AN APPROXIMATION OF THE HANKEL TRANSFORM FOR ABSOLUTELY CONTINUOUS MAPPINGS

N.M. DRAGOMIR, S.S. DRAGOMIR, M. GU, X. GAN, AND R. WHITE

Abstract. Using some techniques developed by Dragomir and Wang in the recent paper [2] in connection to Ostrowski integral inequality, we point out some approximation results for the Henkel’s transform of absolutely continuous mapping.

1. Introduction

Two-dimensional systems may often show circular symmetry, for example optical systems are often constructed from components that, in themselves, are circulary symmetrical.

When circular symmetry exists, that is, when f(x, y) = f(r), r^{2} = x^{2} + y^{2}, then the bidimensional Fourier transform can be represented in the following way [8, p. 244 - p. 250]

(1.1)  ∫_{−∞}^{∞} ∫_{−∞}^{∞} f(x, y)e^{−i2π(xu+yv)}dxdy

= ∫_{0}^{∞} ∫_{0}^{2π} f(r)e^{−i2πqr cos(θ−φ)}rdrdθ = ∫_{0}^{∞} f(r) [∫_{0}^{2π} e^{−i2πqr cos(θ−φ)}dθ] rdr

= 2π ∫_{0}^{∞} f(r)J_{0}(2πqr)rdr

where x + iy = re^{iθ}, u + iv = qe^{iφ}, q^{2} = u^{2} + v^{2} and we have used the relation

(1.2)  J_{0}(z) = \frac{1}{2π} ∫_{0}^{2π} e^{−iz cos β}dβ

We refer to G(f)(q) given by

(1.3)  G(f)(q) = 2π ∫_{0}^{∞} f(r)J_{0}(2πqr)rdr

as the Hankel transform (of zero order) of f(r).

The main aim of the present article is to point out some estimates of the Hankel transform for absolutely continuous mappings defined on an finite interval [a, b] by the use of some techniques developed by Dragomir and Wang in the recent paper [2] in connection to Ostrowski integral inequality. Some adaptive quadrature formulae which will allowe another approach than the classical one will be also derived.

Date: February, 1999.

2. An Integral Representation

Let J_{1} (·) represent the first-order Bessel functions of the first kind, that is,

(2.1)  J_{1} (x) := \frac{1}{2π} ∫_{−π}^{π} e^{ix sin φ−iφ}dφ, x ∈ ℝ.

Define the corresponding Bessel’s mean as:

(2.2)  B_{1} (z, w) = \{ \begin{matrix}z if w = z \\ \frac{zJ_{1}(z)−wJ_{1}(w)}{z−w} if w ≠ z \end{matrix} ; w, z ∈ ℝ.

The following representation of Hankel transform holds.

Theorem 1. Let g : [a, b] → 𝕂 (𝕂 = ℂ, ℝ) be an absolutely continuous mapping on [a, b] . Then we have the representation

(2.3)  G (g) (ρ) = \frac{1}{ρ}B_{1} (2πbρ, 2πaρ) × ∫_{a}^{b} g (s) ds

+\frac{2π}{b − a} ∫_{a}^{b} ∫_{a}^{b} k (r, s) g′ (s) rJ_{0} (2πrρ) drds

for all ρ ∈ [a, b] , ρ ≠ 0, where k (·, ·) : [a, b]^{2} → ℝ is given by

(2.4)  k (u, v) := \{ \begin{matrix}v − a if v ∈ [a, u] \\ v − b if v ∈ (u, b] \end{matrix} ; (u, v) ∈ [a, b]^{2}

and J_{0} (·) is the zeroth-order Bessel function of the first kind, that is,

(2.5)  J_{0} (x) = \frac{1}{2π} ∫_{0}^{2π} e^{−ix cos β}dβ, x ∈ ℝ.

Proof. Using the integration by parts formula for absolutely continuous mappings on [a, b] , we can write

(2.6)  ∫_{a}^{x} (s − a) g′ (s) ds = (x − a) g (x) − ∫_{a}^{x} g (s) ds

and

(2.7)  ∫_{x}^{b} (s − b) g′ (s) ds = (b − x) g (x) − ∫_{x}^{b} g (s) ds

for all x ∈ [a, b] . Adding (2.6) and (2.7) , we end up with (see also [2])

(2.8)  g (x) = \frac{1}{b − a} ∫_{a}^{b} g (s) ds + \frac{1}{b − a} ∫_{a}^{b} k (r, s) g′ (s) ds

for all x ∈ [a, b] , which is of importance in itself, too. Now, consider the Hankel transform of g on the interval [a, b] , that is:

