We study some class of Dunkl L2-multiplier operators; and related to these
operators we establish the Heisenberg-Pauli-Weyl uncertainty principle and Donoho-Stark’s uncertainty principle. We give also an application of the theory of reproducing
kernels to the Tikhonov regularization on the Sobolev-Dunkl spaces.
1. Introduction
In this paper, we consider Rd with the Euclidean inner product ·,· and norm |y|≔y,y. For α∈Rd∖{0}, let σα be the reflection in the hyperplane Hα⊂Rd orthogonal to α:
(1)σαx≔x-2α,xα2α.
A finite set R⊂Rd∖{0} is called a root system, if R∩R. α={-α,α} and σαR=R for all α∈R. We assume that it is normalized by |α|2=2 for all α∈R. For a root system R, the reflections σα, α∈R, generate a finite group G⊂O(d), the reflection group associated with R. All reflections in G correspond to suitable pairs of roots. For a given β∈Rd∖⋃α∈RHα, we fix the positive subsystem R+≔{α∈R:〈α,β〉>0}. Then for each α∈R either α∈R+ or -α∈R+.
Let k:R→C be a multiplicity function on R (i.e., a function which is constant on the orbits under the action of G). As an abbreviation, we introduce the index γ=γk≔∑α∈R+k(α).
Throughout this paper, we will assume that k(α)≥0 for all α∈R. Moreover, let wk denote the weight function wk(x)≔∏α∈R+|〈α,x〉|2k(α), for all x∈Rd, which is G-invariant and homogeneous of degree 2γ.
Let ck be the Mehta-type constant given by
(2)ck≔∫Rde-x2/2wkxdx-1.
We denote by μk the measure on Rd given by dμk(x)≔ckwk(x)dx and by Lp(μk), 1≤p≤∞, the space of measurable functions f on Rd, such that
(3)fLpμk≔∫Rdfxpdμkx1/p<∞,555555555555555555155555551≤p<∞,fL∞μk≔esssupx∈Rdfx<∞.
For f∈L1(μk) the Dunkl transform is defined (see [1]) by
(4)Fkfy≔∫RdEk-ix,yfxdμkx,555555555555555155555555555555y∈Rd,
where Ek(-ix,y) denotes the Dunkl kernel (for more details, see Section 2).
Many uncertainty principles have already been proved for the Dunkl transform, namely, by Rösler [2] and Shimeno [3] who established the Heisenberg-Pauli-Weyl inequality for the Dunkl transform, by showing that, for every f∈L2(μk),
(5)fL2μk2≤22γ+dxfL2μkyFkfL2μk.
Recently, the author [4, 5] proved general forms of the Heisenberg-Pauli-Weyl inequality for the Dunkl transform.
Let m be a function in L2(μk). The Dunkl L2-multiplier operators, Tk,m, are defined, for regular functions f on Rd, by
(6)Tk,mfa,x≔Fk-1maxFkfx,5555515555555555a,x∈0,∞×Rd.
These operators are studied in [6, 7] where the author established some applications (Calderón’s reproducing formulas, best approximation formulas, and extremal functions…).
For m∈L2(μk) satisfying the admissibility condition: ∫0∞|m(ax)|2(da/a)=1, a.e. x∈Rd, then the operators Tk,m satisfy Plancherel’s formula:
(7)Tk,mfL2Ωk2≔∫Rd∫0∞Tk,mfa,x2dΩka,x=fL2μk2,5555555555555555555555555555555555555555f∈L2μk,
where Ωk is the measure on ]0,∞[×Rd given by dΩk(a,x)≔(da/a)dμk(x).
For the operators Tk,m we establish a Heisenberg-Pauli-Weyl uncertainty principle. More precisely, we will show, for f∈L2(μk),
(8)fL2μk2≤22γ+dyFkfL2μk·∫Rd∫0∞x2Tk,mfa,x2daadμkx1/2,
provided m∈L2(μk) satisfying ∫0∞|m(ax)|2(da/a)=1, a.e. x∈Rd.
