Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets
We prove a Calderón reproducing formula for the Dunkl continuous wavelet transform on R. We apply this result to derive new inversion formulas for the dual Dunkl-Sonine integral transform.
Gespeichert in:
| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Datum: | 2009 |
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2009
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/149135 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets / M.A. Mourou // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 16 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-149135 |
|---|---|
| record_format |
dspace |
| spelling |
Mourou, M.A. 2019-02-19T17:36:42Z 2019-02-19T17:36:42Z 2009 Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets / M.A. Mourou // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 16 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 42B20; 42C15; 44A15; 44A35 https://nasplib.isofts.kiev.ua/handle/123456789/149135 We prove a Calderón reproducing formula for the Dunkl continuous wavelet transform on R. We apply this result to derive new inversion formulas for the dual Dunkl-Sonine integral transform. The author is grateful to the referees and editors for careful reading and useful comments. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets |
| spellingShingle |
Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets Mourou, M.A. |
| title_short |
Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets |
| title_full |
Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets |
| title_fullStr |
Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets |
| title_full_unstemmed |
Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets |
| title_sort |
inversion of the dual dunkl-sonine transform on r using dunkl wavelets |
| author |
Mourou, M.A. |
| author_facet |
Mourou, M.A. |
| publishDate |
2009 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We prove a Calderón reproducing formula for the Dunkl continuous wavelet transform on R. We apply this result to derive new inversion formulas for the dual Dunkl-Sonine integral transform.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149135 |
| citation_txt |
Inversion of the Dual Dunkl-Sonine Transform on R Using Dunkl Wavelets / M.A. Mourou // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 16 назв. — англ. |
| work_keys_str_mv |
AT mourouma inversionofthedualdunklsoninetransformonrusingdunklwavelets |
| first_indexed |
2025-11-25T21:08:28Z |
| last_indexed |
2025-11-25T21:08:28Z |
| _version_ |
1850551243388223488 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 071, 12 pages
Inversion of the Dual Dunkl–Sonine Transform on R
Using Dunkl Wavelets
Mohamed Ali MOUROU
Department of Mathematics, Faculty of Sciences, Al-Jouf University,
P.O. Box 2014, Al-Jouf, Skaka, Saudi Arabia
E-mail: mohamed ali.mourou@yahoo.fr
Received March 02, 2009, in final form July 04, 2009; Published online July 14, 2009
doi:10.3842/SIGMA.2009.071
Abstract. We prove a Calderón reproducing formula for the Dunkl continuous wavelet
transform on R. We apply this result to derive new inversion formulas for the dual Dunkl–
Sonine integral transform.
Key words: Dunkl continuous wavelet transform; Calderón reproducing formula; dual
Dunkl–Sonine integral transform
2000 Mathematics Subject Classification: 42B20; 42C15; 44A15; 44A35
1 Introduction
The one-dimensional Dunkl kernel eγ , γ > −1/2, is defined by
eγ(z) = jγ(iz) +
z
2(γ + 1)
jγ+1(iz), z ∈ C,
where
jγ(z) = Γ(γ + 1)
∞∑
n=0
(−1)n (z/2)2n
n! Γ(n+ γ + 1)
is the normalized spherical Bessel function of index γ. It is well-known (see [3]) that the functions
eγ(λ·), λ ∈ C, are solutions of the differential-difference equation
Λγu = λu, u(0) = 1,
where
Λγf(x) = f ′(x) +
(
γ +
1
2
)
f(x)− f(−x)
x
is the Dunkl operator with parameter γ+1/2 associated with the reflection grour Z2 on R. Those
operators were introduced and studied by Dunkl [2, 3, 4] in connection with a generalization
of the classical theory of spherical harmonics. Besides its mathematical interest, the Dunkl
operator Λα has quantum-mechanical applications; it is naturally involved in the study of one-
dimensional harmonic oscillators governed by Wigner’s commutation rules [6, 11, 16].
It is known, see for example [14, 15], that the Dunkl kernels on R possess the following Sonine
type integral representation
eβ(λx) =
∫ |x|
−|x|
Kα,β(x, y) eα(λy) |y|2α+1 dy, λ ∈ C, x 6= 0, (1.1)
mailto:mohamed_ali.mourou@yahoo.fr
http://dx.doi.org/10.3842/SIGMA.2009.071
2 M.A. Mourou
where
Kα,β(x, y) :=
aα,β sgn(x) (x+ y)
(
x2 − y2
)β−α−1
|x|2β+1
if |y| < |x|,
0 if |y| ≥ |x|,
(1.2)
with β > α > −1/2, and
aα,β :=
Γ(β + 1)
Γ(α+ 1) Γ(β − α)
.
