I∈R
f(s)g(s − t)e−iωs
ds (3)

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
The Windowed Fourier Transform
Figure: The Windowed Fourier Transform (Twinf)(ω) with signal f(x)Asaph Keikara Muhumuza Compactly Supported Wavelets Method for Solving Solving PDEs

Outline
Introduction
References
Background
Wavelet Transform
Definition
Windowed Fourier Transform Cont’d
The signal function f(x) is multiplied with the window
function g(t), and the Fourier coefficients of the product
are computed, and the procedure is then repeated for the
translated versions of the window say
g(t − t0), g(t − 2t0), · · · as shown in Figure 5.
Twin
m,n (f) =
I∈R
f(s)g(s − nt0)e−imω0s
ds (4)
If for instance, g is compactly supported, then it is clear
that with appropriately chosen ω0, the Fourier coefficients
Twin
m,n (f) are just sufficient to characterize and, if need be,
to construct f(•)g(• − nt0). Changing amounts to shifting
slice by step and its multiples, allowing recovery from

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Wavelet Transform
The wavelet provides a similar time frequency description,
with few noticeable differences.
The equivalent wavelet transform formulas analogous to
(3) and (4) are
(Twavf)(a, b) = |a|−1
2
I∈R
f(t)ψ
t − b
a
dt (5)
and
(Twav
m,n f)(a, b) = |a0|−m
2
I∈R
f(t)ψ a−m
0 t − nb0 dt (6)
where a is called the scale and b the translation.

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Wavelet/Fourier Transform (Differences)
In both cases it is assumed that ψ satisﬁed orthogonality
conditions i.e.,
I∈R
ψ(t)dt = 0 and
I∈R
ψ(t)ψ(t)dt = 1
Both Equations (3) and (5) take inner products of f with the
family of functions indexed respectively by two labels
gω,t
(s) = eiωs
g(s − t) (7)
and
ψa,b
= |a|−1
2 ψ
t − b
a
(8)

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Wavelet/Fourier Transform (Differences) Cont’d
A typical choice of ψ is
ψ(t) = (1 − t2
) exp(−t2
)/2
which is a family of Gaussian functions and it is smooth,
C∞.
As a changes, the
ψa,0
(s) = |a|−1
2 ψ(s/a)
over different sequence ranges (large values of scaling
parameter |a| correspond to small frequencies, or large
scale ψa,0); small values of |a| correspond to high
frequencies or very ﬁne scale ψa,0.

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Wavelet Transformations

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Admissible Wavelet Functions
The mother wavelet (kernel of the wavelet transform) can
be almost any function.
More rigorously, an L2(R) function (a ﬁnite energy square
integrable function over a range of its independent
variable) which can be either real or complex; is admissible
function if
Cg =
∞
−∞
|G(ω)|2
|ω|
dω < ∞ (9)
where G(ω) is the Fourier transform of g.

Outline
Introduction
References
Background
Wavelet Transform
Definition
Admissible Wavelet Functions Cont’d
A wavelet transform operator say Wφ maps a finite energy
or L2(R) signal that is real or complex valued function as
follows;
Wψ : L2
(R) → L2
(R {0} × R)
Stated mathematically, any finite energy signal is mapped
from time or space domain to a finite energy two
dimensional distribution in the scale transition or wavelet
domain.

Outline
Introduction
References
Background
Wavelet Transform
Definition
General Wavelet Representation
The wavelet transform of a function f w.r.t. given
admissible mother wavelet, ψ is defined as: Wavelet
domain coefficient at scale a and translation b
Wψ(a, b) = |a|−1
2
I∈R
f(x)ψ∗ x − b
a
dx
= f,
1
|a|
ψ
x − b
a
= f, ψa,b
= f, U(a, b)ψ .

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Multiresolution Analysis for One Dimensional
Wavelets
Deﬁnition 1 A multiresolution analysis on R is a sequence
of subspaces {Vj}j∈Z of functions L2(R) satisfying the
following conditions
(i) For all j ∈ Z, Vj ⊂ Vj+1 i.e.,
· · · , V−2 ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ V2 · · ·
(ii) If f(x) is C0
c on R, then f(x) ∈ span{Vj }j∈Z.
• That is given > 0, there exists a j ∈ Z and a function
g(x) ∈ Vj such that ||f − g|| < .
• Equivalently
j∈Z
Vj = L2(R).

