International Journal of Analysis and Applications
ISSN 2291-8639
Volume 9, Number 2 (2015), 90-95
http://www.etamaths.com

TITCHMARSH’S THEOREM FOR THE CHEREDNIK-OPDAM

TRANSFORM IN THE SPACE L2α,β(R)

S. EL OUADIH∗ AND R. DAHER

Abstract. In this paper, we prove the generalization of Titchmarshs theorem for the Cherednik-

Opdam transform for functions satisfying the (ψ, 2)-Cherednik-Opdam Lipschitz condition

in the space L2
α,β

(R).

1. Introduction and Preliminaries

In [3], E. C. Titchmarsh’s characterizes the set of functions in L2(R) satisfying the Cauchy-
Lipschitz condition by means of an asymptotic estimate growth of the norm of their Fourier
transform, namely we have

Theorem 1.1 [3] Let δ ∈ (0, 1) and assume that f ∈ L2(R). Then the following are equivalents
(i) ‖f(t + h) −f(t)‖ = O(hδ), as h → 0,

(ii)

∫
|λ|≥r

|f̂(λ)|2dλ = O(r−2δ) as r →∞,

where f̂ stands for the Fourier transform of f.

In this paper, we prove the generalization of Theorem 1.1 for the Cherednik-Opdam transform
for functions satisfying the (ψ, 2)-Cherednik-Opdam Lipschitz condition in the space L2α,β(R).
For this purpose, we use the generalized translation operator.

In this section, we develop some results from harmonic analysis related to the differential-
difference operator T(α,β). Further details can be found in [1] and [2]. In the following we fix
parameters α, β subject to the constraints α ≥ β ≥−1

2
and α > −1

2
.

Let ρ = α + β + 1 and λ ∈ C. The Opdam hypergeometric functions G(α,β)λ on R are eigenfunc-
tions T(α,β)G

(α,β)
λ (x) = iλG

(α,β)
λ (x) of the differential-difference operator

T(α,β)f(x) = f′(x) + [(2α + 1) coth x + (2β + 1) tanh x]
f(x) −f(−x)

2
−ρf(−x),

that are normalized such that G
(α,β)
λ (0) = 1. In the notation of Cherednik one would write

T(α,β) as

T(k1 + k2)f(x) = f
′(x) +

{
2k1

1 + e−2x
+

4k2
1 −e−4x

}
(f(x) −f(−x)) − (k1 + 2k2)f(x),

with α = k1 +k2− 12 and β = k2−
1
2
. Here k1 is the multiplicity of a simply positive root and k2

the (possibly vanishing) multiplicity of a multiple of this root. By [1] or [2], the eigenfunction

2010 Mathematics Subject Classification. 46L08.

Key words and phrases. Cherednik-Opdam operator; Cherednik-Opdam transform; generalized translation.

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papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

90



TITCHMARSH’S THEOREM FOR THE CHEREDNIK-OPDAM TRANSFORML SPECTRUM 91

G
(α,β)
λ is given by

G
(α,β)
λ (x) = ϕ

α,β
λ (x) −

1

ρ− iλ
∂

∂x
ϕ
α,β
λ (x) = ϕ

α,β
λ (x) +

ρ

4(α + 1)
sinh(2x)ϕ

α+1,β+1
λ (x),

where ϕ
α,β
λ (x) =2 F1(

ρ+iλ
2

; ρ−iλ
2

; α + 1;−sinh2 x) is the classical Jacobi function.

Lemma 1.2. [4] The following inequalities are valids for Jacobi functions ϕ
α,β
λ (x)

(i) |ϕα,βλ (x)| ≤ 1.
(ii) 1 −ϕα,βλ (x) ≤ x

2(λ2 + ρ2).
(iii) there is a constant c > 0 such that

1 −ϕα,βλ (x) ≥ c,
for λx ≥ 1.

Denote L2α,β(R), the space of measurable functions f on R such that

‖f‖2,α,β =
(∫

R
|f(x)|2Aα,β(x)dx

)1/2
< +∞,

where
Aα,β(x) = (sinh |x|)2α+1(cosh |x|)2β+1.

The Cherednik-Opdam transform of f ∈ Cc(R) is defined by

Hf(λ) =
∫
R
f(x)G

(α,β)
λ (−x)Aα,β(x)dx for all λ ∈ C.

The inverse transform is given as

H−1g(x) =
∫
R
g(λ)G

(α,β)
λ (x)

(
1 −

ρ

iλ

) dλ
8π|cα,β(λ)|2

,

here

cα,β(λ) =
2ρ−iλΓ(α + 1)Γ(iλ)

Γ( 1
2
(ρ + iλ))Γ( 1

2
(α−β + 1 + iλ))

.

The corresponding Plancherel formula was established in [1], to the effect that∫
R
|f(x)|2Aα,β(x)dx =

∫ +∞
0

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ),

where f̌(x) := f(−x) and dσ is the measure given by

dσ(λ) =
dλ

16π|cα,β(λ)|2
.

