Some Constructions of Wavelets, Frames and Orthonormal Bases

概要

Some Constructions of Wavelets, Frames and Orthonormal Bases

Hirofumi Hashimoto

February 2023

 

Some Constructions of Wavelets, Frames and Orthonormal Bases

Hirofumi Hashimoto
Doctoral Program in Mathematics

Submitted to the
Degree Programs in Pure and Applied Sciences of the
Graduate School of Science and Technology
in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy in
Science
at the
University of Tsukuba

Acknowledgements
I would like to express my sincere appreciation to my advisor, Associate Professor Tamotu Kinoshita,
for valuable advice, encouragement and kindness. I would also like to thank Associate Professor
Kensuke Fujinoki and Dr. Katsuya Fujii for their collaboration. Finally, I would like to express my
deepest gratitude to my family, including my parents, for their kindness and support.

Contents
1 Fourier analysis
1.1 Fourier transform on Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Uncertainty principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Wavelet analysis
2.1 Continuous wavelet transform . . . . . . . . . . . . . . . . . . . . . .
2.2 Discrete wavelet transform . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Construction of the orthonormal wavelet in the Hardy space H 2 (R) .
2.3.1 Proof of Theorem 2.15 . . . . . . . . . . . . . . . . . . . . . .
2.4 Directional frames having Lipschitz continuous Fourier transforms .
2.4.1 Lipschitz continuous type . . . . . . . . . . . . . . . . . . . .
2.4.2 Numerical simulations . . . . . . . . . . . . . . . . . . . . . .

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1
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23
26

3 Radon transform
3.1 Radon transform on Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Wavelet-like orthonormal basis and its application to two-dimensional Radon transforms
3.2.1 Proof of Theorem 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27
28
30
31

References

33

1

Fourier analysis

1.1

Fourier transform on Euclidean space

Fourier analysis is the most fundamental theory in time-frequency analysis. The Fourier transform of
f ∈ L1 (Rnx ) and the inverse Fourier transform of g ∈ L1 (Rnξ ) are defined by
∫
F[f ](ξ) =
F

−1

f (x)e−ix·ξ dx,

Rn

1
[g](x) =
(2π)n

ξ ∈ Rnξ ,

∫
g(ξ)eix·ξ dξ,
Rn

x ∈ Rnx .

The Fourier transform F maps f on Rnx to F[f ] (or simply fˆ) on Rnξ . Furthermore, when n = 1, the
variables x and ξ are usually interpreted as time and frequency, respectively. Let BU C(Rn ) denote the
Banach space consisting of bounded uniformly continuous functions on Rn . Since the Fourier transform
of f ∈ L1 (Rn ) is bounded as ∥fˆ∥L∞ ≤ ∥f ∥L1 and uniformly continuous, we have fˆ ∈ BU C(Rn ). In
addition, the following formulas hold:
(F1) F[αf + βg] = αF[f ] + βF[g],

α, β ∈ C.

(F2) F[f ∗ g] = F[f ]F[g].
(F3) lim F[f ](ξ) = 0.
|ξ|→∞

(F4) Fx→ξ [f (x − h)] (ξ) = e−ih·ξ Fx→ξ [f ](ξ).
(F5) Fx→ξ [eih·x f (x)](ξ) = Fx→ξ [f ](ξ − h).
(F6) Fx→ξ [f (ax)](ξ) = |a|−n Fx→ξ [f ](a−1 ξ),

a ̸= 0.

Formula (F2) shows that the Fourier transform intertwines the convolution and the product. Let
C0 (Rn ) denote the Banach space consisting of continuous functions vanishing at infinity on Rn . Then,
formula (F3) implies fˆ ∈ C0 (Rn ), which is referred to as the Riemann-Lebesgue lemma. Furthermore,
formulas (F4), (F5) and (F6) show the relations between translation, modulation and dilation and the
Fourier transform, respectively, which are frequently used in Fourier analysis.
Since fˆ ∈ BU C(Rn ), fˆ does not belong to L1 (Rn ) generally. Therefore, we require some additional
conditions and techniques to reconstruct f from fˆ using the inverse Fourier transform F −1 .
Proposition 1.1. Suppose that f, fˆ ∈ L1 (Rn ). Then, the inversion formula holds:
f (x) = F −1 [F[f ]] (x)

a.e. x ∈ Rn .

