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**Unformatted text preview: **Wavelets in Applied and Pure Mathematics
Vladimir V. Kisil A BSTRACT. The course gives an overview of wavelets (or coherent states) construction and its realisations in applied and pure mathematics. After a short introduction to wavelets based on the representation theory of groups we will consider:
• Spaces of analytic functions with reproducing kernels: the Hardy and the
Bergman spaces, etc.;
• The Fock-Segal-Bargmann space and Berezin-Toeplitz quantisation;
• Functional calculus of self-adjoint operators;
• Elements of signal processing.
The variety of applications is essentially grouped just around three groups:
the Heisenberg group, SL2 (R), and ax + b group. Course Outline
List of Figures 5 Preface 7 Preface 7 Lecture 1. What Are Wavelets and What Are They Good for?
1. Fourier Transform and Bases in Hilbert Spaces
2. Complex Analysis and Reproducing Kernels
3. Qantum Mechanics and Quantisation
4. Signal Prosessing
5. Functional Calculus
6. Discussion 9
9
10
12
13
13
13 Lecture 2. Groups and Homogeneous Spaces
1. Basics of Group Theory
2. Homogeneous Spaces and Invariant Measures 17
17
18 Lecture 3. Elements of the Representation Theory
1. Representations of Groups
2. Decomposition of Representations
3. Invariant Operators and Schur’s Lemma 23
23
25
26 Lecture 4. Wavelets on Groups and Square Integrable Representations
1. Wavelet Transform on Groups
2. Square Integrable Representations 29
29
31 Lecture 5. Wavelets on Homogeneous Spaces: the Segal-Bargmann Space
1. Quantum Mechanical Setting
2. Fundamentals of Wavelets on Homogeneous Spaces
3. Advanced Properties 35
35
36
41 Lecture 6. Wavelets in Banach Spaces and Functional Calculus
1. Coherent States for Banach Spaces
2. Wavelets in Operator Algebras
3. Examples
Acknowledgements 43
43
49
55
61 Bibliography 63 Index 67 3 List of Figures
1
2 The Gaussian function
An example of Windowed Fourier Transform 5 12
14 Preface
The purpose of this course is to sketch in ten lectures a huge area related to
wavelets. There are no precise boundaries of this area: it overlaps with many other
subjects in pure mathematics and many applications in physics and engendering.
Moreover there are many different approaches to wavelets based on rather different techniques. However this course is not intended to be complete and encyclopedic. Our main goal is to generate an interest in wavelets and to show that much
of the theory and applications are related to groups and symmetries.
The word “wavelets” came to fashion about 15 years ago and is very popular
now. On the other hand the notion of wavelets resemble coherent states used in
quantum mechanics for 75 years already. Future analysis shows that many classic
objects (e.g. from complex analysis) known at least from XIX century are essentially wavelets-coherent states too. This indicates that a significance of wavelets is
above just a current fashion.
We apply the name “wavelets” to the whole range of related objects to stress
their common origin and nature. Meanwhile the common usage of this term is
much narrower. Objects called “wavelets” by us usually appear as coherent states
(CS) in the literature. While commonly “wavelets” are coherent states related to
ax + b group. 7 LECTURE 1 What Are Wavelets and What Are They Good for?
In this introductory lecture one would only sketch an answer to the above
question: even the whole course could not be enough for that. Now we just list
several instances of wavelets appeared in different areas. All mentioned topics will
be considered in greater details in following lectures.
Most of the listed facts should be well known to reader, we are just presenting
them in a way highlighting the common structure. Similarities and differences of
these instances of wavelets will be discussed in the final section 6.
1. Fourier Transform and Bases in Hilbert Spaces
We start from two basic examples which were at the beginning of harmonic
and functional analysis.
1.1. Fourier Series and Basis in Hilbert Space. Consider the space L2 [−π, π]
of square integrable functions on [−π, π] with the Lebesgue measure. It is a Hilbert
space with a scalar product
Zπ
1
hf1 , f2 i =
(1.1)
f1 (x)f¯2 (x) dx.