Define the Dunkl–Sonine integral transform Xα,β and its dual tXα,β, respectively, by
Xα,βf(x) =
∫ |x|
−|x|
Kα,β(x, y) f(y) |y|2α+1 dy,
tXα,βf(y) =
∫
|x|≥|y|
Kα,β(x, y) f(x) |x|2β+1 dx.
Soltani has showed in [14] that the dual Dunkl–Sonine integral transform tXα,β is a trans-
mutation operator between Λα and Λβ on the Schwartz space S(R), i.e., it is an automorphism
of S(R) satisfying the intertwining relation
tXα,β Λβ f = Λα
tXα,β f, f ∈ S(R).
The same author [14] has obtained inversion formulas for the transform tXα,β involving pseudo-
differential-difference operators and only valid on a restricted subspace of S(R).
The purpose of this paper is to investigate the use of Dunkl wavelets (see [5]) to derive a new
inversion of the dual Dunkl–Sonine transform on some Lebesgue spaces. For other applications
of wavelet type transforms to inverse problems we refer the reader to [7, 8] and the references
therein.
The content of this article is as follows. In Section 2 we recall some basic harmonic analysis
results related to the Dunkl operator. In Section 3 we list some basic properties of the Dunkl–
Sonine integral trnsform and its dual. In Section 4 we give the definition of the Dunkl continuous
wavelet transform and we establish for this transform a Calderón formula. By combining the
results of the two previous sections, we obtain in Section 5 two new inversion formulas for the
dual Dunkl–Sonine integral transform.
2 Preliminaries
Note 2.1. Throughout this section assume γ > −1/2. Define Lp(R, |x|2γ+1dx), 1 ≤ p ≤ ∞, as
the class of measurable functions f on R for which ||f ||p,γ <∞, where
||f ||p,γ =
(∫
R
|f(x)|p|x|2γ+1dx
)1/p
, if p <∞,
and ||f ||∞,γ = ||f ||∞ = ess supx∈R|f(x)|. S(R) stands for the usual Schwartz space.
The Dunkl transform of order γ+1/2 on R is defined for a function f in L1(R, |x|2γ+1dx) by
Fγf(λ) =
∫
R
f(x) eγ(−iλx) |x|2γ+1dx, λ ∈ R. (2.1)
Inversion of the Dual Dunkl–Sonine Transform on R 3
Remark 2.2. It is known that the Dunkl transform Fγ maps continuously and injectively
L1(R, |x|2γ+1dx) into the space C0(R) (of continuous functions on R vanishing at infinity).
Two standard results about the Dunkl transform Fγ are as follows.
Theorem 2.3 (see [1]).
(i) For every f ∈ L1 ∩ L2(R, |x|2γ+1dx) we have the Plancherel formula∫
R
|f(x)|2|x|2γ+1dx = mγ
∫
R
|Fγf(λ)|2|λ|2γ+1dλ,
where
mγ =
1
22γ+2(Γ(γ + 1))2
. (2.2)
(ii) The Dunkl transform Fα extends uniquely to an isometric isomorphism from
L2(R, |x|2γ+1dx) onto L2(R,mγ |λ|2γ+1dλ). The inverse transform is given by
F−1
γ g(x) = mγ
∫
R
g(λ)eγ(iλx)|λ|2γ+1dλ,
where the integral converges in L2(R, |x|2γ+1dx).
Theorem 2.4 (see [1]). The Dunkl transform Fα is an automorphism of S(R).
An outstanding result about Dunkl kernels on R (see [12]) is the product formula
eγ(λx)eγ(λy) = T x
γ (eγ(λ·)) (y), λ ∈ C, x, y ∈ R,
where T x
γ stand for the Dunkl translation operators defined by
T x
γ f(y) =
1
2
∫ 1
−1
f
(√
x2 + y2 − 2xyt
)(
1 +
x− y√
x2 + y2 − 2xyt
)
Wγ(t)dt
+
1
2
∫ 1
−1
f
(
−
√
x2 + y2 − 2xyt
)(
1− x− y√
x2 + y2 − 2xyt
)
Wγ(t)dt, (2.3)
with
Wγ(t) =
Γ(γ + 1)√
π Γ(γ + 1/2)
(1 + t)
(
1− t2
)γ−1/2
.