(2.9)  G (g) (ρ) := 2π ∫_{a}^{b} g (r) rJ_{0} (2πrρ) dr

AN APPROXIMATION OF THE HANKEL TRANSFORM FOR ABSOLUTELY CONTINUOUS MAPPINGS 3

and use the representation (2.8) to get G (g) (ρ) (2.10)

= 2π ∫_{a}^{b} [\frac{1}{b − a} ∫_{a}^{b} g (s) ds + \frac{1}{b − a} ∫_{a}^{b} k (r, s) g′ (s) ds] rJ_{0} (2πrρ) dr
= \frac{1}{b − a} ∫_{a}^{b} g (s) ds · 2π ∫_{a}^{b} rJ_{0} (2πrρ) dr

+\frac{2π}{b − a} ∫_{a}^{b} ∫_{a}^{b} k (r, s) g′ (s) rJ_{0} (2πrρ) dsdr.

Consider the change of variable r′ = 2πrρ. Then r = \frac{r′}{2πρ}, dr = \frac{dr′}{2πρ} and

∫_{a}^{b} rJ_{0} (2πrρ) dr = ∫_{2πaρ}^{2πbρ} \frac{r′}{2πρ}J_{0} (r′) \frac{1}{2πρ}dr′
= \frac{1}{(2πρ)^{2}} ∫_{2πaρ}^{2πbρ} r′J_{0} (r′) dr′.

It is a well-known property of Bessel functions that

(2.11)  ∫_{0}^{x} ξJ_{0} (ξ) dξ = xJ_{1} (x) , x ∈ ℝ.

Consequently,

∫_{2πaρ}^{2πbρ} r′J_{0} (r′) dr′ = ∫_{0}^{2πbρ} r′J_{0} (r′) dr′ − ∫_{0}^{2πaρ} r′J_{0} (r′) dr′
= 2πbρJ_{1} (2πbρ) − 2πaρJ_{1} (2πaρ)

and

\frac{2π}{b − a} ∫_{a}^{b} rJ_{0} (2πrρ) dr = \frac{2π}{b − a} · \frac{1}{(2πρ)^{2}} [2πbρJ_{1} (2πbρ) − 2πaρJ_{1} (2πaρ)]
= \frac{1}{ρ} \frac{2πbρJ_{1} (2πbρ) − 2πaρJ_{1} (2πaρ)}{2πbρ − 2πaρ}
= \frac{1}{ρ}B_{1} (2πbρ, 2πaρ) .

Using (2.10) , we deduce the desired representation (2.1) .

In practical applications we have a = 0 and b = 1. Consequently, we can state the following corollary. Corollary 2. Let g : [0, 1] → 𝕂 (𝕂 = ℂ, ℝ) be an absolutely continuous mapping on [0, 1] . Then we have the representation G (g) (ρ) (2.12)

= \frac{J_{1} (2πρ)}{ρ} × ∫_{0}^{1} g (s) ds + 2π ∫_{0}^{1} ∫_{0}^{1} k̃ (r, s) g′ (s) rJ_{0} (2πrρ) drds, ρ ∈ (0, 1]

where k̃ : [0, 1] → ℝ is given by

k̃ (u, v) = \{ \begin{matrix}v if v ∈ [0, u] \\ v − 1 if v ∈ (u, 1] \end{matrix} .

Now let us define the mapping with real values I (g) : [a, b] → [0, ∞) given by

(2.13)  I (g) (ρ) := |G (g) (ρ)|^{2} , ρ ∈ [a, b] .

Using the well known property of complex numbers

(2.14)  |x + y|^{2} = |x|^{2} + 2 Re (zȳ) + |y|^{2} for any x, y ∈ ℂ

we can state the following corollary:

Corollary 3. With the assumptions from Theorem 1, we have

(2.15) I (g) (ρ) = \frac{1}{ρ^{2}} |B_{1} (2πbρ, 2πaρ)|^{2} |∫_{a}^{b} g (s) ds|^{2}

+\frac{4π}{ρ (b − a)} Re [B_{1} (2πbρ, 2πaρ) ∫_{a}^{b} g (s) ds

× ∫_{a}^{b} ∫_{a}^{b} k (r, s) r\overline{g′ (s)}\overline{J_{0} (2πrρ)}drds]