π
−π Let us introduce the set of functions
1
e0 (x) = , e2n (x) = cos nx, e2n−11 = sin nx,
(1.2)
2
It is a straightforward calculation that
(1.3) where n ∈ N. hei , ej i = δi−j , where δi−j is the Kronecker delta. Moreover the set (1.2) is a maximal family of
functions in L2 [−π, π] with the above property. In fact (1.3) could be taken as a
definition of orthonormal base in a Hilbert space H.
It worths to state main properties of the family (1.2) in the generality of an
arbitrary orthonormal base. For such a base ej the following is true [32, § III.5.1]:
• A base ei define a linear continuous mapping W : H → `2 (Z) of a vector
f ∈ H to a sequence of coefficients in `2 (Z) by the formula:
(1.4)
fˆn = hf, en i . (1.5) • The above mapping is an isometry of the Hilbert spaces (the Parseval(Pythagoras) identity):
∞
X
hf, f 0 i =
fˆj fˆj0 .
j=−∞ 9 10 1. WHAT ARE WAVELETS? • The mapping W could be inverted by an operator M : `2 (Z) → H,
namely we could reconstruct a vector f from its sequence of coefficients
as a linear combination of ej :
(1.6) f= ∞
X fj ej . j=−∞ • From the above we could define a reproducing operator P = MW on H,
symbolically written by the Dirac bra and ket notations:
P= ∞
X |ej i hej | . j=−∞ 1.2. Fourier Transform. It is useful to compare the above properties of the
Fourier series with the Fourier integral transform. The later is defined in L2 (R)
with the scalar product
Z
1
hf1 , f2 i = √
(1.7)
f1 (t)f¯2 (t) dt.
2π R
by means of functions:
(1.8) ea (t) = eiat , a ∈ R. A replacement of the orthonormal property (1.3) is the following identity (cf. (1.10)):
(1.9) hea , eb i = δ(a − b), where δ(a − b) is the Dirac delta function and the identity is true in the sense of
distributions [32, § III.4.4].
The following is true [32, § IV.2.3]:
• Functions ea define a linear continuous mapping W : L2 (Z) → L2 (Z) by
the formula:
Z
1
ˆ
f(a)
= hf, ea i = √
f(t)e−iat dt.
2π R
• The above mapping is an isometry of the Hilbert spaces (the Plancherel
identity):
D
E
ˆ fˆ0 .
hf, f 0 i = f,
• The mapping W could be inverted by an operator M : L2 (Z) → L2 (Z),
i.e. we could reconstruct a function f from its Fourier transform as a
continuous linear combination of ea (t):
Z
1
iat
ˆ
f= √
f(a)e
da.
2π R
• The composition of the above two operators P = MW gives an integral
resolution (in the distributional sense) of the Dirac delta function δ(u−t):
Z
Z
(1.10)
f(u) = f(t) ei(u−t)a da dt.
R R 2. Complex Analysis and Reproducing Kernels
We move to the classic Hilbert spaces in complex analysis which are examples of wavelets in pure mathematics. Particularly the first example named after
G.H. Hardy, probably the purest mathematician of all times and nations. 2. COMPLEX ANALYSIS 11 2.1. The Hardy Space. Let H2 (T) be the Hardy space of L2 functions on the unit
circle T with an analytic continuation inside the unit disk D. The scalar product is
defined as follows:
Z
1
hf, f 0 i =
(2.1)
f(t)f¯0 (t) dt.
2π T
We could consider a set of functions in H2 (T) parametrised by a point a of D:
1
ea (t) =
(2.2)
.
it
¯e − 1
a
Then we could find similarly to cases of the Fourier series and integral that:
• Functions ea (t) define a linear continuous mapping W : H2 (T) → H2 (D)
of a function on T to an analytic function in D:
ˆ
f(a)
= [Wf](a) = hf, ea i
µ
¶
Z
1
1
f(t)
=
dt
¯eit − 1
2π T
a
Z
f(t)
1
ieit dt
=
2πi T a − eit
Z
1
f(t)
=
(2.3)
dz,
2πi T a − z (2.4) This is the Cauchy integral formula, of course.