The Dunkl convolution of two functions f , g on R is defined by the relation
f ∗γ g(x) =
∫
R
T x
γ f(−y)g(y)|y|2γ+1dy. (2.4)
Proposition 2.5 (see [13]).
(i) Let p, q, r ∈ [1,∞] such that 1
p + 1
q−1 = 1
r . If f ∈Lp(R, |x|2γ+1dx) and g ∈Lq(R, |x|2γ+1dx),
then f ∗γ g ∈ Lr(R, |x|2γ+1dx) and
||f ∗γ g||r,γ ≤ 4||f ||p,γ ||g||q,γ . (2.5)
(ii) For f ∈ L1(R, |x|2γ+1dx) and g ∈ Lp(R, |x|2γ+1dx), p = 1 or 2, we have
Fγ(f ∗γ g) = FγfFγg. (2.6)
For more details about harmonic analysis related to the Dunkl operator on R the reader is
referred, for example, to [9, 10].
4 M.A. Mourou
3 The dual Dunkl–Sonine integral transform
Throughout this section assume β > α > −1/2.
Definition 3.1 (see [14]). The dual Dunkl–Sonine integral transform tXα,β is defined for
smooth functions on R by
tXα,βf(y) :=
∫
|x|≥|y|
Kα,β(x, y)f(x)|x|2β+1 dx, y ∈ R, (3.1)
where Kα,β is the kernel given by (1.2).
Remark 3.2. Clearly, if supp (f) ⊂ [−a, a] then supp
(
tXα,βf
)
⊂ [−a, a].
The next statement provides formulas relating harmonic analysis tools tied to Λα with those
tied to Λβ, and involving the operator tXα,β.
Proposition 3.3.
(i) The dual Dunkl–Sonine integral transform tXα,β maps L1(R, |x|2β+1dx) continuously into
L1(R, |x|2α+1dx).
(ii) For every f ∈ L1(R, |x|2β+1dx) we have the identity
Fβ(f) = Fα ◦ tXα,β(f). (3.2)
(iii) Let f, g ∈ L1(R, |x|2β+1dx). Then
tXα,β(f ∗β g) = tXα,βf ∗α
tXα,βg. (3.3)
Proof. Let f ∈ L1(R, |x|2β+1dx). By Fubini’s theorem we have∫
R
tXα,β(|f |)(y)|y|2α+1dy =
∫
R
(∫
|x|≥|y|
Kα,β(x, y)|f(x)||x|2β+1 dx
)
|y|2α+1dy
=
∫
R
|f(x)|
(∫ |x|
−|x|
Kα,β(x, y)|y|2α+1dy
)
|x|2β+1 dx.
But by (1.1),∫ |x|
−|x|
Kα,β(x, y)|y|2α+1dy = eβ(0) = 1. (3.4)
Hence, tXα,βf is almost everywhere defined on R, belongs to L1(R, |x|2α+1dx) and ||tXα,βf ||1,α ≤
||f ||1,β , which proves (i). Identity (3.2) follows by using (1.1), (2.1), (3.1), and Fubini’s theorem.
Identity (3.3) follows by applying the Dunkl transform Fα to both its sides and by using (2.6),
(3.2) and Remark 2.2. �
Remark 3.4. From (3.2) and Remark 2.2, we deduce that the transform tXα,β maps L1(R,
|x|2β+1dx) injectively into L1(R, |x|2α+1dx).
From [14] we have the following result.
Inversion of the Dual Dunkl–Sonine Transform on R 5
Theorem 3.5. The dual Dunkl–Sonine integral transform tXα,β is an automorphism of S(R)
satisfying the intertwining relation
tXα,βΛβf = Λα
tXα,βf, f ∈ S(R).
Moreover tXα,β admits the factorization
tXα,βf = tV −1
α ◦ tVβf for all f ∈ S(R),
where for γ > −1/2, tVγ denotes the dual Dunkl intertwining operator given by
tVγf(y) =
Γ(γ + 1)√
π Γ(γ + 1/2)
∫
|x|≥|y|
sgn(x) (x+ y)
(
x2 − y2
)γ−1/2
f(x) dx.
Definition 3.6 (see [14]). The Dunkl–Sonine integral transform Xα,β is defined for a locally
bounded function f on R by
Xα,βf(x) =
∫ |x|
−|x|
Kα,β(x, y) f(y) |y|2α+1 dy if x 6= 0,
f(0) if x = 0.
(3.5)
Remark 3.7.
(i) Notice that by (3.4), ||Xα,βf ||∞ ≤ ||f ||∞ if f ∈ L∞(R).
