On estimates for the Bessel transform

Prace Naukowe Uniwersytetu Śląskiego nr 3106, Katowice

ON ESTIMATES FOR THE BESSEL TRANSFORM

Ahmed Abouelaz, Radouan Daher, Mohamed El Hamma

Abstract. Using a Bessel translation operator, we obtain a generalization of Theorem 2.2 in [3] for the Bessel transform for functions satisfying the (ψ, δ, β)- Bessel Lipschitz condition in the space L_2,α(R⁺).

1. Introduction and preliminaries

Integral transforms and their inverses (e.g., the Bessel transform) are widely used to solve various problems in calculus, mechanics, mathematical physics, and computational mathematics (see, e.g., [6, 7, 9, 10]).

Let

B = d²

dt² +(2α + 1) t

d dt,

be the Bessel differential operator. For α > −¹₂, we introduce the Bessel normalized function of the first kind j_α defined by

j_α(x) = Γ(α + 1)

∞

X

n=0

(−1)ⁿ(x/2)²ⁿ n!Γ(n + α + 1),

Received: 25.11.2012. Revised: 22.07.2013.

(2010) Mathematics Subject Classification: 42A38.

Key words and phrases: Bessel operator, Bessel transform, Bessel translation operator.

where Γ(x) is the gamma-function (see [5]). The function y = j_α(x) satisfies the differential equation

By + y = 0

with the initial conditions y(0) = 1 and y⁰(0) = 0. The function j_α(x) is infinitely differentiable and even.

Lemma 1.1. The following inequalities are valid for Bessel function jα

(1) |j_α(x)| ≤ 1

(2) 1 − j_α(x) = O(x²), 0 ≤ x ≤ 1.

Proof. (See [1]) 

Lemma 1.2. The following inequality is true

|1 − j_α(x)| ≥ c, with x ≥ 1, where c > 0 is a certain constant.

Proof. The asymptotic formulas for the Bessel function imply that jα(x)

→ 0 as x → ∞. Consequently, a number x₀ > 0 exists such that with x ≥ x₀ the inequality |j_α(x)| ≤ ¹₂ is true. Let m = min_x∈[1,x0]|1 − j_α(x)|. With x ≥ 1 we get the inequality

|1 − jα(x)| ≥ c,

where c = min(¹₂, m). 

Assume that L_2,α(R⁺), α > −¹₂, is the Hilbert space of measurable func- tions f (x) on R⁺ with the finite norm

kf k = kf k_2,α =

Z ∞ 0

|f (t)|²t^2α+1dt

1/2

.

It is well known that the Bessel transform of a function f ∈ L2,α(R⁺) is defined (see [4, 5, 8]) by the formula

f (λ) =b Z ∞

0

f (t)jα(λt)t^2α+1dt, λ ∈ R⁺.

The inverse Bessel transform is given by the formula f (t) = (2^αΓ(α + 1))⁻²

Z ∞ 0

f (λ)jb _α(λt)λ^2α+1dλ.

Theorem 1.3 ([4]). If f ∈ L2,α(R⁺) then we have the Parseval’s equality k bf k = (2^αΓ(α + 1))kf k.

In L_2,α(R⁺), consider the Bessel translation operator T_h

T_hf (t) = c_α Z π

0

f (p

t²+ h²− 2th cos ϕ) sin^2αϕdϕ, where

cα=

Z π 0

sin^2αϕdϕ

−1

= Γ(α + 1) Γ(1/2)Γ(α + ¹₂). It is easy to see that

T₀f (x) = f (x).

The operator Th is linear, homogeneous, and continuous. Below are some properties of this operator (see [5]):

(1) T_hj_α(λx) = j_α(λh)j_α(λx).

(2) T_h is self-adjoint: If f (x) is continuous function such that Z ∞

0

x^2α+1|f (x)|dx < ∞

and g(x) is continuous and bounded for all x ≥ 0, then Z ∞

0

(T_hf (x))g(x)x^2α+1dx = Z ∞

0

f (x)(T_hg(x))x^2α+1dx.

(3) T_hf (x) = T_xf (h).

(4) kThf − f k → 0 as h → 0.

The following relation connect the Bessel translation operator, in [2], we have (1.1) (T\_hf )(λ) = j_α(λh) bf (λ).

For any function f (x) ∈ L_2,α(R⁺) we define differences of the order m such that m ∈ {1, 2, . . .} with a step h > 0 by

(1.2) ∆^m_hf (x) = (T_h− I)^mf (x), where I is the unit operator.

Lemma 1.4. Let f ∈ L^2,α(R⁺). Then k∆^m_h f (x)k²= 1

(2^αΓ(α + 1))² Z ∞

0

|1 − j_α(λh)|^2m| bf (λ)|²λ^2α+1dλ.

Proof. From formulas (1.1) and (1.2), we have (∆\^m_hf )(λ) = (j_α(λh) − 1)^mf (λ).b

By Parseval’s identity, we obtain the result. 

In [3], we have

Theorem 1.5. Let f ∈ L^2,α(R⁺). Then the following are equivalents (1) f ∈ Lip(ψ, α, 2)

(2) R∞

r | bf (λ)|²λ^2α+1sλ = O(ψ(r⁻²) as h → +∞, where Lip(ψ, α, 2) is the ψ-Bessel Lipschitz class.

