An analog of Titchmarsh's theorem of: Jacobi transform

An Analog of Titchmarsh’s Theorem of Jacobi Transform

Radouan Daher

Faculty of Sciences Ain Ckock, Casablanca, Morocco ra daher@yahoo.fr

Mohamed El Hamma

Faculty of Sciences Ain Ckock, Casablanca, Morocco m elhamma@yahoo.fr

Abstract

In this paper, we prove an analog of Titchmarsh’s theorem for the Jacobi transform for functions satisfying the Jacobi-Lipschitz condition in L ² ( R ⁺ , Δ _α,β ( t ) dt ), using a generalized translation operator.

Mathematics Subject Classification: 33C45; 43A90; 42C15

Keywords: Jacobi operator, Jacobi transform, generalized translation op- erator

1 Introduction and notations

In the present paper, we study a the set of functions in L ² ( R ⁺ , Δ _α,β ( t ) dt ) satis- fying the Cauchy Lipschitz condition, we use translation operator τ h associated of Jacobi operators, the operator has played a decisive role in the development of Euclidean harmonic analysis, as evidenced, for example, by landmark paper [3] by H¨ ormander.

Titchmarsh’s [7, Theorem 85] characterized the set of functions L ² ( R ) satisfy- ing the Cauchy Lipschitz condition by means of asymptotic estimate growth of the norm of their Fourier transform, namely we have

Theorem 1.1 [7] Let α ∈ (0 , 1) and assume that f ∈ L ² ( R ). Then the following are eguivalents:

(1) f ( t + h ) − f ( t ) L

(

= O ( h ^α ) as h −→ 0 (2)

|λ|≥r | f ( λ ) | ² dλ = O ( r ^−2α ) r −→ ∞

where f stands for the Fourier transform of f

The main aim of this paper is to establish of Theorem 1.1 in Jacobi opera- tors setting by means the generalized translation operator τ h defined in Section 2. We point out that similar results have been established in the context of noncompact rank 1 Riemannian symmetric spaces [6].

In this section, we briefly collect the pertinnent definitions and facts relevant for Jacobi analysis, can be found in [4].

Let ( a ) ₀ = 1 and ( a ) _k = a ( a + 1) ... ( a + k − 1). The hypergeometric function

F ( a, b, c, z ) =

∞

k=0

( a ) _k ( b ) _k

( c ) _k k ! z ^k , |z| < 1

the function z −→ F ( a, b, c, z ) is the unique solution of the differential equation z (1 − z ) u ”( z ) + ( c − ( a + b + 1) z ) u ( z ) − abu ( z ) = 0

which is regular in 0 and equals 1 there.

The Jacobi function with parametres ( α, β ) is defined by the formula ϕ ^(α,β) _λ ( t ) = F ( 1

2 ( ρ − iλ ) , 1

2 ( ρ + iλ ) , α + 1 , − sinh ² t )

For α ≥ ⁻¹ ₂ , α > β ≥ ⁻¹ ₂ , ρ = α + β + 1, the system {ϕ ^(α,β) _λ } _λ≥0 is a continuous orthonormal system in R ⁺ , with respect to the weight

Δ _α,β ( t ) = (2 sinh t ) ^2α+1 (2 cosh t ) ^2β+1 The Jacobi operator

L = L _α,β = d ²

dt ² + ((2 α + 1) coth t + (2 β + 1) tanh t ) d dt

By means of which the Jacobi function ϕ ^(α,β) _λ may alternatively be charac- terized as the unique solution to

L ϕ + ( λ ² + ρ ² ) ϕ = 0 (1)

on R ⁺ satisfying ϕ ^(α,β) _λ (0) = 1, ϕ

_λ ^(α,β) (0) = 0, and λ −→ ϕ ^(α,β) _λ ( t ) is analytic for all t ≥ 0.

We adhere to the conventions and normalization used [2], the c-function c ( λ ) = 2 ^ρ Γ( iλ )Γ( ¹ ₂ (1 + iλ ))

Γ( ¹ ₂ ( ρ + iλ ))Γ( ¹ ₂ ( ρ + iλ ) − β ) where α > 0 , α > β ≥ ⁻¹ ₂ .

The Jacobi transform of a function f ∈ L ² ( R ⁺ , Δ _α,β ( t ) dt ) is defined by

f ( λ ) =

_∞

0 f ( t ) ϕ ^(α,β) _λ ( t )Δ _α,β ( t ) dt (2) and the inversion formula is statement thet (cf.[4])

f ( t ) = 1 2 π

_∞

0

f ( λ ) ϕ ^(α,β) _λ ( t ) dμ ( λ ) (3) where dμ ( λ ) = |c ( λ ) | ⁻² dλ .