(ii) It follows from (1.1) that
eβ(λx) = Xα,β(eα(λ·)(x) (3.6)
for all λ ∈ C and x ∈ R.
Proposition 3.8.
(i) For any f ∈ L∞(R) and g ∈ L1(R, |x|2β+1dx) we have the duality relation∫
R
Xα,βf(x)g(x)|x|2β+1dx =
∫
R
f(y) tXα,βg(y)|y|2α+1dy. (3.7)
(ii) Let f ∈ L1(R, |x|2β+1dx) and g ∈ L∞(R). Then
Xα,β
(
tXα,βf ∗α g
)
= f ∗β Xα,βg. (3.8)
Proof. Identity (3.7) follows by using (3.1), (3.5) and Fubini’s theorem. Let us check (3.8).
Let ψ ∈ S(R). By using (3.3), (3.7) and Fubini’s theorem, we have∫
R
f ∗β Xα,βg(x)ψ(x)|x|2β+1dx =
∫
R
Xα,βg(x)ψ∗βf
−(x) |x|2β+1dx
=
∫
R
g(y) tXα,β(ψ∗βf
−)(y)|y|2α+1dy =
∫
R
g(y)
(
tXα,βψ ∗α
tXα,βf
−) (y)|y|2α+1dy,
where f−(x) = f(−x), x ∈ R. But an easy computation shows that tXα,βf
− =
(
tXα,βf
)−.
Hence,∫
R
f ∗β Xα,βg(x)ψ(x)|x|2β+1dx =
∫
R
g(y) tXα,βψ ∗α
(
tXα,βf
)− (y)|y|2α+1dy
=
∫
R
tXα,βf ∗αg(y) tXα,βψ(y)|y|2α+1dy =
∫
R
Xα,β
(
tXα,βf ∗α g
)
(x)ψ(x)|x|2β+1dx.
This clearly yields the result. �
6 M.A. Mourou
4 Calderón’s formula for the Dunkl continuous
wavelet transform
Throughout this section assume γ > −1/2.
Definition 4.1. We say that a function g ∈ L2(R, |x|2γ+1dx) is a Dunkl wavelet of order γ, if
it satisfies the admissibility condition
0 < Cγ
g :=
∫ ∞
0
|Fγg(λ)|2dλ
λ
=
∫ ∞
0
|Fγg(−λ)|2dλ
λ
<∞. (4.1)
Remark 4.2.
(i) If g is real-valued we have Fγg(−λ) = Fγg(λ), so (4.1) reduces to
0 < Cγ
g :=
∫ ∞
0
|Fγg(λ)|2dλ
λ
<∞.
(ii) If 0 6= g ∈ L2(R, |x|2γ+1dx) is real-valued and satisfies
∃ η > 0 such that Fγg(λ)−Fγg(0) = O(λη) as λ→ 0+
then (4.1) is equivalent to Fγg(0) = 0.
Note 4.3. For a function g in L2(R, |x|2γ+1dx) and for (a, b) ∈ (0,∞)× R we write
gγ
a,b(x) :=
1
a2γ+2
T−b
γ ga(x),
where T−b
γ are the generalized translation operators given by (2.3), and ga(x) := g(x/a), x ∈ R.
Remark 4.4. Let g ∈ L2(R, |x|2γ+1dx) and a > 0. Then it is easily checked that ga ∈ L2(R,
|x|2γ+1dx), ||ga||2,γ = aγ+1 ||g||2,γ , and Fγ(ga)(λ) = a2γ+2Fγ(g)(aλ).
Definition 4.5. Let g ∈ L2(R, |x|2γ+1dx) be a Dunkl wavelet of order γ. We define for regular
functions f on R, the Dunkl continuous wavelet transform by
Φγ
g (f)(a, b) :=
∫
R
f(x)gγ
a,b(x)|x|
2γ+1dx
which can also be written in the form
Φγ
g (f)(a, b) =
1
a2γ+2
f ∗γ g̃a(b),
where ∗γ is the generalized convolution product given by (2.4), and g̃a(x) := g(−x/a), x ∈ R.
The Dunkl continuous wavelet transform has been investigated in depth in [5] in which precise
definitions, examples, and a more complete discussion of its properties can be found. We look
here for a Calderón formula for this transform. We start with some technical lemmas.
Lemma 4.6. For all f, g ∈ L2(R, |x|2γ+1dx) and all ψ ∈ S(R) we have the identity∫
R
f ∗γ g(x)F−1
γ ψ(x)|x|2γ+1dx = mγ
∫
R
Fγf(λ)Fγg(λ)ψ−(λ)|λ|2γ+1dλ,
where mγ is given by (2.2).