The main aim of this paper is to establish a generalization of Theorem 1.5 in the Bessel transform. For this purpose, we use the Bessel translation operator.

2. Main Results

In this section we give the main result of this paper. We need first to define (ψ, δ, β)-Bessel Lipschitz class.

Definition 2.1. A function f ∈ L2,α(R⁺) is said to be in the (ψ, δ, β)- Bessel Lipschitz class, denote by Lip²(ψ, δ, β), if

k∆^m_hf (t)k = O(h^δψ(h^β)) as h → 0, where

(1) δ > m, β > 0 and m ∈ {1, 2, . . .},

(2) ψ is a continuous increasing function on [0, ∞), (3) ψ(0) = 0 and ψ(ts) = ψ(t)ψ(s) for all t, s ∈ [0, ∞), (4) and

Z 1/h 0

s^2m−2δ−1ψ(s^−2β)ds = O(h^2δ−2mψ(h^2β)) as h → 0.

Theorem 2.2. Let f ∈ L2,α(R⁺). Then the following are equivalent (1) f ∈ Lip²(ψ, δ, β),

(2) R∞

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

Proof. (1) =⇒ (2): Assume that f ∈ Lip²(ψ, δ, β). Then k∆^m_hf (t)k = O(h^δψ(h^β)) as h → 0.

Lemma 1.4 gives

k∆^m_h f (x)k²= 1 (2^αΓ(α + 1))²

Z ∞ 0

|1 − j_α(λh)|^2m| bf (λ)|²λ^2α+1dλ.

If λ ∈ [¹_h,²_h] then λh ≥ 1 and Lemma 1.2 implies that

1 ≤ 1

c^2m|1 − j_α(λh)|^2m. Then

Z 2/h 1/h

| bf (λ)|²λ^2α+1dλ ≤ 1 c^2m

Z 2/h 1/h

|1 − j_α(λh)|^2m| bf (λ)|²λ^2α+1dλ

≤ 1

c^2m Z ∞

0

|1 − j_α(λh)|^2m| bf (λ)|²λ^2α+1dλ

and there exists a positive constant C such that Z 2/h

1/h

| bf (λ)|²λ^2α+1dλ ≤ Ch^2δψ(h^2β).

We obtain

Z 2r r

| bf (λ)|²λ^2α+1dλ ≤ Cr^−2δψ(r^−2β).

So that Z ∞

r

| bf (λ)|²λ^2α+1dλ = hZ 2r r

+ Z 4r

2r

+ Z 8r

4r

+ . . .i

| bf (λ)|²λ^2α+1dλ

≤ C r^−2δψ(r^−2β) + (2r)^−2δψ((2r)^−2β) + . . .

≤ Cr^−2δψ(r^−2β) 1 + 2^−2δψ(2^−2β)

+(2^−2δψ(2^−2β))²+ (2^−2δψ(2^−2β))³+ . . .

≤ CK_δ,βr^−2δψ(r^−2β),

where K_δ,β = (1 − 2^−2δψ(2^−2β))⁻¹ since 2^−2δψ(2^−2β) < 1. This proves that Z ∞

r

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

(2) =⇒ (1): Suppose now that Z ∞

r

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

We have to show that Z ∞

0

|1 − j_α(λh)|^2m| bf (λ)|²λ^2α+1dλ = O(h^2δψ(h^2β)) as h → 0.

We write

Z ∞ 0

|1 − j_α(λh)|^2m| bf (λ)|²λ^2α+1dλ = I₁+ I₂,

where

I₁ = Z 1/h

0

|1 − jα(λh)|^2m| bf (λ)|²λ^2α+1dλ

and

I2= Z ∞

1/h

|1 − jα(λh)|^2m| bf (λ)|²λ^2α+1dλ.

Firstly, we have from (1) in Lemma 1.1

I₂≤ 4^m Z ∞

1/h

| bf (λ)|²λ^2α+1dλ = O(h^2δψ(h^2β)).

Set

g(x) = Z ∞

x

| bf (λ)|²λ^2α+1dλ.

From (1) and (2) of Lemma 1.1 and integration by parts, we obtain

I₁ =

Z 1/h 0

|1 − j_α(λh)|^2m| bf (λ)|²λ^2α+1dλ

≤ 2^m Z 1/h

0

|1 − jα(λh)|^m| bf (λ)|²λ^2α+1dλ

≤ −C₁h^2m Z 1/h

0

x^2mg⁰(x)dx

≤ −C1g(1/h) + 2mC1h^2m Z 1/h

0

x^2m−1g(x)dx

≤ C₂h^2m Z 1/h

0

x^2m−1x^−2δψ(x^−2β)dx

≤ C₂h^2m Z 1/h

0

x^2m−1−2δψ(x^−2β)dx

≤ C3h^2mh^2δ−2mψ(h^2β)

≤ C₃h^2δψ(h^2β),

where C₁, C₂ and C₃ are a positive constants and this ends the proof. 

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Department of Mathematics Faculty of Sciences Aïn Chock University of Hassan II Casablanca, Morocco e-mail: ah.abouelaz@gmail.com e-mail: rjdaher024@gmail.com e-mail: m_elhamma@yahoo.fr