• The above mapping is an isometry of the Hilbert spaces H2 (T) and H2 (D),
where the scalar product on H2 (D) defined as usual:
Zπ
1
0
hf, f i = lim
f(reit )f¯0 (reit ) dt.
r→1 2π
−π (2.5) • The mapping W could be inverted by an operator M : H2 (D) → H2 (T),
with a very simple definition:
ˆ it ).
f(t) = lim f(re
r→1 • From the above we could define a reproducing operator P = MW on
H2 (T), which is essentially the Szeg¨o singular integral operator. Considered on L2 (T) the operator P is an orthogonal projection on its closed
subspace H2 (T).
2.2. The Bergman Space. We consider the Bergman space in a way very similar
to the Hardy space above. Let L2 (D) be the space of square integrable function
on D. There is a closed linear subspace—the Bergman space B2 (D)—of analytic
functions in L2 (D). We define a family of functions
1
,
a ∈ D.
ea (z) =
(¯
az − 1)2 (2.6) • Functions ea (z) define a linear continuous mapping W : L2 (D) → B2 (D) of
a square integrable function on D to an analytic function in D:
ˆ
f(a)
= [Wf](a) = hf, ea i
Z
f(t)
dz,
=
z − 1)2
T (a¯
• The above mapping is an isometry of the Hilbert spaces if restricted to
B2 (D) ⊂ L2 (D), in fact it is the identity operator on B2 (D). 12 1. WHAT ARE WAVELETS? • Consequently W could be trivially “inverted” by the identity operator
M : B2 (D) → B2 (D).
• It is follows from the above that the operator P = MW = M = W is
reproducing on B2 (D) and is orthogonal projection L2 (D) onto B2 (D).
This is the Bergman projection.
R EMARK 2.1. In both cases of the Hardy and the Bergman spaces we meet
orthogonal projection P from the spaces of square integrable functions onto their
subspace of analytic functions. Let Mb be a (bounded) operator on L2 of multiplication by a bounded function b. It is easy to see that for any such b the T¨oplitz
operator Tb = PMb is a bounded operator on the subspace of analytical functions.
We will link later such operators with the wavelet theory.
3. Qantum Mechanics and Quantisation
Now we turn to the object which combines the beauty of the mentioned above
classic spaces of complex analysis and importance in applied area of quantum
mechanics. As in case of the Fourier integral we start from L2 (R) with the scalar
product (1.7). Let us consider the family of functions:
2 ez (t) = e−(¯z
Note that e0 (t) = e −t2 /2 √
+t2 )/2+ 2¯
zt , t ∈ R, z ∈ C. is the celebrated Gaussian shown on Figure 1. All other F IGURE 1. The Gaussian function e−x 2 /2 . functions obtained from it by horizontal shifts and multiplication by a function
eipt which takes value on the unit circle in C. In quantum mechanical language
the function ez (t) with z = q + ip describes a state of a particle with an expectation
of its coordinate equal to q, an expectation of its momentum—p, and the minimal
value of product of coordinate and momentum dispersions [25, § 1.3]. We will
discuss a physical meaning in details letter on.
We again find a similar structure:
• Functions ez (t) define a linear continuous mapping W : L2 (R) → SB2 (C)
of square integrable function f(t) on R to an analytic function in C:
ˆ = [Wf](z) = hf, ez i
f(z)
Z
√
1
2
2
= √
(3.1)
f(t)e−(z +t )/2+ 2zt dt.
2π R
Such analytic functions are square integrable on C with respect to the
2
Gaussian measure dβ(z) = e−|z| dz and form Segal-Bargmann space SB2 (C). 6. DISCUSSION (3.2) 13 • The above mapping is an isometry of the Hilbert spaces L2 (R) and SB2 (C),
where the scalar product on SB2 (C) defined as follows:
Z
0
hf, f i = f(z)f¯0 (z) dβ(z).