Inversion of the Dual Dunkl–Sonine Transform on R 7
Proof. Fix g ∈ L2(R, |x|2γ+1dx) and ψ ∈ S(R). For f ∈ L2(R, |x|2γ+1dx) put
S1(f) :=
∫
R
f ∗γ g(x)F−1
γ ψ(x)|x|2γ+1dx
and
S2(f) := mγ
∫
R
Fγf(λ)Fγg(λ)ψ−(λ)|λ|2γ+1dλ.
By (2.5), (2.6) and Theorem 2.3, we see that S1(f) = S2(f) for each f ∈ L1 ∩L2(R, |x|2γ+1dx).
Moreover, by using (2.5), Hölder’s inequality and Theorem 2.3 we have
|S1(f)| ≤ ||f ∗γ g||∞||F−1
γ ψ||1,γ ≤ 4||f ||2,γ ||g||2,γ ||F−1
γ ψ||1,γ
and
|S2(f)| ≤ mγ ||FγfFγg||1,γ ||ψ||∞ ≤ mγ ||Fγf ||2,γ ||Fγg||2,γ ||ψ||∞ = ||f ||2,γ ||g||2,γ ||ψ||∞,
which shows that the linear functionals S1 and S2 are bounded on L2(R, |x|2γ+1dx). Therefore
S1 ≡ S2, and the lemma is proved. �
Lemma 4.7. Let f1, f2 ∈ L2(R, |x|2γ+1dx). Then f1 ∗γ f2 ∈ L2(R, |x|2γ+1dx) if and only if
Fγf1Fγf2 ∈ L2(R, |x|2γ+1dx) and we have
Fγ(f1 ∗γ f2) = Fγf1Fγf2
in the L2-case.
Proof. Suppose f1 ∗γ f2 ∈ L2(R, |x|2γ+1dx). By Lemma 4.6 and Theorem 2.3, we have for any
ψ ∈ S(R),
mγ
∫
R
Fγf1(λ)Fγf2(λ)ψ(λ)|λ|2γ+1dλ =
∫
R
f1 ∗γ f2(x)F−1
γ ψ−(x)|x|2γ+1dx
=
∫
R
f1 ∗γ f2(x)F−1
γ ψ(x)|x|2γ+1dx = mγ
∫
R
Fγ(f1 ∗γ f2)(λ)ψ(λ)|λ|2γ+1dλ,
which shows that Fγf1Fγf2 = Fγ(f1 ∗γ f2). Conversely, if Fγf1Fγf2 ∈ L2(R, |x|2γ+1dx), then
by Lemma 4.6 and Theorem 2.3, we have for any ψ ∈ S(R),∫
R
f1 ∗γ f2(x)F−1
γ ψ(x)|x|2γ+1dx = mγ
∫
R
Fγf1(λ)Fγf2(λ)ψ̃(λ)|λ|2γ+1dλ
=
∫
R
F−1
γ (Fγf1Fγf2)(x)F−1
γ ψ(x)|x|2γ+1dx,
which shows, in view of Theorem 2.4, that f1 ∗γ f2 = F−1
γ (Fγf1Fγf2). This achieves the proof
of Lemma 4.7. �
A combination of Lemma 4.7 and Theorem 2.3 gives us the following.
Lemma 4.8. Let f1, f2 ∈ L2(R, |x|2γ+1dx). Then∫
R
|f1 ∗γ f2(x)|2|x|2γ+1dx = mγ
∫
R
|Fγf1(λ)|2|Fγf2(λ)|2|λ|2γ+1dλ,
where both sides are finite or infinite.
8 M.A. Mourou
Lemma 4.9. Let g ∈ L2(R, |x|2γ+1dx) be a Dunkl wavelet of order γ such that Fγg ∈ L∞(R).
For 0 < ε < δ <∞ define
Gε,δ(x) :=
1
Cγ
g
∫ δ
ε
ga ∗γ g̃a(x)
da
a4γ+5
(4.2)
and
Kε,δ(λ) :=
1
Cγ
g
∫ δ
ε
|Fγg(aλ)|2da
a
. (4.3)
Then
Gε,δ ∈ L2(R, |x|2γ+1dx), Kε,δ ∈
(
L1 ∩ L2
)
(R, |x|2γ+1dx), (4.4)
and
Fγ(Gε,δ) = Kε,δ.