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Multiresolution Analysis for One Dimensional
Wavelets Cont’d
(iii) For al j ∈ Z, j∈Z Vj = {0}.
(iv) A function f(x) ∈ Vj if and only if f(2−jx) ∈ V0.
(v) A function f(x) ∈ V0 if and only if D2j f(x) ∈ Vj, (i.e.,
f(x − k) ∈ V0 for all k ∈ Z).
(vi) There exists a function Φ in L2 on R called the scaling
function such that the collection Tk ψ(x) is orthonormal
system of translates and
V0 = span{Tk Φ(x)}

Outline
Introduction
References
Background
Wavelet Transform
Definition
The Following Definitions Follow Immediately for
Wavelets
(i) Definition
Given a real number k we define a translation operator Tk
acting on function defined on R by the formula
Tk (f)(x) = f(x − k)
(ii) Definition
Given a integer number j we define a dyadic dilation operator
Dj acting on function defined on R by the formula
Dj
(f)(x) = f(2j
x)Asaph Keikara Muhumuza Compactly Supported Wavelets Method for Solving Solving PDEs

Outline
Introduction
References
Background
Wavelet Transform
Deﬁnition
Important Lemma
1. Lemma 1 (a) For every k ∈ R, the operator Tk is an
isometry on L2(R).
(b) For every integer j ∈ Z the operator 2jDj is an isometry
on L2(R).
2. Result For each j ∈ Z the system
2j/2
Φ(2j/2
x − k)
k∈R
is an orthonormal basis in Vj.

Outline
Introduction
References
Compact Support
Smooth Wavelets
Deﬁnition 2
1. Deﬁnition 2 A sequence of vectors (xn)n∈A in a Hilbert
space H is called a Riesz sequence if there exist constants
0 < c ≤ C such that
c
n∈A
|an|
1
2
≤ ||
n∈A
an|| ≤ C
n∈A
|an|2
1
2
(10)
for all sequence of scalars (an)n∈A. A Riesz sequence is
called a Riesz basis if additionally span(an)n∈A = H

Outline
Introduction
References
Compact Support
Smooth Wavelets
Lemma 2
1. Lemma 2 (a) If we have a Riesz basis (xn)n∈Z, in H, then
there exist biorthogonal functions (x∗
n )n∈Z i.e., vectors in H
such that x∗
n , xm = σn,m. The sequence (x∗
n )n∈Z is also a
Riesz basis in H.
(b) (xn)n∈Z is a Riesz basis in H then there exists
constants 0 < c ≤ C such that
c||x|| ≤
n∈Z
| x, xn |2
1
2
≤ C||x|| ∀ x ∈ H. (11)

Outline
Introduction
References
Compact Support
Smooth Wavelets
Proposition 1
1. Proposition 1 Let Φ be a function in L2(R) and let
0 < a ≤ A be two constants. The following conditions are
equivalent:
(i) for every sequence of scalars (an)n∈Z we have
a
n∈A
|an|2
1
2
≤
+∞
−∞
|
n∈Z
anΦ(x − n)|2
dx
1
2
≤ A
n∈Z
|an|2
(12)
(ii) for almost all ξ ∈ R
a2
2π
≤
l∈Z
|ˆΦ(ξ + 2πl)|2
≤
A2
2π
(13)

Outline
Introduction
References
Compact Support
Smooth Wavelets
Corollary 1
Corollary 1 The system {Φ(t − m)}m∈Z is an orthonormal
system if and only if
i∈Z
|ˆΦ(ξ + 2πl)|2
=
1
2π
(14)
for almost all ξ ∈ R.