C (3.3) • The mapping W could be inverted by an operator M : SB2 (C) → L2 (R)
such that the original functions is a linear combination of ez (t):
Z
√
zt
−(¯
z2 +t2 )/2+ 2¯
ˆ
dβ(z).
f(t) = f(z)e
C • From the above we could define a reproducing operator P = MW on
L2 (R) and P 0 = WM on SB2 (C). The former gives yet another integral
resolution of the delta function, cf. the Fourier integral case. The later is
Segal-Bargmann projection. Considered on L2 (C, dβ(z)) the operator P 0 is
an orthogonal projection on its closed subspace SB2 (C). We again could
¨
consider Toplitz
operator of the form Tb = P 0 Mb for a bounded function
b, cf. Remark 2.1.
4. Signal Prosessing
The Fourier series and integral appeared as a tool for decomposition of an arbitrary oscillation (or signals) into a superposition of harmonic oscillations with
a fixed frequencies. This technique is quite successful in the cases then spectrum
of frequencies is independent from time or changes very slowly. But in many
common situation like music, speech, etc. this is not true and the Fourier transformation is out of help.
To improve performance it is useful to introduce Windowed Fourier Transform
(WFT ). It analyses the spectrum of frequence of not entire signal but only a part
“seen” through a small windows. The position and size of the windows are among
parameters of WFT. An example of such a transformation is shown on Figure 2
which is taken from the book [41], it is also instructional to view other pictures
from this book on-line.
The word “wavelets” is commonly attributed to the area of signal processing.
Decompositions of that type are of huge importance in signal processing and are
under active investigation. We will discuss this topic in details due to course.
5. Functional Calculus
In the above consideration we oftenly meet a decomposition of an arbitrary
function in to linear superposition of elementary ones, cf. (1.6), (2.6), (3.3). Because
functions are used as models for operators such formulas could be employed for
constructions of functional calculi. Particularly the Cauchy integral formula (2.2)
inspires the Riesz-Dunford functional calculus defined by the integral formula:
Z
1
f(t)
f(A) =
dz,
(5.1)
2πi T A − z
for an operator A.
6. Discussion
The above consideration could rise many questions. We list now our answers
to some of them: 14 1. WHAT ARE WAVELETS? F IGURE 2. An example of Windowed Fourier Transform
• Why is there a common pattern in the above different examples?
Opinions vary. Our feeling that the common structure related to the
symmetries. In each of the above case there is a group (or even several
groups) which is represented by transformations in the function spaces.
The groups are:
the Fourier series the group of integers Z;
the Fourier integral the group of reals R;
the Hardy space the SL2 (R) group;
the Bergman space the SL2 (R) group;
the Segal-Bargmann space the Heisenberg group H1 ;
the signal processing the ax + b group. 6. DISCUSSION 15 For example all functions ax , which are essentially wavelets or coherent
states, could be obtained from the function e0 (mother wavelet or vacuum
vector) by means of the corresponding group.
• Why are there significant differences? (e.g. in the Hardy space the inverse
operator M (2.5) is not defined as an integral)
The above group are different with different properties, therefore pictures generated by them even within a common scheme could significant differences. Even the same group could have representations with
very different properties. For example we will see later that the same
group SL2 (R) group could generate analytic function theories of “elliptic” and “hyperbolic” types. By the way the mentioned operator M (2.5)
could be expressed as integral similar to the scalar product (2.4).
• Do groups provide the ultimate explanations in the above examples?
Probably not. One could expect the “ultimate explanation” only in a
very simple situation and we hope that the above examples are more
complicated and consequently interesting. But group do explain many
fundamental properties of the mentioned objects and allow to put many
different cases within a common framework (cf. with the Erlangen program of F. Klein ;-).
• In section 2 we meet the Cauchy integral formula. Are wavelets related to other
objects of complex analysis (Cauchy-Riemann equation, Laplacian, Taylor and
Lorant expansion, etc.)?
Yes. We will see it later. For the moment we will mention that the Taylor
expansion is a close relative of the Fourier series from the range of our
examples.
• Are groups useful in classification of known types of wavelets or they could help
to discover new one?
We already mentioned above few new objects derived from the group
approach: hyperbolic complex analysis and new types of functional calculi of operators.