+\frac{4π^{2}}{(b − a)^{2}} |∫_{a}^{b} ∫_{a}^{b} k (r, s) g′ (s) rJ_{0} (2πrρ) drds|^{2}

for all ρ ∈ [a, b] , ρ ≠ 0. If a = 0, b = 1, then

(2.16)  I (g) (ρ) = \frac{1}{ρ^{2}} |J_{1} (2πρ)|^{2} |∫_{0}^{1} g (s) ds|^{2}

+\frac{4π}{ρ} Re [J_{1} (2πρ) ∫_{0}^{1} g (s) ds

× ∫_{0}^{1} ∫_{0}^{1} k̃ (r, s) r\overline{g′ (s)}\overline{J_{0} (2πrρ)}drds]

+4π^{2} |∫_{0}^{1} ∫_{0}^{1} k̃ (r, s) g′ (s) rJ_{0} (2πrρ) drds|^{2}

for all ρ ∈ (0, 1] .

3. Integral Inequalities

The main aim of this section is to point out an estimate for the remainder

(3.1)  R [g] (ρ) := \frac{2π}{b − a} ∫_{a}^{b} ∫_{a}^{b} k (r, s) g′ (s) rJ_{0} (2πrρ) drds

in formula (2.3) . We can state the following integral inequalities.

AN APPROXIMATION OF THE HANKEL TRANSFORM FOR ABSOLUTELY CONTINUOUS MAPPINGS 5

Theorem 4. Let g be as in Theorem 1. Then

|G (g) (ρ) − \frac{1}{ρ}B_{1} (2πbρ, 2πaρ) × ∫_{a}^{b} g (s) ds| (3.2)

≤ \{ \begin{matrix}\frac{2π}{b−a} ∥g′∥_{∞} ∫_{a}^{b} [\frac{(r−a)^{2}+(b−r)^{2}}{2} ] |r| dr if g′ ∈ L_{∞} [a, b] ; \\ \frac{2π}{b−a} & ∥g′∥ & p & [∫_{a}^{b} [\frac{(r−a)^{q+1}+(b−r)^{q+1}}{q+1} ] |r|^{q} dr] & \frac{1}{q} \\ if g′ ∈ L_{p} [a, b] , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ 2π ∥g′∥_{1} ∫_{a}^{b} |r| dr \end{matrix}

for all ρ ∈ [a, b] , ρ ≠ 0. Proof. Using the representation (2.3) , we get

|G (g) (ρ) − \frac{1}{ρ}B_{1} (2πbρ, 2πaρ) ∫_{a}^{b} g (s) ds|

≤ \frac{2π}{b − a} ∫_{a}^{b} ∫_{a}^{b} |k (r, s)| |r| |g′ (s)| |J_{0} (2πrρ)| drds =: A (ρ)

for all ρ ∈ [a, b] \ {0} . It is obvious that

|J_{0} (2πrρ)| = |\frac{1}{2π} ∫_{0}^{2π} e^{−x cos β}dβ| ≤ \frac{1}{2π} ∫_{0}^{2π} |e^{−x cos β}| dβ = \frac{1}{2π} ∫_{0}^{2π} dβ = 1.

In this way we can state the following inequality

A (ρ) ≤ \frac{2π}{b − a} ∫_{a}^{b} ∫_{a}^{b} |k (r, s)| |r| |g′ (s)| drds := B (ρ) .

It is obvious that

(3.3) B (ρ) ≤ ∥g′∥_{∞} \frac{2π}{b − a} ∫_{a}^{b} ∫_{a}^{b} |k (r, s)| |r| drds
= ∥g′∥_{∞} \frac{2π}{b − a} ∫_{a}^{b} (∫_{a}^{b} |k (r, s)| ds) |r| dr
= ∥g′∥_{∞} \frac{2π}{b − a} ∫_{a}^{b} [∫_{a}^{r} (s − a) ds + ∫_{r}^{b} (b − s) ds] |r| dr
= ∥g′∥_{∞} \frac{2π}{b − a} ∫_{a}^{b} [\frac{(r − a)^{2} + (b − r)^{2}}{2} ] |r| dr,

and the first inequality in (3.2) is obtained. For the second inequality, we use Hölder’s integral inequality for double integrals

6 N.M. DRAGOMIR, S.S. DRAGOMIR, M. GU, X. GAN, AND R. WHITE

to get: B (ρ) (3.4)