Outline
Introduction
References
Compact Support
Smooth Wavelets
Proposition 2
Proposition 2 Suppose that {φ(t − m)}m∈Z is a Riesz
sequence, then
(i) any function g ∈ span{Φ(t − m)}m∈Z we can be written
as
g(x) =
m∈Z
amΦ(x − m)
and the series converges in L2(R). (ii)
g ∈ span{φ(t − m)}m∈Z if and only if
ˆg(ξ) = φ(ξ)ˆΦ(ξ) (15)

Outline
Introduction
References
Compact Support
Smooth Wavelets
Proposition 2 Cont’d
(iii) if
g =
n∈Z
anΦ(x − m)}
and φ(ξ) is given by Equation (15) then
φ(ξ) =
n∈Z
ane−inξ
and conversely
(iv) the norms of g and Φ are related s follows:
A−1
||g||2 ≤
1
2π
2π
0
|φ(ξ)|2
dξ
1
2
≤ a−1
||g||2 (16)

Outline
Introduction
References
Compact Support
Smooth Wavelets
Remarks 1
From Deﬁnition 1 (multiresolution analysis), for conditions
(i) and (vi) it can be seen that for a scaling function Φ in V1,
so (iv) gives Φ(ξ/1) ∈ V0. From (vi) we obtain
φ(ξ/2) =
n∈Z
anφ(x − n) (17)
or equivalently
φ(ξ) =
n∈Z
anφ(2x − n) (18)

Outline
Introduction
References
Compact Support
Smooth Wavelets
Remarks 1 Cont’d
From Proposition 2 above, the above Equations (17) and
(18) can be rewritten equivalently as
ˆΦ(ξ) = mΦ(ξ/2)ˆΦ(ξ/2) (19)
or equivalently
ˆΦ(2ξ) = mΦ(ξ)ˆΦ(ξ) (20)
where mΦ is a function deﬁned by
mΦ(ξ) =
1
2
n∈Z
ane−inξ
(21)

Outline
Introduction
References
Compact Support
Smooth Wavelets
Remarks 1 Cont’d
Naturally, the coefﬁcients an, in (21) are the same as in
(17) and (18). Since ||ˆΦ(ξ/2)|| =
√
2
we infer from (17) that
n∈Z
|an|2
= 2
so that
1
2π
2π
0
|mΦ(ξ)|2
dξ
1
2
=
1
√
2
Each of the equivalent equations (17)-(20) is called a
scaling equation.

Outline
Introduction
References
Compact Support
Smooth Wavelets
Lemma 3
Lemma 3
|mΦ(ξ)|2
+ |mΦ(ξ + π)|2
= 1 (22)
for almost all ξ ∈ R.
Proof Follows from Corollary 1 and using (19) and the fact
that mΦ(xi) is 2π- periodic gives
1
2π
=
l∈Z
Φ(ξ + πl)2
=
l∈Z
|mΦ(ξ/2 + πl)|2
.|ˆΦ(ξ/2 + πl)|2

Outline
Introduction
References
Compact Support
Smooth Wavelets
Lemma 3 Cont’d
Proof Cont’d
1
2π
=
l∈Z
|mΦ(ξ/2 + 2kπ)|2
.|ˆΦ(ξ/2 + 2kπ)|2
+
l∈Z
|mΦ(ξ/2 + π + 2kπ)|2
.|ˆΦ(ξ/2 + π + 2kπ)|2
= |mΦ(ξ/2)|2
.
l∈Z
|ˆΦ(ξ/2 + 2kπ)|2
+ |mΦ(ξ/2 + π)|2
.
l∈Z
|ˆΦ(ξ/2 + π + 2kπ)|2
The result follow immediately from (22).

Outline
Introduction
References
Compact Support
Smooth Wavelets
Theorem 1
Theorem 1 Suppose we have a function Φ in L2(R) such
that
(i) {φ(t − m)}m∈Z is a Riesz sequence in L2(R),
(ii) Φ(x/2) = k∈Z ak Φ(x − k) with the convergence of the
series understood as the norm convergence in L2(R),
(iii) ˆΦ(ξ) is continuous at 0 and ˆΦ(0) = 0.
Then, the space
Vj = span{Φ(2j
x − k)} (23)
with j ∈ Z forms a multiresolution analysis.

Outline
Introduction
References
Compact Support
Smooth Wavelets
Proposition 3
Proposition 3 Suppose we have a function Φ ∈ L2(R)
satisfying condition (i) of Theorem 1. Let the space Vj be
deﬁned by (23) and let Pj be an orthonormal projection
onto Vj.
Then, for each f ∈ L2(R) we have
lim
j→−∞
Pjf = 0,
in particular
j∈Z
Vj = {0}.

Outline
Introduction
References
Compact Support
Smooth Wavelets
Proposition 4
Proposition 4 Suppose we have a function Φ ∈ L2(R)
satisfying conditions (i) and (ii) of Theorem 1.
Then
j∈Z
Vj
is dense in L2(R), where the spaces Vj are deﬁned by (23).

