Gelfand-Shilov空間における連続ウェーブレット変換について

Wavelet Transforms on Gelfand-Shilov Spaces

松江工業高等専門学校 数理科学科 福田 尚広 (Naohiro Fukuda)　National Institute of Technology, Matsue College

筑波大学大学院数理物質科学研究科 木下 保 (Tamotu Kinoshita)Institute of Mathematics, University of Tsukuba

筑波大学大学院数理物質科学研究科 芳野 和久 (Kazuhisa Yoshino)　Institute of Mathematics, University of Tsukuba

1 Wavelet Transform

We are concerned with convolution operators. The FBI transform

Tf(x, ξ;h) =1

21/2(πh)3/4

∫Ry

ei(x−y)ξ/he−(x−y)2/2hf(y)dy

provides an alternative approach to analytic wave front sets in the microlo-cal analysis, which is developed independently by Sato-Kashiwara-Kawai.Thanks to the term eixξ/h, the FBI transform can be rewritten as the convo-lution operator with the function of coherent state. If we remove eixξ/h andreplace e−iyξ/h by e−iyξ, it becomes the short time Fourier transform (STFT).Then, the parameter h plays a role of the size of the window e−(x−y)2/2h. Thewavelet transform is also a convolution operator with different parameters.Let A := R+ × R denote the ax + b group, endowed with the multiplica-tion (a, b)(a′, b′) = (aa′, ab′ + b) The left-invariant Haar measure on A isdµ = da

a2db. Let U be the unitary representation of A on L2(R) defined by

(U(a, b)f

)(x) =

1√af

(x− b

a

).

Based on this representation, the theory of the harmonic analysis on groupsgives a generalized Fourier transform, that is wavelet. The wavelet transformof f ∈ L2(R) with respect to the analyzing wavelet ψ ∈ L2(R) satisfying theadmissible condition

Cψ :=∫R

|ψ(ξ)|2

|ξ|dξ <∞,

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is defined by

Wψf(a, b) =1√Cψ

∫Rf(x)ψa,b(x)dx,

where

ψa,b(x) =1√aψ

(x− b

a

)for (a, b) ∈ A.

The inverse wavelet transform of F ∈ L2(R+ ×R) with respect to the ana-lyzing wavelet ψ ∈ L2(R) is defined by

MψF (x) =1√Cψ

∫R+

∫RF (a, b)ψa,b(x)

dbda

a2(x ∈ R).

Remark: If ψ is real-valued, we have the equality

∫ 0

−∞

|ψ(ξ)|2

|ξ|dξ =

∫ ∞

0

|ψ(ξ)|2

|ξ|dξ (<∞) (1)

which gives the reconstruction formula f =MψWψf and

∥Wψf∥L2(A) = ∥f∥L2(R).

These always hold when the set of a is R instead of R+. In general, without(1) the Cauchy-Schwarz inequality gives

∥Wψf∥L2(A) ≤ C∥f∥L2(R).

If we regard A := R+ × R as not group but set, we have to rewrite thisestimate as

∥a−1Wψf∥L2(R+×R) ≤ C∥f∥L2(R).

This is regarded as the continuity property in L2.

Time - Frequency localization depends on window size. There is a generalrelationship between a and frequency:

Wide window (Stretched wavelet with large a)Poor time localization and Good frequency localization.

Coarse features ⇒ Low frequency

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For the wide window, we can not find out the high frequency, which is aver-aged by integration.

Narrow window (Compressed wavelet with small a)Good time localization and Poor frequency localization.

Rapidly changing details ⇒ High frequencyFor the narrow window, we can not find out the low frequency, which behavesvery slowly.

Remark: Even if we use the narrow window, we can not detect the highfrequency, which exists only locally in the frequency space. There is a limitto the detection with window due to the uncertainty principle, which saysthat the window sizes of time space and frequency space have an inverseproportionality.

Indeed, both STFT and wavelet transform use a window having an inverseproportionality. For STFT, the window size can be changed, but must befixed and applied to all frequencies. A more flexible approach in which win-dow size varies across frequencies would be desirable. So, the wavelet trans-form utilizes different window sizes for each frequency, as a ∼ |ξ|−1. Thatis just the auto focus property of wavelets. The wavelet transform is animproved version rather than a simplified version of STFT.

2 Application

We consider the Cauchy problem on [0, T ]×Rx{∂2t u− A(t)∂2xu = 0,u(0, x) = u0(x), ∂tu(0, x) = u1(x),

(2)

where the coefficient A(t) satisfies the weakly hyperbolic condition

A(t) ≥ 0 for t ∈ [0, T ].

