Inversion Formulas
for the Spherical Radon–Dunkl Transform
⋆Zhongkai LI and Futao SONG
Department of Mathematics, Capital Normal University, Beijing 100048, China E-mail: [email protected],[email protected]
Received October 18, 2008, in final form March 01, 2009; Published online March 03, 2009 doi:10.3842/SIGMA.2009.025
Abstract. The spherical Radon–Dunkl transform Rκ, associated to weight functions in-variant under a finite reflection group, is introduced, and some elementary properties are obtained in terms of h-harmonics. Several inversion formulas of Rκ are given with the aid of spherical Riesz–Dunkl potentials, the Dunkl operators, and some appropriate wavelet transforms.
Key words: spherical Radon–Dunkl transform; h-harmonics; inversion formula; wavelet
2000 Mathematics Subject Classification: 44A12; 33C55; 65R32; 42C40
1
Introduction
Lethx, yi denote the usual Euclidean inner product ofx, y∈Rd+1, and Sd={x: kxk= 1} the unit sphere in Rd+1. We usedωk to denote the surface (Lebesgue) measure on ak-dimensional sphere. The spherical Radon transform R is one of the tools in integral geometry, which is defined, for f ∈C(Sd), by
Rf(x) = Λd−1
Z
hx,yi=0
f(y)dωd−1(y), x∈Sd,
where Λ−d−11 is the surface area of Sd−1. There are a number of papers devoting to the study of the spherical Radon transformR by different methods (see [1,13,14,15,16,17,21,22,23,24,
25, 26, 27, 28]), and to its applications to various problems (see [10, 11]). Some deep results about R were obtained with the aid of spherical harmonics (see [21,22,23,24,25,26,27,28]), and furthermore,R is a special case of the spherical means
Mτf(x) =
Λd−1
(1−τ2)(d−1)/2
Z
hx,yi=τ
f(y)dωd−1(y), x∈Sd,
by takingτ = 0, and also of the spherical Riesz potentials
Iαf(x) = Γ((1−α)/2) 2πd/2Γ(α/2)
Z
Sd|h
x, yi|α−1f(y)dωd(y), x∈Sd,
by taking the limit lim α→0+I
αf =Rfin some sense (see [22]). The former is a tool in approximation
on the sphereSd, and the later is one of the research objectives in harmonic analysis onSd. The purpose of the present paper is to study an analogous model of the spherical Radon transform R in Dunkl’s theory. This is based on the definition of the generalized spherical
means Mτκf(x) due to [33] (instead ofMτκ, the notation Tθκ withτ = cosθ was used there), in terms of the equation
Z 1
−1
Mτκf(x)g(τ)wλκ(τ)dτ =cκ
Z
Sd
f(y)Vκ[g(hx,·i)](y)h2κ(y)dωd(y)
for anyginL1([−1,1];wλκ), wherewλκ(t) = ˜cλκ+1/2(1−t2)λκ−1/2,Vκ is the intertwining
opera-tor associated to a given finite reflection group, andh2κis the related weight function (for details concerning them and other notations in the equation, see the next section). We define Rκ by Rκf =M0κf, and call Rκ the spherical Radon–Dunkl transform. Although Mτκ is defined im-plicitly, it is a proper extension ofMτ andMτ0=Mτ, and moreover, from [2] and [33,34,35,36], Mκ
τ shares many properties with Mτ and plays the same roles in weighted approximation and related harmonic analysis on the sphere Sd. One could expect that the spherical Radon–Dunkl TransformRκ would have similar features toR and be a suitable tool in reconstruction of func-tions in weighted spaces. This is the motivation of the paper. Despite less closed representation, a further work worth doing is to find applications of Rκ in geometry or other fields.
The paper is organized as follows. In Section2, some necessary facts in Dunkl’s theory are reviewed, and in Section 3, the spherical Radon–Dunkl transform Rκ is defined and some of elementary properties are obtained in terms of h-harmonics. Sections 4 and 5 are devoted to inversion formulas of Rκ, which are given by means of spherical Riesz–Dunkl potentials Iκα, the Dunkl operators, and some appropriate wavelet transforms. These conclusions generalize part of those in [21,22,23].