Proof. Using Schwarz inequality for the measure
da
a4γ+5
we obtain
|Gε,δ(x)|2 ≤
1
(Cγ
g )2
(∫ δ
ε
da
a4γ+5
)∫ δ
ε
|ga ∗γ g̃a(x)|2
da
a4γ+5
,
so ∫
R
|Gε,δ(x)|2|x|2γ+1dx ≤ 1
(Cγ
g )2
(∫ δ
ε
da
a4γ+5
)∫ δ
ε
∫
R
|ga ∗γ g̃a(x)|2|x|2γ+1dx
da
a4γ+5
.
By Theorem 2.3, Lemma 4.8, and Remark 4.4, we have∫
R
|ga ∗γ g̃a(x)|2|x|2γ+1dx = mγ
∫
R
|Fγ(ga)(λ)|4|λ|2γ+1dλ
≤ mγ ||Fγ(ga)||2∞
∫
R
|Fγ(ga)(λ)|2|λ|2γ+1dλ
= ||Fγ(ga)||2∞ ||ga||22,γ = a6γ+6 ||Fγg||2∞ ||g||
2
2,γ .
Hence∫
R
|Gε,δ(x)|2|x|2γ+1dx ≤
||Fγg||2∞ ||g||
2
2,γ
(Cγ
g )2
(∫ δ
ε
a2γ+1da
)(∫ δ
ε
da
a4γ+5
)
<∞.
The second assertion in (4.4) is easily checked. Let us calculate Fγ(Gε,δ). Fix x ∈ R. From
Theorem 2.3 and Lemma 4.7 we get
ga ∗γ g̃a(x) = mγ
∫
R
|Fγ(ga)(λ)|2eγ(iλx)|λ|2γ+1dλ,
so
Gε,δ(x) =
mγ
Cγ
g
∫ δ
ε
(∫
R
|Fγ(ga)(λ)|2eγ(iλx)|λ|2γ+1dλ
)
da
a4γ+5
.
As |eγ(iz)| ≤ 1 for all z ∈ R (see [12]), we deduce by Theorem 2.3 that
mγ
∫ δ
ε
∫
R
|Fγ(ga)(λ)|2|eγ(iλx)||λ|2γ+1dλ
da
a4γ+5
Inversion of the Dual Dunkl–Sonine Transform on R 9
≤
∫ δ
ε
||ga||22,γ
da
a4γ+5
= ||g||22,γ
∫ δ
ε
da
a2γ+3
<∞.
Hence, applying Fubini’s theorem, we find that
Gε,δ(x) = mγ
∫
R
(
1
Cγ
g
∫ δ
ε
|Fγg(aλ)|2da
a
)
eγ(iλx)|λ|2γ+1dλ
= mγ
∫
R
Kε,δ(λ)eγ(iλx)|λ|2γ+1dλ
which completes the proof. �
We can now state the main result of this section.
Theorem 4.10 (Calderón’s formula). Let g ∈ L2(R, |x|2γ+1dx) be a Dunkl wavelet of order
γ such that Fγg ∈ L∞(R). Then for f ∈ L2(R, |x|2γ+1dx) and 0 < ε < δ <∞, the function
fε,δ(x) :=
1
Cγ
g
∫ δ
ε
∫
R
Φγ
g (f)(a, b)ga,b(x)|b|2γ+1db
da
a
belongs to L2(R, |x|2γ+1dx) and satisfies
lim
ε→0, δ→∞
∥∥fε,δ − f
∥∥
2,γ
= 0. (4.5)
Proof. It is easily seen that
fε,δ = f ∗γ Gε,δ,
where Gε,δ is given by (4.2). It follows by Lemmas 4.7 and 4.9 that fε,δ ∈ L2(R, |x|2γ+1dx) and
Fγ(fε,δ) = Fγ(f)Kε,δ, where Kε,δ is as in (4.3). From this and Theorem 2.3 we obtain
∥∥fε,δ − f
∥∥2
2,γ
= mγ
∫
R
|Fγ(fε,δ − f)(λ)|2|λ|2γ+1dλ
= mγ
∫
R
|Fγf(λ)|2(1−Kε,δ(λ))2|λ|2γ+1dλ.
But by (4.1) we have
lim
ε→0, δ→∞
Kε,δ(λ) = 1, for almost all λ ∈ R.
So (4.5) follows from the dominated convergence theorem. �
Another pointwise inversion formula for the Dunkl wavelet transform, proved in [5], is as
follows.