Building on the techniques of Donoho and Stark [8], we show a continuous-time principle for the L2 theory. Let E be measurable subset of Rd, let S be measurable subset of ]0,∞[×Rd, and let f∈L2(μk). If f is ɛ-concentrated on E and Tk,mf is η-concentrated on S (see Section 3 for more details), then
(9)μkE1/2∫∫S1a22γ+ddΩka,x1/2≥1-η-ɛmL1μk,
provided m∈L1∩L2(μk) satisfying ∫0∞|m(ax)|2(da/a)=1, a.e. x∈Rd.
Building on the ideas of Saitoh [9, 10], Matsuura et al. [11], and Yamada et al. [12], we give an application of the theory of reproducing kernels to the Tikhonov regularization, which gives the best approximation of the operator Tk,m on the Sobolev-Dunkl spaces Hs(μk). More precisely, for all λ>0, g∈L2(Ωk), the infimum
(10)inff∈HsμkλfHsμk2+g-Tk,mfL2Ωk2
is attained at one function fλ,g*, called the extremal function.
In particular for f∈Hs(μk) and g=Tk,mf, the corresponding extremal functions {fλ*}λ>0 converge to f as λ→0+.
This paper is organized as follows. In Section 2 we define the Dunkl L2-multiplier operators Tk,m, and we give for them Plancherel’s formula. Some examples of Dunkl L2-multiplier operators are given. In Section 3 we establish the Heisenberg-Pauli-Weyl uncertainty principle and Donoho-Stark’s uncertainty principle for the operators Tk,m. In the last section we give an application of the theory of reproducing kernels to the Tikhonov regularization related to the operators Tk,m on the Sobolev-Dunkl spaces Hs(μk).
2. The Dunkl L2-Multiplier Operators on Rd
The Dunkl operators Dj, j=1,…,d, on Rd associated with the finite reflection group G and multiplicity function k are given, for a function f of class C1 on Rd, by
(11)Djfx≔∂∂xjfx+∑α∈R+kααjfx-fσαxα,x.
For y∈Rd, the initial problem Dju(·,y)(x)=yju(x,y), j=1,…,d, with u(0,y)=1 admits a unique analytic solution on Rd, which will be denoted by Ek(x,y) and called Dunkl kernel [13, 14]. This kernel has a unique analytic extension to Cd×Cd (see [15]). In our case (see [1, 13]),
(12)Ekix,y≤1,x,y∈Rd.
The Dunkl kernel gives rise to an integral transform, which is called Dunkl transform on Rd, and was introduced by Dunkl in [1], where already many basic properties were established. Dunkl’s results were completed and extended later by de Jeu [14]. The Dunkl transform of a function f, in L1(μk), is defined by
(13)Fkfy≔∫RdEk-ix,yfxdμkx,555555555555555555555555555555y∈Rd.
We notice that F0 agrees with the Fourier transform F that is given by
(14)Ffy≔2π-d/2∫Rde-ix,yfxdx,x∈Rd.
Some of the properties of Dunkl transform Fk are collected bellow (see [1, 14]).
Theorem 1.
(i) L1-L∞-boundedness: for all f∈L1(μk), Fk(f)∈L∞(μk) and
(15)FkfL∞μk≤fL1μk.
(ii) Inversion theorem: let f∈L1(μk), such that Fk(f)∈L1(μk). Then
(16)fx=FkFkf-x,a.e.x∈Rd.
(iii) Plancherel theorem: the Dunkl transform Fk extends uniquely to an isometric isomorphism of L2(μk) onto itself. In particular,
(17)FkfL2μk=fL2μk.
Let m be a function in L2(μk). The Dunkl L2-multiplier operators, Tk,m, are defined, for regular functions f on Rd, by
(18)Tk,mfa,x≔Fk-1maxFkfx,5555555555555555a,x∈0,∞×Rd.
The operators Tk,m satisfy the following integral representation.
Lemma 2.
If m,f∈L1∩L2(μk), then
(19)Tk,mfa,x=1a2γ+d∫RdHkxa,ya,mfydμky,555555555555555555555555555a,x∈0,∞×Rd,
where
(20)Hkx,y,m=∫RdmzEkix,zEk-iy,zdμkz.
Proof.
The result follows from ((18)) and Theorem 1(ii) using Fubini-Tonnelli’s theorem.