2
Some facts in Dunkl’s theory
LetG be a finite reflection group onRd+1 with a fixed positive root system R+, normalized so that hv, vi = 2 for allv∈R+. It is known that G is a subgroup ofO(d+ 1) generated by {σv : v ∈ R+}, where σv denotes the reflection with respect to the hyperplane perpendicular to v, i.e. xσv =x−2(hx, vi/hv, vi)v forx∈Rd+1. Letκ be a multiplicity functionv7→κv ∈[0,+∞) defined on R+, with invariance under the action ofG. Thus{κv : v ∈R+}has different values only as many as the number ofG-orbits inR+.
The Dunkl operators are a family of first-order differential-reflection operatorsDj, 1≤j≤ d+ 1, defined by (see [5])
Djf(x) :=∂jf(x) +
X
v∈R+ κv
f(x)−f(xσv)
hx, vi hv, eji,
forf ∈C1(Rd+1),where{ei : 1≤i≤d+1}is the usual standard basis ofRd+1. As substitutes of partial differentiations∂j, these operators are mutually commutative. The associated Laplacian, calledh-Laplacian, is defined by ∆h=D12+· · ·+D2d+1, which plays roles similar to that of the usual Laplacian ∆ = ∆0 (see [4]). In terms of the polarspherical coordinatesx =rx′,r =kxk,
the operator ∆h can be expressed as (see [33])
∆h= ∂2
∂r2 +
2λκ+ 1 r
∂ ∂r +
1 r2∆h,0,
where ∆h,0 is the associated Laplace-Beltrami operator on Sd, and λκ = γκ + (d−1)/2 with γκ = P
v∈R+
κv. If γκ = 0, i.e. κv ≡0, then Dj =∂j, 1≤j ≤d+ 1. In the following, we assume
that γκ >0, and soλκ>0.
uniquely by VκPn ⊆ Pn, Vκ1 = 1 and DjVκ = Vκ∂j, 1 ≤ i ≤ d+ 1. Vκ commutes with the group action and is a linear isomorphism on each Pn (n = 0,1, . . .). Moreover it is a positive operator (see [18]) and can be extended to the space of smooth functions and even to the space of distributions (see [29,30]).
The intertwining operator Vκ allows to introduce some useful tools in Dunkl’s theory. For example, the Dunkl transform Fκ is defined in [7] associated with the measure h2κdx on Rd+1, where
hκ(x) = Y v∈R+
|hx, vi|κv.
Fκ is a generalization of the Fourier transform F =F0 and enjoys properties similar to those of F (see [3,7,19]). the spherical h-harmonics of degree n are their restrictions on Sd. The orthogonality theorem in [4] asserts that if P ∈ Pd+1
withCnλκ, the Gegenbauer polynomial of degree nwith parameterλκ. It is noted that (see [33])
∆h,0Yn=−n(n+ 2λκ)Yn, Yn∈ Hh,dn +1. (4)
When κv = 0 for all v ∈R+, we have V0 =id, and hence, Pn(h2κ;x, y) reduces to the usual zonal polynomial for the ordinary spherical harmonicsPn(x, y) = n+((d−d−1)1)/2/2Cn(d−1)/2(hx, yi).