≤ \frac{2π}{b − a} (∫_{a}^{b} ∫_{a}^{b} |k (r, s)|^{q} |r|^{q} drds)^{\frac{1}{q}} (∫_{a}^{b} ∫_{a}^{b} |g′ (s)|^{p} dsdr)^{\frac{1}{p}}
= \frac{2π}{b − a} (b − a)^{\frac{1}{p}} ∥g′∥_{p} (∫_{a}^{b} (∫_{a}^{r} (s − a)^{q} ds + ∫_{r}^{b} (b − s)^{q} ds) |r|^{q} dr)^{\frac{1}{q}}
= \frac{2π}{b − a} (b − a)^{\frac{1}{p}} ∥g′∥_{p} (∫_{a}^{b} [\frac{(r − a)^{q+1} + (b − r)^{q+1}}{q + 1} ] |r|^{q} dr)^{\frac{1}{q}}

and the second inequality is also proved. Finally, we observe that

(3.5)  B (ρ) ≤ \frac{2π}{b − a} sup_{(r,s)∈[a,b]^{2}} |k (r, s)| ∫_{a}^{b} |r| dr · ∫_{a}^{b} |g′ (s)| ds
= \frac{2π}{b − a} (b − a) ∥g′∥_{1} ∫_{a}^{b} |r| dr = 2π ∥g′∥_{1} ∫_{a}^{b} |r| dr

and the theorem is proved.

Remark 1. In practical applications a ≥ 0, so the first bound in (3.2) becomes

\frac{2π}{b − a} ∥g′∥_{∞} ∫_{a}^{b} [\frac{(r − a)^{2} + (b − r)^{2}}{2} ] rdr

= \frac{2π}{b − a} ∥g′∥_{∞} [\frac{1}{2} ∫_{a}^{b} (r − a)^{2} rdr + \frac{1}{2} ∫_{a}^{b} (b − r)^{2} rdr]
= \frac{2π}{b − a} ∥g′∥_{∞} [\frac{1}{24} (b − a)^{3} (3b + a) + \frac{1}{24} (b − a)^{3} (3a + b)]
= \frac{2π}{b − a} ∥g′∥_{∞} \frac{1}{24} (b − a)^{3} (4b + 4a)
= \frac{π}{3} ∥g′∥_{∞} (b − a)^{2} (a + b) .

The second term will be:

\frac{2π}{(b − a) (q + 1)^{\frac{1}{q}}} ∥g′∥_{p} [∫_{a}^{b} (r − a)^{q+1} r^{q}dr + ∫_{a}^{b} (b − r)^{q+1} r^{q}dr]^{\frac{1}{q}} ,

and the third term is:

π ∥g′∥_{1} (b + a) (b − a) .

Consequently, we can state that

|G (g) (ρ) − \frac{1}{ρ}B_{1} (2πbρ, 2πaρ) ∫_{a}^{b} g (s) ds| (3.6)

≤ \{ \begin{matrix}\frac{π}{3} ∥g′∥_{∞} (a + b) (b − a)^{2} if g′ ∈ L_{∞} [a, b] ; \\ \frac{2π}{(b−a)(q+1)^{\frac{1}{q}}} & ∥g′∥ & p & [∫_{a}^{b} (r − a)^{q+1} r^{q}dr + ∫_{a}^{b} (b − r)^{q+1} r^{q}dr] & \frac{1}{q} \\ if g′ ∈ L_{p} [a, b] , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ π & ∥g′∥ & 1 & (b + a) & (b − a) \end{matrix}

for all ρ ∈ [a, b] . Remark 2. If we assume that a = 0 and b > 0 in (3.6) , then we get

|G (g) (ρ) − \frac{J_{1} (2πbρ)}{ρ} × ∫_{0}^{b} g (s) ds| (3.7)

≤ \{ \begin{matrix}\frac{π}{3} ∥g′∥_{∞} b^{3} if g′ ∈ L_{∞} [a, b] \\ π ∥g′∥_{1} b^{2} \end{matrix} .

The above inequality shows that

G (g) (ρ) ≈ \frac{J_{1} (2πbρ)}{ρ} × ∫_{0}^{b} g (s) ds

when b → 0+ and the precision of approximation is 3. Remark 3. Let us observe that the integral

I = ∫_{a}^{b} (b − r)^{2} (r − a)^{2} dr

can be written in a different form. By using the change of variable r = a (1 − t)+bt, t ∈ [0, 1] , we obtain

I = (b − a) ∫_{0}^{1} [b − (1 − t) a − tb]^{q} [(1 − t) a + tb − a]^{s} dt
= (b − a)^{q+s+1} ∫_{0}^{1} (1 − t)^{q} t^{s}dt = (b − a)^{q+s+1} β (q + 1, s + 1)

where β (·, ·) is a Beta function, that is,

β (q, s) = ∫_{0}^{1} (1 − t)^{q−1} t^{s−1}dt; q, s > 0.