We denote by Ωk the measure on ]0,∞[×Rd given by dΩka,x≔(da/a)dμk(x) and by L2(Ωk) the space of measurable functions F on ]0,∞[×Rd, such that
(21)FL2Ωk≔∫Rd∫0∞Fa,x2dΩka,x1/2<∞.
In the following, we give Plancherel formula for the operators Tk,m.
Theorem 3 (Plancherel formula).
Let m be a function in L2(μk) satisfying the admissibility condition:
(22)∫0∞max2daa=1,a.e.x∈Rd.
Then, for f∈L2(μk), one has
(23)Tk,mfL2Ωk=fL2μk.
Proof.
From Fubini-Tonnelli’s theorem, we obtain
(24)∫Rd∫0∞Tk,mfa,x2dΩka,x=∫0∞∫Rdmay2Fkfy2dμkydaa=∫RdFkfy2∫0∞may2daadμky.
Then, the result follows from ((22)) and Theorem 1(iii).
As applications, we give the following examples.
Example 4.
Let mt, t>0, the function is defined by
(25)mtx≔-8tx2e-tx2,x∈Rd.
Then
mt belongs to L1∩L2(μk), and by ((2)), we have
(26)mtL1μk=8t∫Rdx2e-tx2dμkx=-8t∂∂t∫Rde-tx2dμkx=22γ+d2t2γ+d,mtL2μk2=8t2∫Rdx4e-2tx2dμkx=2t2∂2∂t2∫Rde-2tx2dμkx=2γ+dγ+d/2+12t2γ+d.
mt satisfies the admissibility condition ((22)), that is,
(27)∫0∞mtax2daa=8t2x4∫0∞a3e-2tx2a2da=1.
Then the associated operators Tk,mt satisfy Plancherel’s formula ((23)).
To express the operator Tk,mt, we use Lemma 2, then for f∈L1∩L2(μk), we have
(28)Tk,mtfa,x=8ta2γ+d∫Rd∂∂thkxa,ya,tfydμky,55555555555555555555555555555555555555555x∈Rd,
where
(29)hkx,y,t=∫Rde-tz2Ekix,zEk-iy,zdμkz
is the Dunkl-type heat kernel [16, 17]. From [16] this kernel is given by
(30)hkx,y,t=12tγ+d/2e-x2+y2/4tEkx2t,y2t.
Example 5.
Let mt, t>0, be the function defined by
(31)mtx≔-2txe-tx,x∈Rd.
Then one has:
(a) mt belongs to L1∩L2(μk), and
(32)mtL1μk=2t∫Rdxe-txdμkx=-2t∂∂t∫Rde-txdμkx.
Since
(33)e-tx=1π∫0∞e-sse-t2/4sx2ds,
by Fubini-Tonnelli’s theorem and ((2)), we deduce that
(34)∫Rde-txdμkx=1π∫0∞e-ss∫Rde-t2/4sx2dμkxds=1π∫0∞e-ss2st2γ+dds=Γγ+d+1/2π2t2γ+d.
Thus,
(35)mtL1μk=22γ+dΓγ+d+1/2π2t2γ+d.
On the other hand,
(36)mtL2μk2=4t2∫Rdx2e-2txdμkx=∂2∂t2∫Rde-2txdμkx=∂2∂t2Γγ+d+1/2π2t2γ+d.
Thus,
(37)mtL2μk2=42γ+dΓγ+d+3/2π2t2γ+d+2.
(b) mt satisfies the admissibility condition ((22)); that is,
(38)∫0∞mtax2daa=4t2x2∫0∞ae-2txada=1.
Then the associated operators Tk,mt satisfy Plancherel’s formula ((23)).
To express the operators Tk,mt, we use Lemma 2; then for f∈L1∩L2(μk), we have
(39)Tk,mtfa,x=2ta2γ+d∫Rd∂∂tpkxa,ya,tfydμky,
where
(40)pkx,y,t=∫Rde-tzEkix,zEk-iy,zdμkz
is the Dunkl-type Poisson kernel [18]. From ((33)) this kernel is given by
(41)pkx,y,t=1π∫0∞e-sshkx,y,t24sds.
3. Uncertainty Principle for the Operators Tk,m3.1. Heisenberg-Pauli-Weyl Uncertainty Principle
This section is devoted to establish Heisenberg-Pauli-Weyl uncertainty principle for the operators Tk,m; more precisely, we will show the following theorem.
