An analog of titchmarsh’s theorem for the dunkl transform in the space Lp(ℝd;wk(x)dx)

ISSN: 2008-6822 (electronic) http://www.ijnaa.semnan.ac.ir

An Analog of Titchmarsh’s Theorem for the Dunkl Transform in the Space L ² _α ( R )

R. Daher, M. El Hamma^∗

Department of Mathematics, Faculty of Science Ain Chock, University Hassan II, Casablanca, Morocco.

(Communicated by A. Ebadian)

Abstract

In this paper, using a generalized Dunkl translation operator, we obtain an analog of Titchmarsh’s Theorem for the Dunkl transform for functions satisfying the Lipschitz-Dunkl condition in L2,α = L²_α(R) = L²(R,|x|^2α+1dx), α > ⁻¹₂ .

Keywords: Dunkl Operator, Dunkl Transform, Generalized Dunkl Translation.

2010 MSC: 44A15, 46E30.

1. Introduction and Preliminaries

Dunkl operators are differential-difference operators introduced in 1989, by Dunkl [5]. On the real line, these operators, which are denoted by D = Dα, depend on a real parameter α > ⁻¹₂ and they associated with the reflection groupZ2 onR. For more details about these operators see [4, 5, 6, 7, 10]

and the references therein.

Recently in mathematical papers a new class of generalized translations was described and put into use, namely, the generalized Dunkl translations. The generalized Dunkl translations are constructed on the base of certain differential-difference operators which are widely used in mathematical physics (e.g., [5, 9]).

Titchmarsh’s [11, Therem 85] characterized the set of functions in L_2,α satisfying the cauchy Lip- schitz condition by means of an asymptotic estimate growth of the norm of their Fourier transform, we have

∗Corresponding author

Email addresses: [email protected] (R. Daher ),[email protected](M. El Hamma )

Received: January 2012 Revised: February 2012

Theorem 1.1. [11] Let α∈(0,1) and assume that f ∈L²(R). Then the following are equivalents:

(1) kf(t+h)−f(t)k_L²_(R) =O(h^α) as h−→0 (2) R

|λ|≥r|fb(λ)|²dλ=O(r^−2α) r −→ ∞ where fbstands for the Fourier transform of f.

In this paper, we obtain an analog of Theorem 1.1 in the Dunkl operator setting by means of generalized Dunkl translation. We point out that similar results have been established in the context of noncompact rank 1 Riemannian symmetric spaces [8].

The Dunkl operator is differential-difference operator D which satisfies the condition Df(x) = df

dx(x) + (α+1

2)f(x)−f(−x)

x , f ∈L_2,α. Letj_α(x) be a normalized Bessel function of the first kind, i.e.,

j_α(x) = 2^αΓ(α+ 1)J_α(x)

x^α ,

whereJ_α(x) is a Bessel function of the first kind ([2], chap 7). The functionj_α(x) is infinitely differ- entiable and even.

We understand a generalized exponential function as the function

e_α(x) =j_α(x) +ic_αxj_α+1(x), (1.1)

where c_α = (2α+ 2)⁻¹.

The function y=e_α(x) satisfies the equation Dy =y.

Using the correlation j_α⁰(x) =−xjα+1(x) 2(α+ 1) .

we conclude that the function e_α(x) admits the representation

e_α(x) =j_α(x)−ij_α⁰(x) (1.2)

Let L_2,α is the Hilbert space consists of measurable functions f(x) defined on Rwith the norm kfk=kfk_2,α=

Z +∞

−∞

|f(x)|²|x|^2α+1dx ^1/2

. The Dunkl transform is defined by

f(λ) =b Z +∞

−∞

f(x)e_α(λx)|x|^2α+1dx, λ∈R.

The inverse Dunkl transform is defined by the formula f(x) = (2^α+1Γ(α+ 1))⁻²

Z +∞

−∞

f(λ)eb _α(−λx)|λ|^2α+1dλ . The Dunkl transform satisfies the Parseval equality

Z +∞

−∞

|f(x)|²|x|^2α+1dx= (2^α+1Γ(α+ 1))⁻² Z +∞

−∞

|fb(λ)||λ|^2α+1dλ .