Outline
Introduction
References
Compact Support
Smooth Wavelets
Theorem 2
Theorem 2 Suppose that
m(ξ) =
S
k=T
ak e−ikξ
is a trigonometric polynomial such that
|m(ξ)|2
+ |m(ξ + π)|2
= 1 for all ξ ∈ R (24)
m(0) = 1 (25)
m(ξ) = 0 for ξ ∈ −
π
2
,
π
2
(26)

Outline
Introduction
References
Compact Support
Smooth Wavelets
Theorem 2 Cont’d
Then, the ﬁnite product
Θ(ξ) =
∞
j=1
m(2−j
ξ) (27)
converges almost uniformly.
The function Θ(ξ) is thus continuous. Moreover, it is in
L2(R).
The function ˆΦ given by ˆΦ = 1√
2π
θ has a support contained
in [T, S] and is a scaling multiresolution analysis. In
particular it has orthonormal translates.

Outline
Introduction
References
Compact Support
Smooth Wavelets
Remark 2
The formula
Ψ(x) = 2
S
k=T
ak (−1)k
φ(2x + k + 1) (28)
gives a compactly supported wavelet with support
Ψ(x) ⊂
T − S − 1
2
,
S − T − 1
2

Outline
Introduction
References
Compact Support
Smooth Wavelets
Lemma 4
Lemma 4 If m(ξ) is a trigonometric polynomial and
Equations (24) and (25) hold, then the product (27)
converges almost uniformly and
+∞
−∞
|
∞
j=1
m(2−j
ξ)|2
dξ ≤ 2π (29)
In particular Θ(ξ) is continuous and Θ(0) = 1.
If we assume also Equation (26) then for each k ∈ Z
+∞
−∞
|
∞
j=1
m(2−j
ξ)|2
e−2πikξ
dξ =
2π if k = 0
0 otherwise
(30)

Outline
Introduction
References
Compact Support
Smooth Wavelets
Lemma 5
Lemma 5 If
g(ξ) =
T
k=−T
γk e−ikξ
is a non-negative trigonometric polynomial with the γk ’s
real, then there exists a polynomial
m(ξ) =
T
k=0
ak eikξ
with all ak ’s real and such that
|m(ξ)|2
= g(ξ)
for all ξ ∈ R.

Outline
Introduction
References
Compact Support
Smooth Wavelets
Theorem 3
Theorem 3 There exists a constant C such that for each
r = 0, 1, 2, · · · there exists a multiresolution analysis in
L2(R) with scaling function Φ(x) and an associated wavelet
Ψ(x) such that
(i) Φ(x) and Ψ(x) are Cr
(ii) Φ(x) and Ψ(x) are compactly supported and both suppΦ
and suppΨ are contained in [−Cr , Cr ]

Outline
Introduction
References
Regularity of Multivariate Wavelets
Deﬁnition 1 (Tensor Product)
Given d functions of one variable fi(x) for j = 1, · · · , d we
can form a function of d variables
f1
⊗ f2
⊗ · · · ⊗ fd
deﬁned as
d
j=1
fj
(x1, · · · , xd ) =
d
j=1
fj
(xj)
If we have d closed subspaces Xj ⊂ L2(R) for
j = 1, 2, · · · , d we can form a closed subspace of L2(Rd )
denoted by
d
j=1
Xj or f1
⊗ f2
⊗ · · · ⊗ fd

Outline
Introduction
References
Proposition 1
Let (ψi)d be wavelets on R and let
ψ(x1, · · · , xd ) =
d
j=1
ψj(xj)
Then, the system
2
j1+···+jd
2 ψ(2j1
x1 − k1, · · · , 2jd xd − kd ) (31)
for all j1, · · · , jd and k1, · · · , kd in Z forms orthonormal basis
in L2(Rd )
If we have d closed subspaces Xj ⊂ L2(R) for
j = 1, 2, · · · , d we can form a closed subspace of L2(Rd )
denoted by
dAsaph Keikara Muhumuza Compactly Supported Wavelets Method for Solving Solving PDEs