Theorem 6.
Let m be a function in L2(μk) satisfying the admissibility condition ((22)). Then, for f∈L2(μk), one has
(42)fL2μk2≤22γ+dyFkfL2μk·∫Rd∫0∞x2Tk,mfa,x2daadμkx1/2.
Proof.
Let f∈L2(μk).
Assume that yFkfL2(μk)<∞ and ∫Rd∫0∞|x|2|Tk,mf(a,x)|2(da/a)dμk(x)<∞. The inequality ((5)) leads to
(43)∫RdTk,mfa,x2dμkx≤22γ+d∫Rdx2Tk,mfa,x2dμkx1/2·∫Rdy2FkTk,mfa,·y2dμky1/2.
Integrating with respect to (da/a) gives
(44)Tk,mfL2Ωk2≤22γ+d∫0∞∫Rdx2Tk,mfa,x2dμkx1/2·∫Rdy2FkTk,mfa,·y2dμky1/2daa.
From Theorem 3 and Schwarz’s inequality, we get
(45)fL2μk2≤22γ+d∫0∞∫Rdx2Tk,mfa,x2dμkxdaa1/2·∫0∞∫Rdy2FkTk,mfa,·y2dμkydaa1/2.
But by ((18)), Fubini-Tonnelli’s theorem, and ((22)), we have
(46)∫0∞∫Rdy2FkTk,mfa,·y2dμkydaa=∫0∞∫Rdy2may2Fkfy2dμkydaa=∫Rdy2Fkfy2dμky.
This yields the result and completes the proof of the theorem.
3.2. Uncertainty Principle of Concentration Type
Let E be a measurable subset of Rd. We say that a function f∈L2(μk) is ɛ-concentrated on E, if
(47)f-χEfL2μk≤ɛfL2μk,
where χE is the indicator function of the set E.
Let S be a measurable subset of ]0,∞[×Rd and let f∈L2(μk). We say that Tk,mf is η-concentrated on S, if
(48)Tk,mf-χSTk,mfL2Ωk≤ηTk,mfL2Ωk.
Donoho-Stark’s uncertainty principle for the operators Tk,m is obtained.
Theorem 7.
Let f∈L2(μk) and let m∈L1∩L2(μk) satisfying ((22)). If f is ɛ-concentrated on E and Tk,mf is η-concentrated on S, then
(49)mL1μkμkE1/2∫∫S1a22γ+ddΩka,x1/2≥1-η-ɛ.
Proof.
Let f∈L2(μk). Assume that μk(E)<∞ and ∫∫S1/a2(2γ+d)dΩk(a,x)<∞. From ((47)), ((48)), and Theorem 3 it follows that
(50)Tk,mf-χSTk,mχEfL2Ωk≤Tk,mf-χSTk,mfL2Ωk+χSTk,mf-χEfL2Ωk≤ηTk,mfL2Ωk+Tk,mf-χEfL2Ωk≤η+ɛfL2μk.
Then the triangle inequality shows that
(51)Tk,mfL2Ωk≤χSTk,mχEfL2Ωk+Tk,mf-χSTk,mχEfL2Ωk≤χSTk,mχEfL2Ωk+η+ɛfL2μk.
But
(52)χSTk,mχEfL2Ωk=∫∫STk,mχEfa,x2dΩka,x1/2
and since m,χEf∈L1∩L2(μk), then, by Lemma 2, we have
(53)Tk,mχEfa,x≤1a2γ+dmL1μkfL2μkμkE1/2.
Thus,
(54)χSTk,mχEfL2Ωk≤mL1μkfL2μkμkE1/2·∫∫S1a22γ+ddΩka,x1/2,Tk,mfL2Ωk≤mL1μkfL2μkμkE1/2·∫∫S1a22γ+ddΩka,x1/2+η+ɛfL2μk.
By applying Theorem 3, we obtain
(55)mL1μkμkE1/2∫∫S1a22γ+ddΩka,x1/2≥1-η-ɛ,
which gives the desired result.
Remark 8.
If S⊂{(a,x)∈]0,∞[×Rd:a≥δ} for some δ>0, one supposes that α=max{1/a:(a,x)∈Sforsomex∈Rd}. Then by Theorem 7 we deduce that
(56)α2γ+dmL1μkμkE1/2ΩkS1/2≥1-η-ɛ.