Now, coming back to the second bound in (3.6) for a = 0, we can observe that

∫_{0}^{b} r^{q+1}r^{q}dr = ∫_{0}^{b} r^{2q+1}dr = \frac{b^{2q+2}}{2q + 2}

and

∫_{0}^{b} (b − r)^{q+1} r^{q}dr = b^{2q+2}β (q + 2, q + 1) .

Consequently, we have the inequality

|G (g) (ρ) − \frac{J_{1} (2πbρ)}{ρ} ∫_{0}^{b} g (s) ds| (3.8)

≤ \frac{2π}{b (q + 1)^{\frac{1}{q}}} ∥g′∥_{p} [\frac{b^{2q+2}}{2q + 2} + b^{2q+2}β (q + 2, q + 1)]^{\frac{1}{q}}

= \frac{2πb^{1+ \frac{2}{q}}}{(q + 1)^{\frac{1}{q}}} [\frac{1}{2q + 2} + β (q + 2, q + 1)]^{\frac{1}{q}}

for all ρ ∈ (0, b] .

In some practical applications the upper limit of integration is b = 1, therefore the following corollary is required. Corollary 5. Let g : [0, 1] → 𝕂 (𝕂 = ℂ, ℝ) be an absolutely continuous mapping on [0, 1] . Then we have the inequality

|G (g) (ρ) − \frac{J_{1} (2πρ)}{ρ} × ∫_{0}^{1} g (s) ds| (3.9)

≤ \{ \begin{matrix}\frac{π}{3} ∥g′∥_{∞} if g′ ∈ L_{∞} [a, b] \\ \frac{2π}{(q+1)^{\frac{1}{q}}} & [\frac{1}{2q+2} + β (q + 2, q + 1)] & \frac{1}{q} & ∥g′∥ & p \\ if g′ ∈ L_{p} [0, 1] , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ π & ∥g′∥ & 1 \end{matrix}

for all ρ ∈ (0, 1] .

Using the inequality (3.2) which provides upper bounds for the remainder R [g] (ρ) , we will point out the following inequality which approximates the mapping I (g) (ρ) (c.f.(2.13)). Corollary 6. Let g : [a, b] → 𝕂 (𝕂 = ℂ, ℝ) be an absolutely continuous mapping on [a, b] . Then we have the inequality

|I (g) (ρ) − \frac{1}{ρ^{2}} |B_{1} (2πbρ, 2πaρ)|^{2} |∫_{a}^{b} g (s) ds|^{2}|

≤ [\frac{2}{ρ} |B_{1} (2πbρ, 2πaρ)| |∫_{a}^{b} g (s) ds| + E (g) (ρ)] E (g) (ρ)

where

E (g) (ρ) := \{ \begin{matrix}\frac{2π}{b−a} ∥g′∥_{∞} ∫_{a}^{b} [\frac{(r−a)^{2}+(b−r)^{2}}{2} ] |r| dr if g′ ∈ L_{∞} [a, b] ; \\ \frac{2π}{b−a} & ∥g′∥ & p & [∫_{a}^{b} [\frac{(r−a)^{q+1}+(b−r)^{q+1}}{q+1} ] |r|^{q} dr] & \frac{1}{q} \\ if g′ ∈ L_{p} [a, b] , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ 2π ∥g′∥_{1} ∫_{a}^{b} |r| dr \end{matrix}

for all ρ ∈ [a, b) , ρ ≠ 0.

The proof is obvious by Corollary 3 and Theorem 4.

4. A Quadrature Formula

In this section we point out a quadrature formula for the Hankel’s transform

G (g) (ρ) := 2π ∫_{0}^{1} g (s) rJ_{0} (2πrρ) dr

where g is assumed to be absolutely continuous on [0, 1] .