Theorem 4.11. Let g ∈ L2(R, |x|2γ+1dx) be a Dunkl wavelet of order γ. If both f and Fγf are
in L1(R, |x|2γ+1dx) then we have
f(x) =
1
Cγ
g
∫ ∞
0
(∫
R
Φγ
g (f)(a, b)gγ
a,b(x)|b|
2γ+1db
)
da
a
, a.e.,
where, for each x ∈ R, both the inner integral and the outer integral are absolutely convergent,
but possibly not the double integral.
10 M.A. Mourou
5 Inversion of the dual Dunkl–Sonine transform
using Dunkl wavelets
From now on assume β > α > −1/2. In order to invert the dual Dunkl–Sonine transform, we
need the following two technical lemmas.
Lemma 5.1. Let 0 6= g ∈ L1∩L2(R, |x|2α+1dx) such that Fαg ∈ L1(R, |x|2α+1dx) and satisfying
∃ η > β − 2α− 1 such that Fαg(λ) = O (|λ|η) as λ→ 0. (5.1)
Then Xα,βg ∈ L2(R, |x|2β+1dx) and
Fβ(Xα,βg)(λ) =
mα
mβ
Fαg(λ)
|λ|2(β−α)
.
Proof. By Theorem 2.3 we have
g(x) = mα
∫
R
Fαg(λ)eα(iλx)|λ|2α+1dλ, a.e.
So using (3.6), we find that
Xα,βg(x) = mβ
∫
R
hα,β(λ)eβ(iλx)|λ|2β+1dλ, a.e. (5.2)
with
hα,β(λ) :=
mα
mβ
Fαg(λ)
|λ|2(β−α)
.
Clearly, hα,β ∈ L1(R, |x|2β+1dx). So it suffices, in view of (5.2) and Theorem 2.3, to prove that
hα,β belongs to L2(R, |x|2β+1dx). We have∫
R
|hα,β(λ)|2|λ|2β+1dλ =
(
mα
mβ
)2 ∫
R
|λ|4α−2β+1|Fαg(λ)|2dλ
=
(
mα
mβ
)2
(∫
|λ|≤1
+
∫
|λ|≥1
)
|λ|4α−2β+1|Fαg(λ)|2dλ := I1 + I2.
By (5.1) there is a positive constant k such that
I1 ≤ k
∫
|λ|≤1
|λ|2η+4α−2β+1dλ =
k
η + 2α− β + 1
<∞.
From Theorem 2.3, it follows that
I2 =
(
mα
mβ
)2 ∫
|λ|≥1
|λ|2(α−β)|Fαg(λ)|2|λ|2α+1dλ
≤
(
mα
mβ
)2 ∫
|λ|≥1
|Fαg(λ)|2|λ|2α+1dλ ≤
(
mα
mβ
)2
||Fαg||22,α =
mα
(mβ)2
||g||22,α <∞
which ends the proof. �
Inversion of the Dual Dunkl–Sonine Transform on R 11
Lemma 5.2. Let 0 6= g ∈L1 ∩L2(R, |x|2α+1dx) be real-valued such that Fαg ∈L1(R, |x|2α+1dx)
and satisfying
∃ η > 2(β − α) such that Fαg(λ) = O(λη) as λ→ 0+. (5.3)
Then Xα,βg ∈ L2(R, |x|2β+1dx) is a Dunkl wavelet of order β and Fβ(Xα,βg) ∈ L∞(R).
Proof. By combining (5.3) and Lemma 5.1 we see that Xα,βg ∈ L2(R, |x|2β+1dx), Fβ(Xα,βg) is
bounded and
Fβ(Xα,βg)(λ) = O
(
λη−2(β−α)
)
as λ→ 0+.
Thus, in view of Remark 4.2, Xα,βg satisfies the admissibility condition (4.1) for γ = β. �
Remark 5.3. In view of Remark 4.2, each function satisfying the conditions of Lemma 5.1 is
a Dunkl wavelet of order α.
Lemma 5.4. Let g be as in Lemma 5.2. Then for all f ∈ L1(R, |x|2β+1dx) we have
Φβ
Xα,βg(f)(a, b) =
1
a2(β−α)
Xα,β
[
Φα
g
(
tXα,βf
)
(a, ·)
]
(b).
Proof. By Definition 4.5 we have
Φβ
Xα,βg(f)(a, b) =
1
a2β+2
f ∗β
˜(Xα,βg)a(b).