The following lecture should give answer to more questions. But before we
could proceed we will need a short overview of the representation theory. LECTURE 2 Groups and Homogeneous Spaces
The group theory and the representation theory are two enormous and interesting subjects themselves. However they are auxiliary in our consideration and
we are forced to restrict ourselves only to brief and very dry overview.
Besides introduction to that areas presented in [42, 56] we recommend additionally the books [31, 55]. The representation theory intensively uses tools of
functional analysis and on the other hand inspires its future development. We use
the book [32] for references on functional analysis here and recommend it as a nice
reading too.
1. Basics of Group Theory
We start from the definition of central object which formalizes the universal
notion of symmetries.
D EFINITION 1.1. A transformation group G is a nonvoid set of mappings of a
certain set X into itself with the following properties:
(i) if g1 ∈ G and g2 ∈ G then g1 g2 ∈ G;
(ii) if g ∈ G then g−1 exists and belongs to G.
E XERCISE 1.2. List all transformation groups on a set of three elements.
E XERCISE 1.3. Verify that the following are groups in fact:
(i) Group of permutations of n elements;
(ii) Group of n × n matrixes with non zero determinant over a field F under
matrix multiplications;
(iii) Group of rotations of the unit circle T;
(iv) Groups of shifts of the real line R and plane R2 ;
(v) Group of linear fractional transformations of the extended complex plane.
D EFINITION 1.4. An abstract group (or simply group) is a nonvoid set G on
which there is a law of group multiplication (i.e. mapping G × G → G) with the
properties
(i) associativity: g1 (g2 g3 ) = (g1 g2 )g3 ;
(ii) the existence of identity: e ∈ G such that eg = ge = g for all g ∈ G;
(iii) the existence of inverse: for every g ∈ G there exists g−1 ∈ G such that
gg−1 = g−1 g = e.
E XERCISE 1.5. Check that any transformation group is an abstract group.
E XERCISE 1.6. Check that the following transformation groups (cf. Example 1.3) have the same law of multiplication, i.e. are equivalent as abstract groups:
(i) The group of isometric mapping of an equilateral triangle onto itself;
(ii) The group of all permutations of a set of free elements;
(iii) The group of invertible matrix of order 2 with coefficients in the field of
integers modulo 2;
(iv) The group of linear fractional transformations of the extended complex
plane generated by the mappings z 7→ z−1 and z 7→ 1 − z.
17 18 2. GROUPS AND HOMOGENEOUS SPACES E XERCISE∗ 1.7. Expand the list in the above exercise.
It is simpler to study groups with the following additional property.
D EFINITION 1.8. A group G is commutative if for all g1 , g2 ∈ G, we have g1 g2 =
g2 g1 .
However, most of interesting and important groups are noncommutative.
E XERCISE 1.9.
(i) Which groups among found in Exercise 1.2 are commutative?
(ii) Which groups among listed in Exercise 1.3 are noncommutative?
Groups could have some additional analytical structures, e.g. they could be a
topological sets with a corresponding notion of limit. We always assume that our
groups are locally compact [31, § 2.4].
D EFINITION 1.10. If for a group G the group multiplication and the taking of
inverse are continuous mappings then G is continuous group.
Even a better structure could be found among Lie groups [31, § 6], e.g. groups
with a differentiable law of multiplication. Investigating such groups we could
employ the whole arsenal of analytical tools, thereafter most of groups studied in
this notes will be Lie groups.
E XERCISE 1.11. Check that the following are noncommutative Lie (and thus
continuous) groups:
(i) [55, Chap. 7] The ax + b group: set of elements (a, b), a ∈ R+ , b ∈ R with
the group law:
(a, b) ∗ (a 0 , b 0 ) = (aa 0 , ab 0 + b).
The identity is (1, 0), and (a, b)−1 = (a−1 , −b/a).
(ii) The Heisenberg group [26], [55, Chap. 1]: a set of triples of real numbers
(s, x, y) with the group multiplication:
1
(1.1)
(s, x, y) ∗ (s 0 , x 0 , y 0...

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- Spring '12
- margarettatker
- Math, Vector Space, The Land, Hilbert space, Hilbert Spaces, Square Integrable Representations