Firstly, let us assume that in formula (3.6) we have 0 ≤ a ≤ b ≤ 1. Then we have the coarser upper bound

|G (g) (ρ) − \frac{1}{ρ}B_{1} (2πbρ, 2πaρ) × ∫_{a}^{b} g (s) ds| (4.1)

≤ \{ \begin{matrix}\frac{2π}{3} ∥g′∥_{∞} (b − a)^{2} if g′ ∈ L_{∞} [a, b] ; \\ \frac{2^{1+ \frac{1}{q}} π∥g′∥_{p}}{[(q+1)(q+2)]^{\frac{1}{q}}} (b − a)^{\frac{2}{q}} if g′ ∈ L_{p} [a, b] , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ 2π ∥g′∥_{1} (b − a) . \end{matrix}

Indeed, from (3.6) we have

|G (g) (ρ) − \frac{1}{ρ}B_{1} (2πbρ, 2πaρ) × ∫_{a}^{b} g (s) ds|

≤ \{ \begin{matrix}\frac{π}{3} ∥g′∥_{∞} (a + b) (b − a)^{2} ; \\ _{\frac{2π}{(b−a)(q+1)^{\frac{1}{q}}}} ∥g′∥_{p} [∫_{a}^{b} (r − a)^{q+1} r^{q}dr + ∫_{a}^{b} (b − r)^{q+1} r^{q}dr]^{\frac{1}{q}} ; \\ π ∥g′∥_{1} (b + a) (b − a) . \end{matrix}

As 0 ≤ a ≤ b ≤ 1, then a + b ≤ 2 and hence:

\frac{π}{3} ∥g′∥_{∞} (a + b) (b − a)^{2} ≤ \frac{2π}{3} ∥g′∥_{∞} (b − a)^{2}

and

π ∥g′∥_{1} (b + a) (b − a) ≤ 2π ∥g′∥_{1} (b − a) .

On the other hand,

∫_{a}^{b} (r − a)^{q+1} r^{q}dr ≤ ∫_{a}^{b} (r − a)^{q+1} dr = \frac{(b − a)^{q+2}}{q + 2}

and

∫_{a}^{b} (b − r)^{q+1} r^{q}dr ≤ ∫_{a}^{b} (b − r)^{q+1} dr = \frac{(b − a)^{q+2}}{q + 2} .

Consequently, we get

\frac{2π}{(b − a) (q + 1)^{\frac{1}{q}}} ∥g′∥_{p} [∫_{a}^{b} (r − a)^{q+1} r^{q}dr + ∫_{a}^{b} (b − r)^{q+1} r^{q}dr]^{\frac{1}{q}}

≤ \frac{2π}{(b − a) (q + 1)^{\frac{1}{q}}} ∥g′∥_{p} [\frac{2 (b − a)^{q+2}}{q + 2} ]^{\frac{1}{q}}

= \frac{2^{1+ \frac{1}{q}} π ∥g′∥_{p} (b − a)^{\frac{2}{q}}}{[(q + 1) (q + 2)]^{\frac{1}{q}}}

and the second inequality in (4.1) is also proved.

Now, consider I_{n} : 0 = x_{0} < x_{1} < ... < x_{n−1} < x_{n} = 1 a division of the interval [0, 1] . Put h_{i} := x_{i+1} − x_{i}, (i = 0, ..., n − 1) and υ (h) := max {h_{i}|i = 0, ..., n − 1} the norm of the division. Construct the sums

H (g, I_{n}, ρ) (4.2)

: = \frac{1}{ρ} ∑_{i=0}^{n−1} B_{1} (2πx_{i+1}ρ, 2πx_{i}ρ) × ∫_{x_{i}}^{x_{i+1}} g (s) ds

= \frac{1}{ρ} ∑_{i=0}^{n−1} \frac{1}{h_{i}} [x_{i+1}J_{1} (2πx_{i+1}ρ) − x_{i}J_{1} (2πx_{i}ρ)] × ∫_{x_{i}}^{x_{i+1}} g (s) ds.

We can state the following theorem concerning the approximation of Hankel’s transform in terms of the quadrature formula (4.2) .

Theorem 7. Let g : [0, 1] → 𝕂 (𝕂 = ℂ, ℝ) be an absolutely continuous mapping on [0, 1] . Then we have

(4.3)  G (g) (ρ) = H (g, I_{n}, ρ) + R (g, I_{n}, ρ) , ρ ∈ (0, 1]

where H (g, I_{n}, ρ) is as given by the formula (4.2) and the remainder R (g, I_{n}, ρ) satisfies the estimate

(4.4)  |R (g, I_{n}, ρ)| ≤ \{ \begin{matrix}\frac{2π}{3} ∥g′∥_{∞} ∑_{i=0}^{n−1} h_{i}^{2} if g′ ∈ L_{∞} [a, b] ; \\ \frac{2^{1+ \frac{1}{q}} π}{[(q+1)(q+2)]^{\frac{1}{q}}} & ∥g′∥ & p & (∑_{i=0}^{n−1} h_{i}^{2}) & \frac{1}{q} \\ if g′ ∈ L_{p} [a, b] , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ 2πν (h) ∥g′∥_{1} \end{matrix}

for all ρ ∈ (0, 1] .