The Jacobi transform a unitary isomorphism from L ² ( R ⁺ , Δ _α,β ( t ) dt ) onto L ² ( R ⁺ , _2π ¹ dμ ( λ )), i.e

f = f _L

₍

_,Δ

_(t)dt) = f _L

₍

_,

2π

dμ(λ)) (4)

The limiting case α = β = ⁻¹ ₂ is the Fourier-cosine transform, which we will not study.

We have

L f ( λ ) = − ( λ ² + ρ ² ) f ( λ ) (5) The generalized translation operator was defined by Flensted-Jensen and Koornwinder [2, Formula (5.1)] given by

τ y f ( x ) =

_∞

0 f ( z ) K ( x, y, z )Δ _α,β ( z ) dz with kernel

K ( x, y, z ) = 2 ^−2ρ Γ( α + 1)(cosh x cosh cosh z ) ^{−α−β−1}

Γ( ¹ ₂ )Γ( α + ¹ ₂ )(sinh x sinh y sinh z ) ^2α (1 − B ² ) ^α−

× F (( α + β, α − β, α + 1 2 , 1

2 (1 − B )) for |x − y| < z < x + y and K ( x, y, z ) = 0 elsewhere and

B = cosh ² x + cosh ² y + cosh ² z − 1 2 cosh x cosh y cosh z in [1], we have

τ h f ( λ ) = ϕ ^(α,β) _λ ( h ) f ( λ ) (6)

For α ≥ ⁻¹ ₂ , we introduce the Bessel normalized function of the first kind

j α defined by

j α ( z ) = Γ( α + 1)

∞

n=0

( − 1) ⁿ ( ^z ₂ ) ²ⁿ

n !Γ( n + α + 1) , z ∈ C (7) Moreover, from (7) we see that

z−→0 lim

j α ( z ) − 1

z ² = 0 (8)

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

|z| ≤ η = ⇒ |j α ( z ) − 1 | ≥ c|z| ² (9) Lemma 1.2 The following inequalities are valid for Jacobi functions ϕ ^(α,β) _λ ( t )

(1) |ϕ ^(α,β) _λ ( t ) | ≤ 1 (10)

(2) 1 − ϕ ^(α,β) _λ ( t ) ≤ t ² ( λ ² + ρ ² ) (11) Proof: analog (see [lemmas 3.1-3.2, 3])

Lemma 1.3 Let α > ⁻¹ ₂ , α ≥ β ≥ ⁻¹ ₂ . Then for |η| ≤ ρ , there existes a positive constant c 1 such that

| 1 − ϕ ^(α,β) _μ+iη ( t ) | ≥ c 1 | 1 − j α ( μt ) | (12) Proof: (see[Lemma 9, 1])

2 An analog of Titchmarsh’s Theorem

In this section we give the main resultat of this paper, we need first to define Jacobi-Lipschitz class.

Definition 2.1 Let δ ∈ (0 , 1). A function f ∈ L ² ( R ⁺ , Δ _α,β ( t ) dt ) is said to be in the Jacobi-Lipschitz class, denote by Lip ( δ, 2), if

τ _h f ( x ) − f ( x ) = O ( h ^δ ) , as h −→ 0

Theorem 2.2 Let f ∈ L ² ( R ⁺ , Δ _α,β ( t ) dt ). Then the following are equiva- lents

1. f ∈ Lip ( δ, 2).

2. _∞

r | f ( λ ) | ² dμ ( λ ) = O ( r ^−2δ ) as r −→ + ∞ .

Proof: 1 = ⇒ 2:Assume that f ∈ Lip ( δ, 2). Then we have τ h f ( x ) − f ( x ) = O ( h ^δ ) , as h −→ 0 formulas (4) and (6) gives

τ h f ( x ) − f ( x ) ² =

_∞

0 | 1 − ϕ ^(α,β) _λ ( h ) | ² | f ( λ ) | ² dμ ( λ ) Since (12) and λ ∈ R ⁺ , we have

h 2hη

| 1 − ϕ ^(α,β) _λ ( h ) | ² | f ( λ ) | ² dμ ( λ ) ≥ c 1

h 2hη

| 1 − j α ( λh ) | ² | f ( λ ) | ² dμ ( λ ) From (9), we obtain

h 2hη

| 1 − ϕ ^(α,β) _λ ( h ) | ² | f ( λ ) | ² dμ ( λ ) ≥ c 1 c ² η ⁴ 16

h 2hη

| f ( λ ) | ² dμ ( λ ) There exists then a positive constant K such that

h 2hη

| f ( λ ) | ² dμ ( λ ) ≤ K

_∞

0 | 1 − ϕ ^(α,β) _λ ( h ) | ² | f ( λ ) | ² dμ ( λ )