But ˜(Xα,βg)a = Xα,β (g̃a) by virtue of (1.2) and (3.5). So using (3.8) we find that
Φβ
Xα,βg(f)(a, b) =
1
a2β+2
f ∗β [Xα,β (g̃a)] (b)
=
1
a2β+2
Xα,β
[
tXα,βf ∗α g̃a
]
(b) =
1
a2(β−α)
Xα,β
[
Φα
g
(
tXα,βf
)
(a, ·)
]
(b),
which gives the desired result. �
Combining Theorems 4.10, 4.11 with Lemmas 5.2, 5.4 we get
Theorem 5.5. Let g be as in Lemma 5.2. Then we have the following inversion formulas for
the dual Dunkl–Sonine transform:
(i) If both f and Fβf are in L1(R, |x|2β+1dx) then for almost all x ∈ R we have
f(x) =
1
Cβ
Xα,βg
∫ ∞
0
(∫
R
Xα,β
[
Φα
g
(
tXα,βf
)
(a, ·)
]
(b)
(
Xα,βg
)β
a,b
(x)|b|2β+1db
)
da
a2(β−α)+1
.
(ii) For f ∈ L1 ∩ L2(R, |x|2β+1dx) and 0 < ε < δ <∞, the function
fε,δ(x) :=
1
Cβ
Xα,βg
∫ δ
ε
∫
R
Xα,β
[
Φα
g
(
tXα,βf
)
(a, ·)
]
(b)
(
Xα,βg
)β
a,b
(x)|b|2β+1db
da
a2(β−α)+1
satisfies
lim
ε→0, δ→∞
∥∥fε,δ − f
∥∥
2,β
= 0.
Acknowledgements
The author is grateful to the referees and editors for careful reading and useful comments.
12 M.A. Mourou
References
[1] de Jeu M.F.E., The Dunkl transform, Invent. Math. 113 (1993), 147–162.
[2] Dunkl C.F., Differential-difference operators associated with reflections groups, Trans. Amer. Math. Soc.
311 (1989), 167–183.
[3] Dunkl C.F., Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), 1213–1227.
[4] Dunkl C.F., Hankel transforms associated to finite reflection groups, Contemp. Math. 138 (1992), 123–138.
[5] Jouini A., Dunkl wavelets and applications to inversion of the Dunkl intertwining operator and its dual, Int.
J. Math. Math. Sci. 6 (2004), 285–293.
[6] Kamefuchi S., Ohnuki Y., Quantum field theory and parastatistics, Springer-Verlag, Berlin, 1982.
[7] Mourou M.A., Trimèche K., Calderón’s formula associated with a differential operator on (0,∞) and inver-
sion of the generalized Abel transform, J. Fourier Anal. Appl. 4 (1998), 229–245.
[8] Mourou M.A., Trimèche K., Inversion of the Weyl integral transform and the Radon transform on Rn using
generalized wavelets, Monatsh. Math. 126 (1998), 73–83.
[9] Mourou M.A., Trimèche K., Calderon’s reproducing formula related to the Dunkl operator on the real line,
Monatsh. Math. 136 (2002), 47–65.
[10] Mourou M.A., Trimèche K., Transmutation operators and Paley–Wiener associated with a singular
differential-difference operator on the real line, Anal. Appl. (Singap.) 1 (2003), 43–70.
[11] Rosenblum M., Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint
Operators and Related Topics (Beer Sheva, 1992), Oper. Theory Adv. Appl., Vol. 73, Birkhäuser, Basel,
1994, 369–396, math.CA/9307224.
[12] Rösler M., Bessel-type signed hypergroups on R, in Probability Measures on Groups and Related Struc-
tures, XI (Oberwolfach, 1994), Editors H. Heyer and A. Mukherjea, Oberwolfach, 1994, World Sci. Publ.,
River Edge, NJ, 1995, 292–304.
[13] Soltani F., Lp-Fourier multipliers for the Dunkl operator on the real line, J. Funct. Anal. 209 (2004), 16–35.
[14] Soltani F., Sonine transform associated to the Dunkl kernel on the real line, SIGMA 4 (2008), 092, 14 pages,
arXiv:0812.4666.
[15] Xu Y., An integral formula for generalized Gegenbauer polynomials and Jacobi polynomials, Adv. in Appl.
Math. 29 (2002), 328–343.
[16] Yang L.M., A note on the quantum rule of the harmonic oscillator, Phys. Rev. 84 (1951), 788–790.
http://arxiv.org/abs/math.CA/9307224
http://arxiv.org/abs/0812.4666
1 Introduction
2 Preliminaries
3 The dual Dunkl-Sonine integral transform
4 Calderón's formula for the Dunkl continuous wavelet transform
5 Inversion of the dual Dunkl-Sonine transform using Dunkl wavelets
References
|