4. Extremal Functions for the Operators Tk,m4.1. Sobolev-Dunkl Spaces
Let s≥0. We define the Sobolev-Dunkl space of order s, which will be denoted by Hs(μk), as the set of all f∈L2(μk) such that (1+|z|2)s/2Fk(f)∈L2(μk). The space Hs(μk) is provided with the inner product
(57)f,gHsμk=∫Rd1+z2sFkfzFkgz¯dμkz
and the norm
(58)fHsμk=∫Rd1+z2sFkfz2dμkz1/2.
The space Hs(μk) satisfies the following properties.
Consider H0(μk)=L2(μk).
For all s>0, the space Hs(μk) is continuously contained in L2(μk) and fL2μk≤fHs(μk).
For all s,t>0, such that t>s, the space Ht(μk) is continuously contained in Hs(μk) and fHsμk≤fHt(μk).
The space Hs(μk), s≥0, provided with the inner product 〈·,·〉Hs(μk) is a Hilbert space.
Remark 9.
For s>γ+d/2, the function y→(1+|z|2)-s/2 belongs to L2(μk).
Hence for all f∈Hs(μk), one has FkfL2μk≤fHs(μk), and by Hölder’s inequality
(59)FkfL1μk≤∫Rddμkz1+z2s1/2fHsμk.
Then the function Fk(f) belongs to L1∩L2μk, and therefore
(60)fx=∫RdEkix,zFkfzdμkz,5555555555555555555555555a.e.x∈Rd.
Let λ>0. We denote by 〈·,·〉λ,Hs(μk) the inner product defined on the space Hs(μk) by
(61)f,gλ,Hsμk≔λf,gHsμk+Tk,mf,Tk,mgL2Ωk,
and the norm fλ,Hs(μk)≔〈f,f〉λ,Hs(μk).
Next we suppose that m∈L2(μk) satisfying ((22)). By Theorem 3, the inner product 〈·,·〉λ,Hs(μk) can be written as
(62)f,gλ,Hsμk=λf,gHsμk+f,gL2μk.
Theorem 10.
Let λ>0 and s>γ+d/2 and let m∈L2(μk) satisfying ((22)). The space (Hs(μk),〈·,·〉λ,Hs(μk)) has the reproducing kernel
(63)Ksx,y=∫RdEkix,zEk-iy,z1+λ1+z2sdμkz;
that is,
for all y∈Rd, the function x→Ks(x,y) belongs to Hs(μk);
the reproducing property, for all f∈Hs(μk) and y∈Rd, is
(64)f,Ks·,yλ,Hsμk=fy.
Proof.
(i) Let y∈Rd. From ((12)), the function Φy:z→Ek(-iy,z)/1+λ1+z2s belongs to L1∩L2(μk). Then, the function Ks is well defined and by Theorem 1(ii), we have
(65)Ksx,y=Fk-1Φyx,x∈Rd.
From Theorem 1(iii), it follows that Ks(·,y) belongs to L2(μk), and we have
(66)FkKs·,yz=Ek-iy,z1+λ1+z2s,z∈Rd.
Then by ((12)), we obtain
(67)FkKs·,yz≤1λ1+z2s,Ks·,yHsμk2≤1λ2∫Rddμkz1+z2s<∞.
This proves that for all y∈Rd the function Ks(·,y) belongs to Hs(μk).
(ii) Let f∈Hs(μk) and y∈Rd. From ((62)) and ((66)), we have
(68)f,Ks·,yλ,Hsμk=∫RdEkiy,zFkfzdμkz,
and from Remark 9, we obtain the following reproducing property:
(69)f,Ks·,yλ,Hsμk=fy.
This completes the proof of the theorem.
4.2. Tikhonov Regularization
The main result of this subsection can then be stated as follows.
Theorem 11.
Let s>γ+d/2 and let m∈L2(μk) satisfying ((22)). For any g∈L2(Ωk) and for any λ>0, there exists a unique function fλ,g*, where the infimum
(70)inff∈HsμkλfHsμk2+g-Tk,mfL2Ωk2
is attained. Moreover, the extremal function fλ,g* is given by
(71)fλ,g*y=∫Rd∫0∞∫Rdga,xmaz¯Ek-ix,zEkiy,z1+λ1+z2s·dμkzdΩka,x.