A useful integration formula for the intertwining operatorVκ is
Z
on the sphere Sd which are restrictions of functions in L1(Bd+1; (1− |x|2)γκ−1), and moreover, the formula (5) is true for these functions too and Vκf ∈ L1(Sd;h2κ). In particular, if φ ∈ L1([−1,1];wλκ), then for eachy∈Sd,
Z
Bd+1
φ(hx, yi) 1− |x|2γκ−1
dx= π
(d+1)/2Γ(γ
κ) Γ(λκ+ 1)
Z 1
−1
φ(t)wλκ(t)dt, (6)
i.e. f(x) = φ(hx, yi) ∈ L1(Bd+1; (1− |x|2)γκ−1), and hence Vκ[φ(h·, yi)] is well defined and in L1(Sd;h2κ). In addition, we have the following symmetric relation
Vκ[φ(h·, yi)](x) =Vκ[φ(hx,·i)](y), for a.e. (x, y)∈Sd×Sd. (7)
The validity of (7) for polynomials and for all (x, y)∈Sd×Sdfollows from Gegenbauer expansions and the symmetry of the reproducing kernels Pn(h2κ;x, y), If φ ∈ L1([−1,1];wλκ) and φ1 is
a univariate polynomial, then applying (5) and (6),
Z
Sd
Z
Sd|
Vκ[φ(h·, yi)](x)−Vκ[φ(hx,·i)](y)|h2κ(x)h2κ(y)dωd(x)dωd(y)
=
Z
Sd
Z
Sd|
Vκ[(φ−φ1)(h·, yi)](x)−Vκ[(φ−φ1)(hx,·i)](y)|hκ2(x)h2κ(y)dωd(x)dωd(y)
≤2c−κ2
Z 1
−1|
φ(t)−φ1(t)|wλκ(t)dt,
which implies (7) by the density of polynomials inL1([−1,1];w
λκ). Following the above remarks, the Funk–Hecke formula for h-harmonics proved in [32] (for continuous functions there only) holds also for φ∈L1([−1,1];wλκ), that is
cκ
Z
Sd
Vκ[φ(h·, yi)](x)Hn(x)hκ2(x)dωd(x) =Ln(φ)Hn(y) (8)
for each Hn∈ Hh,dn +1 and y∈Sd, where
Ln(φ) =
Z 1
−1
φ(t)C λκ n (t) Cnλκ(1)
wλκ(t)dt. (9)
The convolution f∗κφ of two functions f ∈ L1(Sd;h2κ) and φ ∈ L1([−1,1];wλκ) is defined in [34], by
f ∗κφ(x) =cκ
Z
Sd
f(y)Vκ[φ(hx,·i)](y)h2κ(y)dωd(y). (10)
The Young inequality concerning such convolution is proved in [34], that is, forp, q, r ≥1 with r−1 =p−1+q−1−1,
kf ∗κφkκ,r≤ kfkκ,pkφkλκ,q. (11)
A typical example of Dunkl’s theory is the case whenG=Z2d+1, for which, the functionhκ(x) has the formhκ(x) =|x1|κ1· · · |xd+1|κd+1 and the intertwining operator Vκ is given by
Vκf(x) = ˜cκ
Z
[−1,1]d+1
f(x1t1, . . . , xd+1td+1)
d+1
Y
i=1
(1 +ti)(1−t2i)κi−1dt1· · ·dtd+1, (12)
3
The spherical Radon–Dunkl transform
Forf ∈L1(Sd;h2
κ), its generalized spherical meansMτκf(x) due to [33] is defined by the equation
Z 1
−1
Mτκf(x)φ(τ)wλκ(τ)dτ =f ∗κφ(x) (13)
for anyφin L1([−1,1];wλκ). Since, forφ∈L∞(Sd),
Z 1
−1
Mτκf(x)φ(τ)wλκ(τ)dτ
≤ k
fkκ,1kφk∞,
it follows that, for eachx∈Sd, the functionψx(τ) =Mτκf(x)∈L1([−1,1];wλκ). This shows that for almost allτ ∈[−1,1], Mκ
τf is well defined. To give further illustration ofMτκ, we introduce the space Wm(p Sd;hκ2)(⊆ Lp(Sd;h2κ)) of functions for m ≥ 0, such that for f ∈ Wm(p Sd;h2κ), there exist some g∈Lp(Sd;h2κ) satisfying Y0(h2κ;f) =Y0(h2κ;g) and [n(n+ 2λκ)]m/2Yn(h2κ;f) = Yn(h2κ;g) for all n= 1,2, . . .. In view of (4), we formally write g = (−∆h,0)m/2f. It is noted
that for evenm,Cm(Sd)⊆Wm(p Sd;h2
κ) (1≤p≤ ∞). The following properties ofMτκare proved in [33,34].