≤ Kh ^2δ For all h > 0. Then

_2r

r

| f ( λ ) | ² dμ ( λ ) ≤ Kr ^−2δ for all r > 0. Furthermore, we have

_∞

r

| f ( λ ) | ² dμ ( λ ) =

∞

i=0

₂

_r

2

r

| f ( λ ) | ² dμ ( λ )

≤ K

∞

i=0

(2 ⁱ r ) ^−2δ

≤ Kr ^−2δ This prove

_∞

r

| f ( λ ) | ² dμ ( λ ) = O ( r ^−2δ ) as r −→ + ∞

2 = ⇒ 1: Suppose now that

_∞

r

| f ( λ ) | ² dμ ( λ ) = O ( r ^−2δ ) as r −→ + ∞ we write

τ _h f ( x ) − f ( x ) ² = I ₁ + I ₂ where

I ₁ =

h

0 | 1 − ϕ ^(α,β) _λ ( h ) | ² | f ( λ ) | ² dμ ( λ ) and

I ₂ =

_∞

h1

| 1 − ϕ ^(α,β) _λ ( h ) | ² | f ( λ ) | ² dμ ( λ ) Estimate the summands I ₁ and I ₂ .

we have from (10)

I ₂ ≤ 4

_∞

1h

| f ( λ ) | ² dμ ( λ ) = O ( h ^2δ ) To estimate I ₁ , we use the inequality (10).

I ₁ =

h

0 | 1 − ϕ ^(α,β) _λ ( h ) | ² | f ( λ ) | ² dμ ( λ ) ≤ 2

h

0 | 1 − ϕ ^(α,β) _λ ( h ) || f ( λ ) | ² dμ ( λ ) From the inequality (11), we have

I ₁ ≤ 2 h ²

h

0

( λ ² + ρ ² ) | f ( λ ) | ² dμ ( λ )

= 2 ρ ² h ²

h

0 | f ( λ ) | ² dμ ( λ ) + 2 h ²

h

0 λ ² | f ( λ ) | ² dμ ( λ ) Note that

2 ρ ² h ²

h

0 | f ( λ ) | ² dμ ( λ ) ≤ 2 ρ ² h ²

_∞

0 | f ( λ ) | ² dμ ( λ )

= 2 ρ ² f ²

= O ( h ^2δ ) since 2 δ < 2

We put

ψ ( r ) =

_∞

r

| f ( λ ) | ² dμ ( λ ) Integrating by parts, we obtain

2 h ²

h

0 λ ² | f ( λ ) | ² dμ ( λ ) = 2 h ²

h

0

( −r ² ψ

( r ) dr

= 2 h ² ( − 1 h ² ψ ( 1

h ) + 2

h

0 rψ ( r ) dr )

= − 2 ψ ( 1

h ) + 4 h ²

h

0 rψ ( r ) dr

≤ 4 Ch ²

h

0 r ^1−2δ dr

= O ( h ^2δ ) Finally, then

τ h f ( x ) − f ( x ) = O ( h ^δ ) , as h −→ 0 which complets the proof of Theorem .

References

[1] W.O. Bray, M.A. Pinsky, Growth properties of Fourier transforms via moduli of continuity, Journal of Functional Analysis 255(2008) 2265-2285.

[2] M. Flensted-Jensen and T. Koornwinder, The convolution structure for Jacobi function expansions, Ark. Mat. 11 (1973), 245-262.

[3] L. H¨ ormander, Estimates for translation invariant operators in L _p spaces, Acta Math. 104, (1960) 93-140.

[4] T.H. Koornwinder, Jacobi functions and analysis on noncompact semisim- ple Lie groups, Special functions, group theoretical aspects and applica- tions, Math. Appl., Dordrecht, pp, 1-85 (1984).

[5] S.S. Platonov, Approximation of functions in L ₂ -metric on noncompact rank 1 symmetric spaces, Algebra i Analiz, 11, No,1, (1999) 244-270.

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

[7] E.C.Titchmarsh, Introduction to the theory of Fourier Integrals , Oxford University Press, Amen House, London, E.C.4, 1948.

Received: September, 2011