Outline
Introduction
References
Theorem 1
Suppose that on R we are given two multiresolution
analyses say
· · · ⊂ Vi
−1 ⊂ Vi
0 ⊂ Vi
1 ⊂ · · ·
with scaling functions Φi(x) and corresponding wavelets
ψi(x) where i = 1, 2. Deﬁning subspaces Fj ⊂ L2(R2) as
Fj = V1
j ⊗ V2
j (32)
The sequence of subspaces (Fj)j∈Z has the following
properties
· · · ⊂ F−1 ⊂ F0 ⊂ F1 ⊂ · · · (33)

Outline
Introduction
References
Theorem 1 Cont’d
j∈Z
Fj = L2(R2
) ⊗ V2
j (34)
j∈Z
Fj = {0} (35)
f(x, y) ∈ Fj if and only if f(2−j
x, 2−j
y) ∈ F0 (36)
f(x, y) ∈ F0 if and only if f(x − k, y − l) ∈ F0 ∀ k, l ∈ Z (37)

Outline
Introduction
References
Theorem 1 Cont’d
the system
{Φ1(x − k)Φ2(y − l)}k,l∈Z (38)
is an orthonormal basis in F0.
Using the properties (33) - (38) we obtain functions
f1 = ψ1 ⊗ ψ2, f2 = ψ1 ⊗ Φ2, f3 = Φ1 ⊗ ψ2
such that the system
{fi(2j
x − k, 2j
y − l)}k,l∈Z (39)
with j, k, l ∈ Z and i = 1, 2, 3 is an orthonormal basis in
L2(R2)

Outline
Introduction
References
Proposition 2
Suppose we have d multiresolution analyses in L2(R) with
scaling functions Φ0,j(x) and associated wavelets Φ1,j(x)
for j = 1, 2, · · · , d. Let E = {0, 1}d (0, 0, · · · , 0)
×d
.
For e1, · · · , ed ∈ E let Ψc = d
j=1 Φcj ,j
.
Then
{2
dj
2 Ψc
(2j
x − γ)}c∈E,j∈Z,γ∈Zd (40)
is an orthonormal basis in L2(Rd )-
Here we use the notation
2j
x − γ = 2j
x − γ1, · · · , 2j
x − γd

Outline
Introduction
References
Deﬁnition 2
The function F on Rd is r − regular, if F is of Cr −class,
r = −1, 0, 1, 2, · · · and
∂α
∂xα
F(x) ≤
Ck
(1 + |x|)k
for each k = 0, 1, 2, · · · and each multiindex α with
|α| ≤ max(r, 0) and some Ck .
For e1, · · · , ed ∈ E let Ψc = d
j=1 Φcj ,j
.
As usual class C−1 means measurable function and C0
means continuous function.

Outline
Introduction
References
Lemma 1
If F(x) is an r−regular function on Rd and we deﬁne G(x)
by the condition
ˆG(ξ) = m(ξ) ˆF(ξ)
for a C∞, 2πZd −periodic function m(ξ), then G(x) is
r−regular.
Corollary 1 If Φ(x) is an r−regular function on Rd such
that {Φ(t − γ)}γ∈Zd is Riesz sequence, then the function
Φ1(x) deﬁned by
Φ1(ξ) =


l∈Zd
|ˆΦ(ξ + 2π)|2 ˆΦ(ξ)

 is also r − regular.

Outline
Introduction
References
I., Daubechies. Ten lectures on wavelets. Number 61 in
CBMS-NSF. Series in Applied mathematics. SIAM
Publications, Philadelphia, 1992
I., Daubechies. Orthogonal bases of compactly supported
wavelets. Comm. Pure and Appl. Math., 41:909–996, 1988.
C.K., Chi. An introduction to wavelets. Academic Press.,
1992
C.K., Chi., and J.Z., Wang. On Compactly supported
splines and wavelets. J. Approx. Th. 71(3): 263–304, 1992.
A., Cohen, and I., Daubechies. On the instability of arbitrary
biorthogonal wavelet packets. Preprint.
S.S., Burns, R.A., Gopinath, and H., Guo. An introduction
to wavelets and wavelet transforms. Printice Hall, 1998.Asaph Keikara Muhumuza Compactly Supported Wavelets Method for Solving Solving PDEs

Outline
Introduction
References
Figure: THANKS FOR LISTENNING

Wavelets presentation

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