Proof. Apply formula (4.1) on the subintervals [x_{i}, x_{i+1}] (i = 0, ..., n − 1) to get

|2π ∫_{x_{i}}^{x_{i+1}} g (r) rJ_{0} (2πrρ) dr − \frac{1}{ρ}B_{1} (2πx_{i+1}ρ, 2πx_{i}ρ) × ∫_{x_{i}}^{x_{i+1}} g (s) ds|

≤ \{ \begin{matrix}\frac{2π}{3} sup_{r∈[x_{i},x_{i+1}]} |g′ (r)| h_{i}^{2} \\ \frac{2^{1+ \frac{1}{q}} π}{[(q+1)(q+2)]^{\frac{1}{q}}} (∫_{x_{i}}^{x_{i+1}} |g′ (s)|^{p} ds)^{\frac{1}{p}} hi^{\frac{2}{q}} \\ 2π (∫_{x_{i}}^{x_{i+1}} |g′ (s)| ds) h_{i}. \end{matrix}

Summing over i from 0 to n − 1 and using the generalized triangle inequality, we get

|R (g, I_{n}, ρ)|
≤ ∑_{i=0}^{n−1} |2π ∫_{x_{i}}^{x_{i+1}} g (r) rJ_{0} (2πrρ) dr − \frac{1}{ρ}B_{1} (2πx_{i+1}ρ, 2πx_{i}ρ) × ∫_{x_{i}}^{x_{i+1}} g (s) ds|

≤ \{ \begin{matrix}\frac{2π}{3} ∑_{i=0}^{n−1} sup_{r∈[x_{i},x_{i+1}]} |g′ (r)| h_{i}^{2} \\ \frac{2^{1+ \frac{1}{q}} π}{[(q+1)(q+2)]^{\frac{1}{q}}} ∑_{i=0}^{n−1} (∫_{x_{i}}^{x_{i+1}} |g′ (s)|^{p} ds)^{\frac{1}{p}} hi^{\frac{2}{q}} \\ 2π ∑_{i=0}^{n−1} (∫_{x_{i}}^{x_{i+1}} |g′ (s)| ds) h_{i} \end{matrix} .

Now, observe that

∑_{i=0}^{n−1} sup_{r∈[x_{i},x_{i+1}]} |g′ (r)| h_{i}^{2} ≤ ∥g′∥_{∞} ∑_{i=0}^{n−1} h_{i}^{2}

and then the first inequality in (4.4) is proved. Using Hölder’s discrete inequality, we deduce

∑_{i=0}^{n−1} (∫_{x_{i}}^{x_{i+1}} |g′ (s)|^{p} ds)^{\frac{1}{p}} hi^{\frac{2}{q}}

≤ (∑_{i=0}^{n−1} [(∫_{x_{i}}^{x_{i+1}} |g′ (s)|^{p} ds)^{\frac{1}{p}} ]^{p})^{\frac{1}{p}} × [∑_{i=0}^{n−1} (hi^{\frac{2}{q}} )^{q}]^{\frac{1}{q}}

= (∑_{i=0}^{n−1} (∫_{x_{i}}^{x_{i+1}} |g′ (s)|^{p} ds))^{\frac{1}{p}} × (∑_{i=0}^{n−1} h_{i}^{2})^{\frac{1}{q}}

= ∥g′∥_{p} (∑_{i=0}^{n−1} h_{i}^{2})^{\frac{1}{q}} .

Finally, let us observe that

2π ∑_{i=0}^{n−1} (∫_{x_{i}}^{x_{i+1}} |g′ (s)| ds) h_{i} ≤ 2πν (h) ∑_{i=0}^{n−1} (∫_{x_{i}}^{x_{i+1}} |g′ (s)| ds)
= 2πν (h) ∥g′∥_{1}

and the theorem is completely proved.

Remark 4. It is obvious that

∑_{i=0}^{n−1} h_{i}^{2} ≤ ν (h) ∑_{i=0}^{n−1} h_{i}

and then by (4.4) we deduce the coarser upper bound

(4.5)  |R (g, I_{n}, ρ)| ≤ \{ \begin{matrix}\frac{2π}{3} ∥g′∥_{∞} ν (h) ; \\ \frac{2^{1+ \frac{1}{q}} π}{[(q+1)(q+2)]^{\frac{1}{q}}} ∥g′∥_{p} [ν (h)]^{\frac{1}{q}} ; \\ 2πν (h) ∥g′∥_{1} . \end{matrix}

Now, observe that if ν (h) → 0, then every branch on the right hand side of (4.5) goes to 0, which proves that H (g, I_{n}, ρ) approximates the Hankel’s transform with any accuracy.