Let us denote by Gs(R) (1 ≤ s < ∞) the space of Gevrey functions f(x)satisfying

supx∈K

|∂nxf(x)| ≤ CKrnKn!

s

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for any compact set K ⊂ R, n ∈ N. [2] gave the assumption that A ∈Cα[0, T ] (0≤α≤1) and proved the well-posedness in Gs for

1 ≤ s < 1 +α

2. (3)

Counter Example: [3] gave the following example of the ill-posedness:

Define that T0 = 0, Tj =j∑

n=1

2−(n−1)/20 (j ≥ 1),

A(t) = 2−j/10Θ((221j/20(t− Tj)

)for t ∈ [Tj, Tj+1] (j ≥ 0),

where

Θ(τ) =2− 2 cos 2πτ

2 + 3Γ3 sin 2πτ + (Γ− 9Γ2) cos 2πτ

and

Γ = (1 + 2√7)1/3 − 3

(1 + 2√7)1/3

.

Then, the Cauchy problem (2) with A(t) ∈ C0[0, T ] which is non-negativeand degenerates at t=Tj (j≥0), is ill-posed in Gs for s > 11/10 ∼ 1.

To know the behaviour of the coefficient concerned with the frequency, thestandard Fourier transform is not good, because the coefficients are usuallynot defined in the whole interval Rt. Therefore, it is natural to considerSTFT:

TwA(ξ, b) =∫Re−itξA(t)w(t− b)dt

and the wavelet transform:

WψA(a, b) =1√a

∫RA(t)ψ

(t− b

a

)dt.

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T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13

b

1

a

The slopes of both figures indicate that a peak moves toward the blow-uppoint T∞ as the frequency increases, which possibly causes the ill-posedness.

Remark 2.1 If a equals some power of 2, the form of the counter exampleresembles the wavelet. Generally for a function F

(t−b′a′

), the wavelet trans-

form with ψ(t−ba

)detects a ∼ a′ and b ∼ b′. The above figure means that

a ∼ 2−21j/20 and b = Tj are conspicuous since A(t) = 20−j/10Θ(

t−Tj2−21j/20

)for

each interval.

Remark 2.2 Amplitudes of oscillating coefficients are flattened by the de-generacy. Regularities depend on not only frequency but also amplitude (de-generacy). For example, according to [1] let us consider

high frequencysmall amplitude

f1(t) =

0 for t = 0,sin(log |t|)(log |t|)2 + 1

otherwise,

higher frequencysmaller amplitude

f2(t) =

0 for t = 0,sin(exp 1

|t|)

exp 1|t|

otherwise.

Then, we find that

• f1 belongs not ∪0<α<1Cα but BV ,

• f2 belongs to not BV but ∩0<α<1Cα.

The counter example corresponds to the case of f1, so the regularity is onlyC0 (around t = T∞) and not BV.

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STFT would require some graphs to adjust the brightness of the spectrogram.On the other hand, such an arrangement is not necessary for the wavelettransform. For this case, the wavelet transform will be useful.

3 Gelfand-Shilov Space and Continuity

From the point of view of the uncertainty principle, we are interested inbetter Time-Frequency localization. In this sense, the Schwartz space S willbe preferable, because it has an arbitrary polynomial decay in both time andfrequency spaces. For instance, very famous Mexican hat wavelet belongsto the Schwartz space S. But in fact, the Mexican hat wavelet like theGaussian satisfies an exponential decay. Therefore, we shall introduce theGelfand-Shilov space which is an interpolation between arbitrary polynomialdecay and exponetial decay, that is sub-exponetial decay in both time andfrequency spaces. For positive constants µ, ν and h such that ν + µ ≥ 1, wedefine the Banach Gelfand-Shilov space

Sµν,h(R) ={f ∈ S ; ∥xα∂βxf(x)∥L∞(R) ≤ Chα+βα!νβ!µ for all α, β ∈ N

}with the norm

∥f∥Sµν,h

(R) = supα,β∈N

∥xα∂βxf(x)∥L∞(R)

hα+βα!νβ!µ,

and the (non-Banach) Gelfand-Shilov space Sµν (R)

Sµν (R) = ind limh>0

Sµν,h(R)

with the inductive limit topology. The Gelfand-Shilov spaces have oftenappeared in the study of functional analysis and PDE’s (see [8], etc.). For thediscrete wavelet case requiring strong additional conditions, [4] constructedthe wavelets belonging to the Gelfand-Shilov spaces.

Ax Gsx Cr

x

Aξ nonexistence nonexistence Battle-Lemarie, DaubechiesGsξ Meyer [FKU] [HWW]

Crξ Meyer [FKU] [HWW]

As for the continuous wavelet transform requiring only the admissiblecondition, there are many possibilities to choose analyzing wavelets. Re-cently, [9] proved some estimates concerned with the continuity (bounded-ness) property of wavelet transforms in the (non-Banach) Gelfand-Shilov type

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of space Sµ,+ν (R) which is restricted to the half space ξ > 0 as the Hardyspace. For example, the Bessel wavelet ψ(x) defined by ψ(ξ) = e−ξ− ξ−1

for ξ > 0 and = 0 for ξ ≤ 0 belongs to S1,+2 (R). In fact, we know that

ψ(x) = 1π√1−ixK1(2

√1− ix), where K1 is the first modified Bessel function

of the second kind (see [6]).In this paper we assume vanishing moment conditions for not only ψ

but also f . Paying the attention to the parameter h, we try to derive somedetailed estimates. Our purpose is to show the continuity (boundedness)property of wavelet transforms in the (Banach) Gelfand-Shilov space Sµν,h(R).Moreover, we also compute the wavelet transforms of concrete functions inthe Gelfand-Shilov spaces and show the optimality of our results.