Now, L1 (R) is a commutative Banach algebra under the convolution satisfying
∥f ∗ g∥L1 ≤ ∥f ∥L1 ∥g∥L1 .
By Proposition 1.1 and formulas (F1), (F2) and (F3), the Fourier transform is an algebra isomorphism of L1 (R) to C0 (R). It is interesting to note that every complex (non-trivial) homomorphism
φ : L1 (R) → C can be written as
∫
∞

φ(f ) =

f (x)e−ixt dx

−∞

for a unique t ∈ R (see [33]).
The Fourier transform of f is also defined if f belongs to the Schwartz class S(Rn ) ⊂ L1 (Rn ), and
the domain D(F) and the range R(F) can be characterized.

1

Proposition 1.2. The Fourier transform F is a continuous bijection operator from S(Rn ) to S(Rn ).
Moreover, the inversion formula and Parseval’s identity hold:
f (x) = F −1 [F[f ]] (x), x ∈ Rn , f ∈ S(Rn ),
∫
∫
1
F[f ](ξ)F[g](ξ)dξ, f, g ∈ S(Rn ).
f (x)g(x)dx =
(2π)n Rn
Rn

(1)

Next, we will generalize the Fourier transform to other function spaces. It is also useful to consider
the Fourier transform of a distribution such as the “Dirac delta” δ. By the duality S-S ′ , the Fourier
transform of a tempered distribution T can be defined by
(F[T ], φ)S ′ ×S = (T, F[φ])S ′ ×S ,

φ ∈ S(Rn ).

Proposition 1.2 and the duality enable us to characterize the domain and the range.
Proposition 1.3. The Fourier transform F is a continuous bijection operator from S ′ (Rn ) to S ′ (Rn ).
Moreover, the inversion formula holds:
T = F −1 [F[T ]] ,

T ∈ S ′ (Rn ).

By identifying f ∈ Lp (Rn ) with a tempered distribution for 1 ≤ p ≤ ∞, we can obtain the Fourier
transform of an Lp -function. In particular, Hilbert space L2 (Rn ) plays an important role in Fourier
analysis.
Now, suppose that f, g ∈ S(Rn ). Since S(Rn ) ⊂ L2 (Rn ), Parseval’s identity (1) can be written as
an inner product on L2 (Rn ):
⟨f, g⟩ =

1
1
f =g
ˆ
⟨fˆ, gˆ⟩ =⇒ ∥f ∥L2 =
n ∥f ∥L2 .
n
(2π)
(2π) 2

Then, by the density argument, the Fourier transform F can be uniquely extended to the bounded
linear operator FL2 , which is called L2 -Fourier transform, in L2 (Rn ). The following proposition is
obtained immediately.
Theorem 1.4 (Plancherel theorem). The L2 -Fourier transform FL2 is a unitary operator on L2 (Rn ),
up to (2π)−n/2 .
Unfortunately, we still only know the existence of the L2 -Fourier transform, so the explicit form
of FL2 remains non-trivial. We want it to have the same form as the Fourier transform on L1 (Rn ),
and in fact the following holds: For f ∈ L1 (Rn ) ∩ L2 (Rn ),
∫
f (x)e−ix·ξ dx a.e. ξ ∈ Rn .
FL2 [f ](ξ) =
Rn

Using this result, we can determine the explicit L2 -Fourier transform.
Proposition 1.5. For f ∈ L2 (Rn ), we have the following:
∫
f (x)e−ix·ξ dx
FL2 [f ](ξ) = l.i.m
R→∞

(2)

|x|≤R


2
∫




−ix·ξ
F
[f
](ξ)
−
f
(x)e
dx
2
 L
 dξ → 0


n
R
|x|≤R

∫
⇐⇒

as R → ∞.