Proof.
The existence and unicity of the extremal function fλ,g* satisfying ((70)) are given by Kimeldorf and Wahba [19], Matsuura et al. [11], and Saitoh [20]. Moreover, by Theorem 10 we deduce that
(72)fλ,g*y=g,Tk,mKs·,yL2Ωk,
where Ks is the kernel given by ((63)).
But by Theorems 1(ii) and ((66)), we have
(73)Tk,mKs·,ya,x=∫RdmazFkKs·,yzEkix,zdμkz=∫RdmazEkix,zEk-iy,z1+λ1+z2sdμkz.
This clearly yields the result.
Remark 12.
The extremal function fλ,g* satisfies the following inequality:
(74)fλ,g*y≤gL2Ωk2λ∫Rddμkz1+z2s1/2.
Proof.
From ((72)) and Theorem 3, we have
(75)fλ,g*y≤gL2ΩkTk,mKs·,yL2Ωk≤gL2ΩkKs·,yL2μk.
Then, by Theorems 1(iii) and ((66)),
(76)fλ,g*y≤gL2ΩkFkKs·,yL2μk≤gL2Ωk∫Rddμkz1+λ1+z2s21/2.
Using the fact that [1+λ(1+|z|2)s]2≥4λ(1+|z|2)s, we obtain the result.
If we take in ((72)), g=Tk,mf, where f∈Hs(μk), we denote by fλ*=fλ,Tk,mf*.
Then by Theorem 3, we have
(77)fλ*y=Tk,mf,Tk,mKs·,yL2Ωk=f,Ks·,yL2μk.
Corollary 13.
Let s>γ+d/2, f∈Hs(μk), and λ>0. The extremal function fλ* given by ((77)) satisfies the following properties:
fλ*(y)=∫RdEkiy,z(Fkfz/(1+λ(1+z2)s))dμk(z);
Fk(fλ*)(z)=Fkfz/1+λ1+z2s;
fλ*Hsμk≤1/2λfL2(μk).
Proof.
(i) It follows from ((77)) by using Theorems 1(iii) and ((66)).
(ii) The function F:z→Fk(f)(z)/(1+λ(1+|z|2)s) belongs to L1∩L2(μk). Then by Theorem 1(ii), we have
(78)fλ*y=Fk-1Fy.
From Theorem 1(iii), it follows that fλ* belongs to L2(μk), and
(79)Fkfλ*z=Fkfz1+λ1+z2s.
(iii) By relation (ii) we have
(80)fλ*Hsμk2=∫Rd1+z2sFkfλ*z2dμkz=∫Rd1+z2s1+λ1+z2s2Fkfz2dμkz.
Using inequality [1+λ(1+|z|2)s]2≥4λ(1+|z|2)s, we obtain
(81)fλ*Hsμk2≤14λ∫RdFkfz2dμkz=14λfL2μk2.
This completes the proof of the corollary.
Corollary 14.
Let s>γ+d/2, f∈Hs(μk), and λ>0. The extremal function fλ* given by ((77)) satisfies
(82)limλ→0+fλ*-fHsμk=0.
Moreover, {fλ*}λ>0 converges uniformly to f as λ→0+.
Proof.
From Corollary 13(ii),
(83)Fkfλ*-fz=-λ1+z2s1+λ1+z2sFkfz.
Consequently,
(84)fλ*-fHsμk2=∫Rdλ21+z23sFkfz21+λ1+z2s2dμkz.
Using the dominated convergence theorem and the fact that
(85)λ21+z23sFkfz21+λ1+z2s2≤1+z2sFkfz2,
we deduce that
(86)limλ→0+fλ*-fHsμk2=0.
On the other hand, from Remark 9, the function Fk(f)∈L1∩L2(μk). Then by ((83)) and Theorem 1(ii),
(87)fλ*y-fy=∫Rd-λ1+z2sFkfz1+λ1+z2sEkiy,zdμkz.
So
(88)fλ*-fL∞μk≤∫Rdλ1+z2sFkfz1+λ1+z2sdμkz.