According to [2] there exists a family of signed measures µ
(α,β)
x,y such that the product formula

G
(α,β)
λ (x)G

(α,β)
λ (y) =

∫
R
G

(α,β)
λ (z)dµ

(α,β)
x,y (z),

holds for all x,y ∈ R and λ ∈ C, where

dµ(α,β)x,y (z) =



Kα,β(x,y,z)Aα,β(z)dz if xy 6= 0

dδx(z) if y = 0
dδy(z) if x = 0

and

Kα,β(x,y,z) = Mα,β|sinh x. sinh y. sinh z|−2α
∫ π
0

g(x,y,z,χ)
α−β−1
+

× [1 −σχx,y,z + σ
χ
x,z,y + σ

χ
z,y,x +

ρ

β + 1
2

coth x. coth y. coth z(sin χ)2] × (sin χ)2βdχ



92 OUADIH AND DAHER

if x,y,z ∈ R\{0} satisfy the triangular inequality ||x|−y|| < |z| < |x|+|y|, and Kα,β(x,y,z) = 0
otherwise. Here

∀x,y,z ∈ R,χ ∈ [0, 1],σχx,y,z =




cosh x+cosh y−cosh z cos χ
sinh x sinh y

if xy 6= 0

0 if xy = 0

and g(x,y,z,χ) = 1 − cosh2 x− cosh2 y. cosh2 z + 2 cosh x. cosh y. cosh z. cos χ.

Lemma 1.3. [2] For all x,y ∈ R, we have
(i) Kα,β(x,y,z) = Kα,β(y,x,z).
(ii) Kα,β(x,y,z) = Kα,β(−x,z,y).
(iii) Kα,β(x,y,z) = Kα,β(−z,y,−x).

The product formula is used to obtain explicit estimates for the generalized translation op-
erators

τ(α,β)x f(y) =

∫
R
f(z)dµ(α,β)x,y (z).

It is known from [2] that

Hτ(α,β)x f(λ) = G
(α,β)
λ (x)Hf(λ),(1.1)

for f ∈ Cc(R).

2. Main Result

In this section we give the main result of this paper. We need first to define (ψ, 2)-Cherednik-
Opdam Lipschitz class.
Denote Nh by

Nh = τ
(α,β)
h + τ

(α,β)
−h − 2I,

where I is the unit operator in the space L2α,β(R).

Definition 2.1. A function f ∈ L2α,β(R) is said to be in the (ψ, 2)-Cherednik-Opdam Lip-
schitz class, denoted by Lip(ψ, 2), if

‖Nhf(x)‖2,α,β = O(ψ(h)) as h → 0,

where ψ is a continuous increasing function on [0,∞),ψ(0) = 0 , ψ(ts) = ψ(t)ψ(s) for all
t,s ∈ [0,∞) and this function verify∫ 1/h

0

sψ(s−2)ds = O(h−2ψ(h2)), h → 0.

Lemma 2.2. If f ∈ Cc(R), then

Hτ̌(α,β)x f(λ) = G
(α,β)
λ (−x)Hf̌(λ).(2.1)

Proof. For f ∈ Cc(R), we have

Hτ̌(α,β)x f(λ) =
∫
R
τ(α,β)x f(−y)G

(α,β)
λ (−y)Aα,β(y)dy

=

∫
R
τ(α,β)x f(y)G

(α,β)
λ (y)Aα,β(y)dy

=

∫
R

[∫
R
f(z)Kα,β(x,y,z)Aα,β(z)dz

]
G

(α,β)
λ (y)Aα,β(y)dy

=

∫
R
f(z)

[∫
R
G

(α,β)
λ (y)Kα,β(x,y,z)Aα,β(y)dy

]
Aα,β(z)dz.



TITCHMARSH’S THEOREM FOR THE CHEREDNIK-OPDAM TRANSFORML SPECTRUM 93

Since Kα,β(x,y,z) = Kα,β(−x,z,y), it follows from the product formula that

Hτ̌(α,β)x f(λ) = G
(α,β)
λ (−x)

∫
R
f(z)G

(α,β)
λ (z)Aα,β(z)dz

= G
(α,β)
λ (−x)

∫
R
f(−z)G(α,β)λ (−z)Aα,β(z)dz

= G
(α,β)
λ (−x)Hf̌(λ).

Lemma 2.3. For f ∈ L2α,β(R), then

‖Nhf(x)‖22,α,β = 4
∫ +∞
0

|ϕα,βλ (h) − 1|
2
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ).

Proof. From formulas (1) and (2), we have

H(Nhf)(λ) = (G
(α,β)
λ (h) + G

(α,β)
λ (−h) − 2)H(f)(λ),

and

H(Ňhf)(λ) = (G
(α,β)
λ (−h) + G

(α,β)
λ (h) − 2)H(f̌)(λ).

Since

G
(α,β)
λ (h) = ϕ

α,β
λ (h) +

ρ

4(α + 1)
sinh(2h)ϕ

α+1,β+1
λ (h),

and ϕ
α,β
λ is even, then

H(Nhf)(λ) = 2(ϕ
α,β
λ (h) − 1)H(f)(λ)

and

H(Ňhf)(λ) = 2(ϕ
α,β
λ (h) − 1)H(f̌)(λ).