Proposition 1.
(i) If f0(x)≡1, then Mτκf0(x)≡1.
(ii) For each τ ∈ [−1,1], there is an extension of Mτκ to Lp(Sd;h2κ) (1 ≤ p < ∞), or C(Sd) (p=∞), such that
kMτκfkκ,p≤ kfkκ,p, τ ∈[−1,1].
(iii) For f ∈L1(Sd;h2
κ),
Yn(h2κ;Mτκf;x) =
Cnλκ(τ) Cnλκ(1)
Yn h2κ;f;x
,
and in particular, ∆h,0(Mτκf) =Mτκ(∆h,0f) if ∆h,0f ∈L1(Sd;h2κ).
Proof . Here we give an independent, but simpler proof for part (ii) as follows. For all φ ∈
L1([−1,1];w
λκ) and g∈Lp
′
(Sd;h2
κ), it follows from (13) that
Z 1
−1
φ(τ)ξ(τ)wλκ(τ)dτ =cκ
Z
Sd
(f∗κφ)(x)g(x)h2κ(x)dωd(x),
whereξ(τ) =cκRSdMτκf(x)·g(x)h2κ(x)dωd(x). By using (7) and (10), the right-hand side above becomes cκRSdf(y)(g∗κ φ)(y)h2κ(y)dωd(y), and then, by applying H¨older’s inequality and the Young inequality (11), its absolute value is dominated by kfkκ,pkgkκ,p′kφkλκ,1. This gives that
supτ∈[−1,1]|ξ(τ)| ≤ kfkκ,pkgkκ,p′, which means that, for almost all τ ∈ [−1,1], kMτκfkκ,p ≤ kfkκ,p. If f is a polynomial, part (iii) implies thatMτκf(x) is a continuous function of (τ, x)∈ [−1,1]×Sd, so that kMκ
τfkκ,p ≤ kfkκ,p is true for all τ ∈ [−1,1] in this case. Finally, from density of the set of polynomials, for each τ ∈ [−1,1], Mτκ can be extended to all functions in Lp(Sd;h2
κ) (1 ≤p <∞), or C(Sd). Following this, part (iii) also holds for f ∈L1(Sd;h2κ) and each τ ∈[−1,1], and moreover,
Mτκf ∼
∞
X
n=0
Cλκ n (τ) Cnλκ(1)
Yn(h2κ;f;x). (14)
The following proposition gives a pointwise description ofMτκ for a larger class of functions.
Proposition 2. For f ∈ W2
m(Sd;h2κ) with m > λκ + 1, Mτκf(x) is a continuous function of (τ, x)∈[−1,1]×Sd and
Mτκf(x) =
∞
X
n=0
Cnλκ(τ) Cnλκ(1)
Yn h2κ;f;x
,
the series on the right-hand side being absolutely and uniformly convergent.
Proof . It is noted that for t ∈ [−1,1], |Cλκ
n (t)| ≤ Cnλκ(1) = (2λκ)n/n! ≃ n2λκ−1 [8, p. 19]. From (2), for g ∈ L2(Sd;h2κ) and all x ∈ Sd we have |Yn(h2κ;g;x)| ≤ kgkκ,2kPn(hκ;x,·)kκ,2. In view of orthogonality ofh-harmonics and from (3),
kPn(hκ;x,·)k2κ,2 =Pn(hκ;x, x)≤λ−κ1(n+λκ)Cnλκ(1)≃λ−κ1n2λκ.
Therefore |Yn(h2κ;g;x)| ≤ckgkκ,2nλκ. If f ∈Wm2(Sd;h2κ), then forn≥1, Yn(h2κ;f;x) = [n(n+ 2λκ)]−m/2Yn(h2κ; (−∆h,0)m/2f;x), so that |Yn(h2κ;f;x)| ≤ ck(−∆h,0)m/2fkκ,2nλκ−m. Hence,
whenm > λκ+1, the series in (14) converges absolutely and uniformly for (τ, x)∈[−1,1]×Sd. In view of the uniqueness ofh-harmonic expansion following from its Ces`aro summability (see [31]),
the conclusions in the proposition are proved.