Thus, we can calculate the L2 -Fourier transform from the right-hand side of expression (2):
∫
f (x)e−ix·ξ dx in L2 (Rn ).
FL2 [f ](ξ) = lim
R→∞ |x|≤R

Hereafter, we will rewrite FL2 simply as F to avoid confusion. For more details, see e.g. [5, 26, 29, 34].
2

1.2

Uncertainty principle

By the Plancherel theorem, properties of a function f ∈ L2 (R) can be translated into properties of its
Fourier transform fˆ ∈ L2 (R). Although we can obtain information of all frequencies by integration
over R, we can obtain little frequency information of a neighborhood of a point x = b. The simplest
way to deal with this problem is to multiply eixξ by a window function g ∈ L2 (R) localized in a
neighborhood of a point x = b.
Definition 1.6 (Windowed Fourier transform). A function g ∈ L2 (R) is called a window function if
it satisfies
∫
|xg(x)|2 dx < ∞.
R

Then, we define the windowed Fourier transform (WFT) of f ∈ L2 (R) by the window function g as
follows:
∫
Vg [f ](b, ξ) =
f (x)ub,ξ (x)dx,
R

g(x − b)eixξ .

where ub,ξ (x) =
Fourier transform (STFT).

If g and gˆ are window functions, the operator Vg is called the short-time

We note that the definition of the window function yields | · |1/2 g ∈ L2 (R) and g ∈ L1 (R), that is,
gˆ ∈ BU C(R) ∩ C0 (R). Using (F4) and (F5), we have
−iq(ξ−p)
ud
gˆ(ξ − p).
q,p (ξ) = e

Hence, by the Plancherel theorem, we obtain
⟨f, uq,p ⟩ =

1 ˆ
⟨f , ud
q,p ⟩.
2π

(3)

Equality (3) says that the value ⟨f, uq,p ⟩ in a neighborhood of a point x = q is equal to the value
(2π)−1 ⟨fˆ, ud
q,p ⟩ in a neighborhood of a point ξ = p by the Fourier transform, which means that
information of the pair (f, fˆ) in a neighborhood of the point (x, ξ) = (q, p) on the time-frequency plane
(phase plane in physics) Rx × Rξ is extracted by the window function g. Therefore, characterizing
the localization of the window function g in the time space and the localization of gˆ in the frequency
space is an important problem.
Definition 1.7. The center x∗ and radius △g of a window function g ∈ L2 (R) are defined by
1
x =
∥f ∥2L2
∗

∫
2

x|f (x)| dx,
R

1
△g =
∥f ∥L2

{∫
R

∗ 2

}1

(x − x ) |f (x)| dx
2

2

.

The value 2△g is called the width of g.
Therefore, the interval which is essentially localized by a window function g is [x∗ − △g , x∗ + △g ].
Since g and gˆ are window functions for the STFT, we consider the region called the time-frequency
window:
[q + x∗ − △g , q + x∗ + △g ] × [p + ξ ∗ − △gˆ, p + ξ ∗ + △gˆ],
(4)
which is extracted by equality (3). Most importantly, the shape of the time-frequency window (4)
depends only on the choice of a window function g, and its measure is 4△g △gˆ (Figure 1).

3

Figure 1: Time-frequency windows of g.
Hence, in order to localize time and frequency information simultaneously, the value △g △gˆ should
be as small as possible, but we know that there is a lower bound for this value.
Theorem 1.8 (Heisenberg uncertainty principle). If g ∈ S(R), then △g △gˆ ≥ 1/2. Equality holds if
and only if
g(x) = ceiax gα (x − b), a, b ∈ R, c ̸= 0,
where gα is the Gaussian function
x2
1
gα (x) = √ e− 4α ,
2 πα

α > 0.

Proof. Using change of variables, we can assume that x∗ = ξ ∗ = 0. ...