Now by Plancherel Theorem, we have the result.

Theorem 2.4. Let f ∈ L2α,β(R). Then the following are equivalents

(a) f ∈ Lip(ψ, 2),

(b)

∫ +∞
r

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ) = O(ψ(r−2)), as r →∞.

Proof. (a) ⇒ (b) Let f ∈ Lip(ψ, 2). Then we have

‖Nhf(x)‖2,α,β = O(ψ(h)) as h → 0.

From Lemma 2.2, we have

‖Nhf(x)‖22,α,β = 4
∫ +∞
0

|1 −ϕα,βλ (h)|
2
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσλ.

If λ ∈ [ 1
h
, 2
h

], then λh ≥ 1 and (iii) of Lemma 1.2 implies that

1 ≤
1

c2
|1 −ϕα,βλ (h)|

2.

Then∫ 2
h

1
h

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ) ≤

1

c2

∫ 2
h

1
h

|1 −ϕα,βλ (h)|
2
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ)

≤
1

c2

∫ +∞
0

|1 −ϕα,βλ (h)|
2
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ)

≤
1

4c2
‖Nhf(x)‖22,α,β

= O(ψ(h2)).



94 OUADIH AND DAHER

We obtain ∫ 2r
r

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ) ≤ Cψ(r−2), r →∞,

where C is a positive constant. Now,∫ +∞
r

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ) =

∞∑
i=0

∫ 2i+1r
2ir

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ)

≤ Cψ(r−2)
∞∑
i=0

(ψ(2−2))i

≤ CCδψ(r−2),
where Cδ = (1 −ψ(2−2))−1 since ψ(2−2) < 1.
Consequently ∫ +∞

r

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ) = O(ψ(r−2)), as r →∞.

(b) ⇒ (a). Suppose now that∫ +∞
r

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ) = O(ψ(r−2)), as r →∞,

and write

‖Nhf(x)‖22,α,β = 4(I1 + I2),
where

I1 =

∫ 1
h

0

|1 −ϕα,βλ (h)|
2
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσλ,

and

I2 =

∫ +∞
1
h

|1 −ϕα,βλ (h)|
2
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσλ.

Firstly, we use the formula |ϕα,βλ (h)| ≤ 1 and

I2 ≤ 4
∫ +∞

1
h

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ) = O(ψ(h2)), as h → 0.

To estimate I1, we use the inequalities (i) and (ii) of Lemma 1.2

I1 =

∫ 1
h

0

|1 −ϕα,βλ (h)|
2
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσλ

≤ 2
∫ 1

h

0

|1 −ϕα,βλ (h)|
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσλ

≤ 2h2
∫ 1

h

0

(λ2 + ρ2)
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσλ = I3 + I4,

where

I3 = 2h
2ρ2
∫ 1

h

0

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσλ,

and

I4 = 2h
2

∫ 1
h

0

λ2
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσλ.

Note that

I3 ≤ 2h2ρ2
∫ +∞
0

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσλ

= 2h2ρ2‖f‖22,α,β = O(ψ(h
2)), as h → 0.



TITCHMARSH’S THEOREM FOR THE CHEREDNIK-OPDAM TRANSFORML SPECTRUM 95

For a while, we put

φ(s) =

∫ +∞
s

(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσ(λ).

Using integration by parts, we find that

h2
∫ 1/h
0

λ2
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσλ = h2

∫ 1/h
0

−s2φ′(s)ds

= h2

(
−

1

h2
φ(

1

h
) + 2

∫ 1/h
0

sφ(s)ds

)

= −φ(
1

h
) + 2h2

∫ 1/h
0

sφ(s)ds.

Since φ(s) = O(ψ(s−2)), we have sφ(s) = O(sψ(s−2)) and∫ 1/h
0

sφ(s)ds = O

(∫ 1/h
0

sψ(s−2)ds

)
= O(h−2ψ(h2)).

Then

h2
∫ 1/h
0

λ2
(
|Hf(λ)|2 + |Hf̌(λ)|2

)
dσλ ≤ 2C1h2h−2ψ(h2),

where C1 is a positive constant.
Finally

I4 = O(ψ(h
2)),

which completes the proof of the theorem. �

References

[1] E. M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175

(1995), no. 1, 75C121.

[2] J. P. Anker, F. Ayadi, and M. Sifi, Opdams hypergeometric functions: product formula and convolution
structure in dimension 1, Adv. Pure Appl. Math. 3 (2012), no. 1, 11C44.

[3] E. C. Titchmarsh , Introduction to the Theory of Fourier Integrals . Claredon , oxford, 1948, Komkni-

ga.Moxow.2005.
[4] S. S. Platonov, Approximation of functions in L2-metric on noncompact rank 1 symmetric space . Algebra

Analiz .11(1) (1999), 244-270.

Department of Mathematics, Faculty of Sciences Äın Chock, University Hassan II, Casablanca,
Morocco

∗Corresponding author