Now we define the transformRκ by
Rκf =M0κf,
and call Rκ the spherical Radon–Dunkl transform. By Proposition 1 (ii), Rκf is well defined forf ∈L1(Sd;h2
κ), and moreover, from Propositions1 and 2, we have the following corollary.
Corollary 1.
(i) For f ∈Lp(Sd;h2κ) (1≤p <∞), or f ∈C(Sd) (p=∞), we have kRκfkκ,p≤ kfkκ,p, and
Rκf ∼
∞
X
n=0
bnYn h2κ;f;x
, (15)
where
bn=
(
(−1)n2 Γ(λκ+1/2) Γ(1/2)
Γ((n+1)/2)
Γ(λκ+(n+1)/2), for n even;
0, for n odd. (16)
(ii) For f ∈Wm2(Sd;hκ2) withm > λκ+ 1, Rκf(x) is a continuous function on Sd and
Rκf(x) =
∞
X
n=0
bnYn h2κ;f;x
,
where the series on the right-hand side is absolutely and uniformly convergent. The numbersbn=Cnλκ(0)/Cnλκ(1) are computed by using 10-9(3) and 10-9(19) in [9]. The following is a nontrivial example of Rκ. We consider the group G = Z2d+1, with κ =
(κ1,0, . . . ,0) and κ1 >0. In this case, hκ(x) = |x1|κ1 and the intertwining operator Vκ in (12) reduces to
Vκf(x) = ˜cκ1
Z 1
−1
f(x1t,x)(1 +˜ t) 1−t2
κ1−1 dt,
We shall show that, forf ∈C(Sd) and for x1 6= 0,
4
Inversion formulas for
R
κby means of spherical Riesz–Dunkl potentials
Forf ∈L1(Sd;h2 moreover, we have the following statements:
The conclusions in the proposition are contained in Proposition 2.9 of [34]. Here we give a short presentation. Since Iα
κf =cf∗κφ withφ(t) =|t|α−1∈L1([−1,1];wλκ), part (i) follows properties of the gamma function, to get the stated value of bn,α.
It is easy to see that (19) and (20) allow us to extend the family{Iκα : ℜα >0, α6= 1,3,5, . . .}
To go further, forr∈Z+ (nonnegative integers) we define
Pr,α(∆h,0) =
the identity operator, r= 0,
Lemma 1. Ifα∈Π, and r∈Z+ such that 2r−2λκ−α∈Π, then for even nandYn∈ Hh, dn +1, Pr,α(∆h,0)Iκ2r−2λκ−αIκαYn=Yn.
Proof . From (19) and (20), we have
Iκ2r−2λκ−αIκαYn=
Γ((n+ 2λκ−2r+α+ 1)/2)Γ((n+ 1−α)/2) Γ((n+ 2r−α+ 1)/2)Γ((n+ 2λκ+α+ 1)/2)Yn.
Furthermore, from (4),
Pr,α(∆h,0)Yn= r
Y
j=1
n+ 2λκ+α−1
2 −r+j
n+ 1−α
2 +r−j
Yn.
Using the properties of Γ-functions, the result is obtained.
The following theorem is a direct consequence of the above lemma.
Theorem 2. If α ∈Π, and r ∈Z+ such that 2r−2λκ−α ∈Π and r ≥λκ+ℜα/2, then for
even f ∈C∞(Sd) and g=Iα
κf, we have the inversion formula f =Pr,α(∆h,0)Iκ2r−2λκ−αg.