In L_2,α, we define the operator of the generalized Dunkl translation (see [3])

T_hf(x) =C(

Z π

0

f_e(G(x, h, ϕ))h^e(x, h, ϕ)sin^2αϕdϕ + Z π

0

f₀(G(x, h, ϕ))h⁰(x, h, ϕ)sin^2αϕdϕ) where

C = Γ(α+ 1)

Γ(¹₂)Γ(α+¹₂), G(x, h, ϕ) = p

x²+h²−2|xh|cosϕ h^e(x, h, ϕ) = 1−sgn(xh)cosϕ .

and

h⁰(x, h, ϕ) = (x+h)h^e(x, h, ϕ)

G(x, h, ϕ) for (x, h)6= (0,0) h⁰(x, h, ϕ) = 0 for (x, h) = (0,0)

f_e(x) = 1

2(f(x) +f(−x)), f₀(x) = 1

2(f(x)−f(−x)) From formula (1.1), we have

|1−j_α(x)| ≤ |1−e_α(x)|.

Forα≥ ⁻¹₂ , we introduce the Bessel normalized function of the first kind j_α defined by j_α(x) = Γ(p+ 1)

∞

X

n=0

(−1)ⁿ

n!Γ(n+α+ 1)(x

2)²ⁿ (1.3)

Moreover, from (1.3) we see that

x−→0lim

j_p(x)−1 x² 6= 0,

by consequence, there exist c >0 and η >0 satisfying

|x| ≤η=⇒ |j_α(x)−1| ≥c|x|² (1.4)

In [3], we obtain

(Td_hf)(λ) =e_α(λh)fb(λ) (1.5)

2. Main Result

In this section we give the main result of this paper, We need first to define the Dunkl-Lipschitz class.

Definition 2.1. Let β ∈ (0,1). A function f ∈ L2,α is said to be in the Dunkl-Lipschitz class, denote by Lip(β,2, α), if

kT_hf(x) + T−hf(x)−2f(x)k=O(h^β), as h−→0.

Theorem 2.2. Let f ∈L_2,α. Then the following are equivalents (1) f ∈Lip(β,2, α).

(2) R

|λ|≥r|fb(λ)|²|λ|^2α+1dλ=O(r^−2β) as r−→+∞.

Proof . 1 =⇒2: Assume that f ∈Lip(β,2, α). Then we have kT_hf(x) + T−hf(x)−2f(x)k=O(h^β), as h−→0.

From formulas (1.2) and (1.5), we have the Dunkl transform of T_hf(x) + T−hf(x) − 2f(x) is 2(jα(λh)−1).