Lemma 3.1 There exists C > 0 and h0 > 0 such that

∥eh0|x|1/νf∥L∞(R) + ∥eh0|ξ|1/µ f∥L∞(R) ≤ C,

if and only if f ∈ Sµν,h(R).

Taking Lemma 3.1 into account, we also introduce the Banach Gelfand-Shilovspace combining with the infinite vanishing moments condition |f(ξ)| ≤Ce−h|ξ|

−1/δ,

Sµ,δν,h(R)={f ∈ S; ∥eh|x|1/νf∥L∞ + ∥ehmax{|ξ|1/µ,|ξ|−1/δ}f∥L∞<∞

}.

We remark that Sµν,h(R) (without the infinite vanishing moments condition)

corresponds to Sµ,δν,h(R) with δ = ∞, i.e.,

Sµ,∞ν,h (R) ={f ∈ S ; ∥eh|x|1/νf∥L∞ + ∥eh|ξ|1/µ f∥L∞ <∞

}.

Then, we get the following theorem (see [5]):

Theorem 3.2 Let µ, ν, h and δ be positive constants such that µ + ν ≥ 1.Define that d(λ) = λ(λ− 1)−1+1/λ. Then for the wavelet transform Wψ with

ψ ∈ Sµ,δν,h(R), the following estimates hold:

(i) if ν > 1∥∥∥∥∥∥eh|b/(a+1)|1/ν

a1/2 + 1Wψf

∥∥∥∥∥∥L∞(R+×R)

≤ C∥∥∥eh|x|1/νf∥∥∥

L∞(R),

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(i)′ if ν ≤ 1∥∥∥eh′21−1/ν |b/(a+1)|1/νWψf∥∥∥L∞(R+×R)

≤ C∥∥∥eh|x|2f∥∥∥

L∞(R)(0 < h′ < h),

(ii) if µ > 1∥∥∥∥∥∥a1/2ehd(δ/µ+1)1/µa−1/(µ+δ)

a+ 1Wψf

∥∥∥∥∥∥L∞(R+×R)

≤ C∥∥∥eh|ξ|1/µ f∥∥∥

L∞(R),

(iii) if µ > 1∥∥∥∥∥∥a1/2ehd(δ/µ+1)1/µ(max{a,a−1})1/(µ+δ)

a+ 1Wψf

∥∥∥∥∥∥L∞(R+×R)

≤ C∥∥∥ehmax{|ξ|1/µ,|ξ|−1/δ}f

∥∥∥L∞(R)

.

Example: Let us consider the Mexican hat wavelet

ψ(x) =2

π1/4√3(1− x2)e−x

2/2, ψ(ξ) =2√2π

π1/4√3ξ2e−ξ

2/2.

We see that ψ ∈ S1/2,∞1/2,h (R) with 0 < h < 1/2. In particular when f(x) =

e−x2/2, we can get

Wψf(a, b) =2√2π1/4a5/2(a2 − 1− b2)√

3Cψ(a2 + 1)5/2e−b

2/(2a2+2).

Then, (i)′ in Theorem 3.2 becomes∥∥∥eh′2−1|b/(a+1)|2Wψf∥∥∥L∞(R+×R)

≤ C∥∥∥eh|x|2f∥∥∥

L∞(R),

where 0 < h′ < h. This implies that the exponent in (i)′ is almost optimalwith respect to a and b, since

h′2−1|b/(a+ 1)|2 ∼ 1

2· b2/(2a2 + 2).

Thus, we see that (i)′ can not be improved anymore.

Moreover, we define the following weighted L∞(R+ ×R) space which isa subspace of L2(R+ ×R) as far as h is positive:

V µ,δν,h (R+ ×R)

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={F ∈L2(R+ ×R);

∥∥∥ehmax{|b/(a+1)|1/ν , a1/µ, a−1/δ}F∥∥∥L∞(R+×R)

<∞}.

We remark that when µ = ∞

V ∞,δν,h (R+ ×R)

={F ∈L2(R+ ×R);

∥∥∥ehmax{|b/(a+1)|1/ν , a−1/δ}F∥∥∥L∞(R+×R)

<∞}.

Theorem 3.2 gives the following continuity results:

Corollary 3.3 Let µ > 1, ν > 1, h > 0 and δ > 0. Then for ψ ∈ Sµ,δν,h(R),the wavelet transform

Sµ,∞ν,h (R) ∋ f 7→Wψf ∈ V ∞,µ+δν,h (R+ ×R)

is continuous. If f also satisfies the infinite vanishing moments condition,the wavelet transform

Sµ,δν,h(R) ∋ f 7→ Wψf ∈ V µ+δ,µ+δν,h (R+ ×R)

is continuous.

References

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