Now we turn to the inversion problem of the spherical Radon–Dunkl transform Rκ. From (15), (16), (20) and (22), we see that
Rκf =π−1/2Γ(λκ+ 1/2)Iκ0f. (23)
This consistency can be also seen from the following equalities
Γ(λκ+ 1)Γ(α/2)
√
πΓ((1−α))/2I α
κf(x) =f ∗κφ=
Z 1
−1
Mτκf(x)φ(τ)wλκ(τ)dτ (24)
in view of (13) and (18), where φ(t) = |t|α−1 ∈L1([−1,1];w
λκ). Assume that f ∈Wm2(Sd;h2κ) with m > λκ + 1. By Proposition 2, for each x ∈ Sd, Mκ
τf(x) is a continuous function of τ ∈[−1,1]. Dividing each part of (24) by Γ(α/2) and taking limit for α → 0+, we regain the relation (23).
From Theorems 1 and 2, we obtain the inversion formulas for the spherical Radon–Dunkl transformRκ.
Theorem 3. Rκ is an isomorphism betweenWm,e2 (Sd;h2κ) andWm2+λκ,e(Sd;h2κ) withm≥0, and
R−κ1=
√
π Γ(λκ+ 1/2)
Iκ−2λκ.
Theorem 4. If r ∈ Z+ such that 2r−2λκ ∈ Π and r ≥ λκ, then for even f ∈ C∞(Sd) and
g=Rκf, we have the inversion formula
f =
√
π Γ(λκ+ 1/2)
Pr,0(∆h,0)Iκ2r−2λκg.
Corollary 2. If λκ is a positive integer, then an evenf ∈C∞(Sd) can be recovered by (i) f =c′Pr,0(∆h,0)RκRκf,
with r=λκ, c′ =π/Γ(λκ+ 1/2)2, and
Pr,0(∆h,0) = 4−r
r
Y
j=1
[−∆h,0+ (2j−1)(2r−2j+ 1)];
and
(ii) f =c′′Pr,0(∆h,0)
Z
Sd
Rκf(y)Vκ(|hx,·i|)(y)h2κ(y)dωd(y)
,
with r=λκ+ 1, c′′=−2π3/2cκ/[Γ(λκ+ 1)Γ(λκ+ 1/2)2], and
Pr,0(∆h,0) = 4−r
r
Y
j=1
[−∆h,0+ (2j−3)(2r−2j+ 1)].
5
Inversion formulas for
R
κby means of associated wavelets
In this section, we shall use, for a suitably chosenψdefined on [0,∞), the wavelet-like transform
Wκf(t, x) =f∗κψt(x), ψt(τ) =t−1ψ(τ /t), (25)
for (t, x) ∈(0,∞)×Sd, to present the inverse of the spherical Radon–Dunkl transform Rκ and itself. Although Rκ is defined implicity and the intertwining operator Vκ is involved in the definition ofWκ, the approaches in studying the usual spherical Radon transform (see [21], for example) could be transplanted to Rκ.
The first lemma below reveals a relation of the spherical Radon–Dunkl transform Rκ with the one-dimensional fractional integral, and the second gives a representation of the successive action ofRκandWκto a function. We shall use a modified notation of the fractional integral as
Bδφ(u) = 2 Γ(δ)
Z √u
0
φ(v) u−v2δ−1
dv, u >0, (26)
forδ >0, which will simplify some expressions.
Lemma 2. For even function f ∈L1(Sd;h2κ) and 0< s <1, we have
Msκ(Rκf) = λκπ−1 wλκ(s)
Bλκ(Mτκf) 1−s2
, (27)
where the action of Bλκ to Mτκf is associated with τ-variable.
Proof . From the product formula of the Gegenbauer polynomialC2λκn (see [8, p. 203]), we have
C2λκn(s) C2λκn(1)
C2λκn(0) C2λκn(1) = 2
Z 1
0
C2λκn(u√1−s2)
C2λκn(1) wλκ−1/2(u)du.
By Proposition1(iii), the three quotients above are the coefficients of a memberY2n inHh,d2n+1 under action ofMκ
s,Rκ(=M0κ), andMuκ√1−s2, respectively. Therefore,
Msκ(RκY2n) = 2
Z 1
0
Muκ√
1−s2Y2n
wλκ−1/2(u)du.