Parseval’s identity gives

kT_hf(x) + T−hf(x)−2f(x)k= Z +∞

−∞

|2(1−j_α(λh))|²|fb(λ)|²|λ|^2α+1dλ . From (1.4), we have

Z

η 2h≤|λ|≤^η

h

|1−j_α(λh)|²|fb(λ)|²|λ|^2α+1dλ ≥ c²η⁴ 16

Z

η 2h≤|λ|≤^η

h

|f(λ)|b ²|λ|^2α+1dλ . There exists then a positive constant C such that

Z

η 2h≤|λ|≤^η

h

|fb(λ)|²|λ|^2α+1dλ ≤ C Z +∞

−∞

|1−jα(λh)|²|f(λ)|b ²|λ|^2α+1dλ

≤ Ch^2β For all h >0, we have

Z

r≤|λ|≤2r

|fb(λ)|²|λ|^2α+1dλ ≤Cr^−2β, for all r >0. Furthermore, we obtain

Z

|λ|≥r

|f(λ)|b ²|λ|^2α+1dλ =

∞

X

i=0

Z

2ⁱs≤|λ|≤2ⁱ⁺¹s

|f(λ)|b ²|λ|^2α+1dλ

≤ C

∞

X

i=0

(2ⁱr)^−2β

≤ Cr^−2β This prove that

Z

|λ|≥r

|f(λ)|b ²|λ|^2α+1dλ=O(r^−2β) as r −→+∞. 2 =⇒1: Suppose now that

Z

|λ|≥r

|f(λ)|b ²|λ|^2α+1dλ=O(r^−2β) as r −→+∞. We write

Z +∞

−∞

|1−jα(λh)|²|f(λ)|b ²|λ|^2α+1dλ = Z

|λ|<¹

h

|1−jα(λh)|²|fb(λ)|²|λ|^2α+1dλ +

Z

|λ|≥¹

h

|1−j_α(λh)|²|fb(λ)|²|λ|^2α+1dλ Firstly, we use the formula |j_α(x)| ≤1

Z

|λ|≥¹_h

|1−j_α(λh)|²|f(λ)|b ²|λ|^2α+1dλ≤4 Z

|λ|≥_h¹

|fb(λ)|²|λ|^2α+1dλ . Then

Z

|λ|≥¹

h

|1−j_α(λh)|²|f(λ)|b ²|λ|^2α+1dλ=O(h^2β) as h−→0. Set

φ(x) = Z ∞

x

|f(λ)|b ²|λ|^2α+1dλ . An integration by parts, we obtain

Z x

0

λ²|f(λ)|b ²|λ|^2α+1dλ = Z x

0

−λ²φ⁰(λ)dλ =−x²φ(x) + 2 Z x

0

λφ(λ)dλ

≤ 2 Z x

0

O(λ^1−2β)dλ

= O(x^2−2β).

We use the formula 1−j_α(x) =O(x²), 0≤x≤1 see [1]. Then

Z +∞

−∞

|1−j_α(λh)|²|f(λ)|b ²|λ|^2α+1dλ = O(h² Z

|λ|<¹

h

λ²|fb(λ)|²|λ|^2α+1dλ+O(h^2β)

= O(h²h^−2+2β) +O(h^2β)

= O(h^2β) +O(h^2β)

= O(h^2β).

and this end the proof.

References

[1] V. A. Abilov and F. V. Abilova,Approximation of Functions by Fourier-Bessel Sums, Izv. Vyssh. Uchebn. Zaved., Mat., (8) (2001), 3–9.

[2] H. Bateman and A. Erd´elyi, Higher Transcendental Functions, vol. II (McGraw-Hill, New York, 1953; Nauka, Moscow, 1974).

[3] E. S. Belkina and S. S. Platonov, Equivalence of K-Functionals and Modulus of Smoothness Constructed by Generalized Dunkl Translations, Izv. Vyssh. Uchebn. Zaved. Mat., (8) (2008) 3–15.

[4] M. F. E. de Jeu, The Dunkl transform, Invent. Math., 113 (1993) 147–162.

[5] C. F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc., 311 (1) (1989) 167–183.

[6] C. F. Dunkl, Hankel transforms associated to finite reflection groups. Hypergeometric functions on domains of positivity, Jack polynomials, and applications, Contemp. Math., 138 (1992) 123–138.

[7] L. Lapointe and L. Vinet,Exact operator solution of the Calogero-Sutherland model, J. Phys., 178 (1996) 425–452.

[8] S. S. Platonov,The Fourier transform of functions satisfying the Lipschitz condition on rank 1 symmetric spaces, Siberian Math. J., 46 (6) (2005) 1108–1118.

[9] M. R¨osler,Dunkl Operators: Theory and Applications, Lect. Notes in Math., 1817 (2002) 93–135.

[10] M. R¨osler,Positivity of the Dunkl intertwining operator, Duke Math. J., 98 (3) (1999) 445–463.

[11] E. C. Titchmarsh,Introduction to the theory of Fourier Integrals , Claredon, Oxford, 1948, Komkniga, Moscow, 2005.