Again, by dominated convergence theorem and the fact that
(89)λ1+z2sFkfz1+λ1+z2s≤Fkfz,
we deduce that
(90)limλ→0+fλ*-fL∞μk=0,
which ends the proof.
As application, we give the following examples.
Example 15.
If mt(x)≔-8t|x|2e-t|x|2 and s>γ+d/2, then
(91)fλ,g*y=-8t∫Rd∫0∞∫Rda2e-ta2z2z2ga,x1+λ1+z2s·Ek-ix,zEkiy,zdμk·zdΩka,x,
and by ((28)), ((72)), and the fact that Ks(y,z)=Ks(z,y)¯ we obtain
(92)fλ,g*y=8t∫Rd∫0∞∫Rdga,xa2γ+d∂∂thkxa,za,t·Ksy,zdμkzdΩka,x.
Example 16.
If mt(x)≔-2t|x|e-t|x| and s>γ+d/2, then
(93)fλ,g*y=-2t∫Rd∫0∞∫Rdae-tazzga,x1+λ1+z2s·Ek-ix,zEkiy,zdμk·zdΩka,x,
and by ((39)) and ((72)) we deduce that
(94)fλ,g*y=2t∫Rd∫0∞∫Rdga,xa2γ+d∂∂tpkxa,za,t·Ksy,zdμkzdΩka,x.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author is partially supported by Deanship of Scientific Research, Jazan University.
DunklC. F.Hankel transforms associated to finite reflection groups1992138123138Zbl0789.33008RöslerM.An uncertainty principle for the Dunkl transform199959335336010.1017/S0004972700033025MR16980452-s2.0-17644441809ShimenoN.A note on the uncertainty principle for the Dunkl transform2001813342MR1818904ZBLl0976.33015SoltaniF.Heisenberg-Pauli-Weyl uncertainty ine quality for the Dunkl transform on ℝd201387231632510.1017/S0004972712000780MR30407152-s2.0-84876467351SoltaniF.A general form of Heisenberg-Pauli-Weyl uncertainty inequality for the Dunkl transform201324540140910.1080/10652469.2012.699966MR30555282-s2.0-84877865535SoltaniF.Best approximation formulas for the Dunkl L2-multiplier operators on Rd201242130532810.1216/RMJ-2012-42-1-305MR28762822-s2.0-84873826255SoltaniF.Multiplier operators and extremal functions related to the dual Dunkl-Sonine operator201333243044210.1016/S0252-9602(13)60010-7MR30306302-s2.0-84875112248DonohoD. L.StarkP. B.Uncertainty principles and signal recovery198949390693110.1137/0149053MR997928ZBLl0689.42001SaitohS.The Weierstrass transform and an isometry in the heat equation19831611610.1080/00036818308839454MR705341SaitohS.Best approximation, Tikhonov regularization and reproducing kernels200528235936710.2996/kmj/1123767016MR2153923ZBLl1087.65053MatsuuraT.SaitohS.TrongD. D.Approximate and analytical inversion formulas in heat conduction on multidimensional spaces2005133–647949310.1163/156939405775297452MR21886252-s2.0-30344437916YamadaM.MatsuuraT.SaitohS.Representations of inverse functions by the integral transform with the sign kernel2007102161168MR2351657DunklC. F.Integral kernels with reflection group invariance19914361213122710.4153/CJM-1991-069-8MR1145585de JeuM. F. E.The Dunkl transform1993113114716210.1007/BF01244305MR12232272-s2.0-0000945341OpdamE. M.Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group1993853333373MR1214452RöslerM.VoitM.Markov processes related with Dunkl operators199821457564310.1006/aama.1998.0609MR16521822-s2.0-0002327987SoltaniF.Inversion formulas in the Dunkl -type heat conduction on R d200584654155310.1080/00036810410001731492MR2151667SoltaniF.Littlewood-Paley g-function in the Dunkl analysis on Rd200563, article 8413MR2164325KimeldorfG.WahbaG.Some results on Tchebycheffian spline functions197133829510.1016/0022-247X(71)90184-3MR0290013ZBLl0201.397022-s2.0-0015000439SaitohS.Approximate real inversion formulas of the Gaussian convolution200483772773310.1080/00036810410001657198MR2072423