Making substitution of variables u=v/√1−s2, (27) is proved for Y
2n. By Proposition 1 (ii), both sides of (27) are bounded operators inL1(Sd;h2
Lemma 3. For even f ∈L1(Sd;h2κ) and ψ∈L1([0,∞);dx), we have
provided the integral on the right-hand side exists with |f| and |ψ| instead of f, ψ.
Proof . From (13), (25) and (27), we have
and then, substituting the formula for Bλκ(Mτκf) from (26), and making changes of variables,
we prove the equality in (28).
where the constant c is independent of ǫ. The key step is to rewrite Tǫ into a convolution operator with an approximate identity, that is,
Indeed, applying Lemma3 to (32) gives that
In order to proveTǫf to be convergent almost everywhere, we need the associated maximal functionT∗f(x) = sup
0<ǫ≤1|
Tǫf(x)|. We shall show thatT∗f is dominated by the maximal function
introduced in [35]
which implies the following estimate forKǫ(cosθ)
Kǫ(cosθ) =O(mǫ(θ)), mǫ(θ) =
ǫ2ρ(sinθ)−1 (ǫ+ sinθ)2λκ+2ρ,
From (13) and (35),
|Tǫf(x)| ≤c
Z π/2
0
(Mcosκ θ|f|)(x)mǫ(θ)(sinθ)2λκdθ, (40)
where the evenness ofMτκf is used. Splitting the interval [0, π/2] intoS
j[2jǫ,2j+1ǫ], we evaluate each integral Uj =R2
j+1ǫ
2jǫ separately. Forj≤0, since ǫ+ sinθ≍ǫ, we have
Uj ≤ c2− j ǫ2λκ+1
Z 2j+1ǫ
0
(Mcosκ θ|f|)(x)(sinθ)2λκdθ≤c22λκjMκf(x);
and for j >0, sinceǫ+ sinθ≍θ,
Uj ≤
cǫ2ρ (2jǫ)2λκ+2ρ+1
Z 2j+1ǫ
0
(Mcosκ θ|f|)(x)(sinθ)2λκdθ ≤c2−2ρjMκf(x).
Collecting these estimates into (40) yields (39).
By Theorem 2.1 in [2], T∗ is of weak (1,1), and strong (p, p) boundedness. Combining with the uniformly convergence ofTǫ forh-harmonics, for general f ∈L1(Sd;h2κ),Tǫf converges tof almost everywhere. The proof of Theorem 5is completed.
In the following, we state two theorems, without proof, which are analogs of Theorems 1.2 and 1.4 in [21]. One is about the reproducing property of the spherical Radon–Dunkl trans-form Rκ, and the other illustrates the range Rκ(L1(Sd;h2κ)).
Theorem 6. Let
Z ∞
0
ψ(s)ds= 0,
Z ∞
0 |
ψ(s) logs|ds <∞.
Then for f ∈Lp(Sd;h2
κ) (1≤p <∞), or C(Sd) (p=∞), we have lim
ǫ→0+k
˜
Tǫf−Rκfkκ,p= 0,
where Tǫf˜ (x) = ¯Cψ−1R∞
ǫ t−1(Wκf)(t, x)dt (ǫ >0), withC¯ψ = 2cλκ
R∞
0 ψ(s) log1sds.
Theorem 7. Let ψ satisfy conditions (29) and (30), g ∈ Lp(Sd;h2
κ) (1 ≤ p < ∞), or C(Sd) (p=∞), and C˜φ6= 0 be the constant in (33). Then the following statements are equivalent:
(i) g∈Rκ(Lp(Sd;h2κ)); (ii) the integrals Sǫg=
R∞
ǫ t−2λκ−1(Wκg)(t, x)dt converge in theLp(Sd;h2κ)-norm. If 1< p <∞, then (i) and(ii) are equivalent to
(iii) sup ǫ>0k
Sǫgkκ,p <∞.
Acknowledgments
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