In practical problems the interval [0, 1] is devided in equidistant subintervals by the division I_{n} : x_{i} = \frac{i}{n}, i = 0, ..., n.

If we consider the sum

H_{n} (g, ρ) (4.6)

: = \frac{1}{ρ} ∑_{i=0}^{n−1} B_{1} (\frac{2π (i + 1) ρ}{n}, \frac{2πiρ}{n}) × ∫ \frac{i+1}{n} ; \frac{i}{n} g (s) ds

= \frac{1}{ρ} ∑_{i=0}^{n−1} [(i + 1) J_{1} (2π \frac{i + 1}{n}ρ) − iJ_{1} (2π \frac{i}{n}ρ)] × ∫ \frac{i+1}{n} ; \frac{i}{n} g (s) ds,

then we can state the following corollary. Corollary 8. Let g be as in Theorem 7. Then we have

(4.7)  G (g) (ρ) = H_{n} (g, ρ) + R_{n} (g, ρ) , ρ ∈ (0, 1]

where H_{n} (g, ρ) is as given in (4.6) and the remainder R_{n} (g, ρ) satisfies the estimate

(4.8) |R_{n} (g, ρ)| ≤ \{ \begin{matrix}\frac{2π}{3n} ∥g′∥_{∞} if g′ ∈ L_{∞} [0, 1] ; \\ \frac{2^{1+ \frac{1}{q}} π}{[(q+1)(q+2)]^{\frac{1}{q}} n^{\frac{1}{q}}} ∥g′∥_{p} if g′ ∈ L_{p} [0, 1] , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{2π}{n} ∥g′∥_{1} . \end{matrix}

Remark 5. Supposing that we know ∥g′∥_{∞} and would like to approximate G (g) (ρ) with a given error ε > 0, then we have to divide the interval [0, 1] into at least n_{ε} ∈ ℕ^{∗} points, where

n_{ε} := [\frac{2π}{3ε} ∥g′∥_{∞}] + 1

where [x] is the integer part of x ∈ ℝ.

Using the tools provided in papers [1], [3]-[7] the authors are going to point out other quadrature formulae for Hankel transform for mappings of bounded variation, monotonic, of r-Hölder’s type etc..

5. A Numerical Example

An example of the approximation of the Hankel Transform is shown below. The quadrature formula developed for the Hankel Transform was applied to the function f(x)=e^{−x}. The graphs displayed in fig 1 and fig 2 show the input function and the Hankel Transform quadrature formula output function.

Figure 1. Input function f(r) = e^{−r}.

Figure 2. Hankel transform of f(r) = e^{−r}.

The two dimensional representation of the Hankel Transform in fig 3 shows the exact Hankel Transform and the Hankel Transform approximation calculated using the quadrature formula developed above for 200 iterations.

Figure 3. Comparison of Hankel transform and approximation. Step number = 200.

The graph shown in fig 4 demonstrates the maximum error of the Hankel Transform approximation as a function the number of steps used in the approximation.

Figure 4. Error vs. step size

References

[1] S.S. Dragomir, On the Ostrowski’s Integral inequality to Lipschitz mappings and applications, submitted.

[2] S.S. Dragomir and S. Wang, Applications of Ostrowski’s inequality for the estimation of error bounds for some special means and for some numerical quadrature rules, Applied Mathematics Letters 11 (1) (1998), 105 -109.

[3] S.S. Dragomir and S. Wang, A new inequality of Ostrowski’s type in Lp norm, Indian Journal of Mathematics, (in press)

[4] S.S. Dragomir, Ostrowski’s inequality for monotonic mapping and applications., J. KSIAM, accepted.

[5] N.S. Barnett and S.S. Dragomir, An inequality of Ostrowski’s type for cumulative distribution functions, Kyungpook Mathematical Journal, accepted.

[6] S.S. Dragomir, On the Ostrowski’s inequality for mappings of bounded variation, submitted.

[7] S.S. Dragomir, N.S. Barnett and S. Wang, An Ostrowski type inequality for a random variable whose probability density function belongs to L_{p} [a, b] , p > 1, Mathematical Inequalities & Applications, accepted.

[8] R.N. Bracewell, The Fourier Transform and Its Applications, Second Edition, Revised, McGraw-Hill, Inc. , 1986.

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