AN APPROXIMATION OF HANKEL TRANSFORM FOR THE FUNCTIONS OF BOUNDED VARIATION

N.M. DRAGOMIR, S.S. DRAGOMIR, AND G.W. BAXTER

Abstract. Using some techniques developed by S.S. Dragomir in connection to Ostrowski’s integral inequality, we point out some approximation results for the Hankel Transform of functions of bounded variation.

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 [2, 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 functions of bounded variation on an finite interval [a, b] by the use of some techniques developed by S.S. Dragomir [6] in connection to Ostrowski’s integral inequality. Some adaptive quadrature formulae will be also derived.

For recent results in approximating the Hankel transform of absolutely continuous functions on using Ostrowsky type inequalities, see [3].

Date: March 16, 2004.

1991 Mathematics Subject Classification. Primary 42Xxx; 43Xxx; Secondary 26D15, 26D99.

Key words and phrases. Hankel Transform, Quadrature Rules.

2. An Integral Representation

Let J_{1} (·) denote the first-order Bessel function of the first kind. That is,

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

We define the corresponding Bessel’s mean as:

(2.2)  B_{1} (z, w) := \frac{zJ_{1} (z) − wJ_{1} (w)}{z − w} if w ≠ z, w, z ∈ ℂ.

The following representation of the Hankel transform for mappings of bounded variation holds.

Theorem 1. Let g : [a, b] → 𝕂 (𝕂 = ℝ, ℂ) be of bounded variation 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 (g) (r) rJ_{0} (2πrρ) dr,

where the kernel A (g) : [a, b] → 𝕂 is given by the Riemann Stieltjes integral

(2.4)  A (g) (r) := ∫_{a}^{r} (s − a) dg (s) + ∫_{r}^{b} (s − b) dg (s) ,

for all ρ ∈ [a, b] , ρ ≠ 0.

Proof. Using the integration by parts formula for the Riemann-Stieltjes integral (see also [6]), we have that

(2.5)  ∫_{a}^{x} (s − a) dg (s) = (x − a) g (x) − ∫_{a}^{x} g (s) ds

and

(2.6)  ∫_{x}^{b} (s − b) dg (s) = (b − x) g (x) − ∫_{x}^{b} g (s) ds,

for all x ∈ [a, b] . Adding (2.5) and (2.6) , and diving by (b − a) , we get the representation [6]

(2.7) g (x) = \frac{1}{b − a} ∫_{a}^{b} g (s) ds + \frac{1}{b − a} ∫_{a}^{x} (s − a) dg (s) + \frac{1}{b − a} ∫_{x}^{b} (s − b) dg (s) ,

for all x ∈ [a, b] . Now, using the definition of Hankel transform, we have:

(2.8)  G (g) (ρ) = 2π ∫_{a}^{b} g (r) rJ_{0} (2πρr) dr

= 2π ∫_{a}^{b} [\frac{1}{b − a} ∫_{a}^{b} g (s) ds + \frac{1}{b − a} ∫_{a}^{r} (s − a) dg (s)

+ \frac{1}{b − a} ∫_{r}^{b} (s − b) dg (s)] rJ_{0} (2πρr) dr

= 2π \frac{1}{b − a} ∫_{a}^{b} g (s) ds × ∫_{a}^{b} rJ_{0} (2πρr) dr + \frac{2π}{b − a} ∫_{a}^{b} A (g) (r) rJ_{0} (2πρr) dr.

Let us compute ∫_{a}^{b} rJ_{0} (2πρr) dr. Consider the change of variable r′ = 2πrρ. Then r = \frac{r′}{2πρ}, dr = \frac{dr′}{2πρ} and

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

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

It is a well known property of Bessel functions that

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

where J_{1} is the first-order Bessel function defined above. Hence we obtain

∫_{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) .

Consequently, we can state that

∫_{a}^{b} rJ_{0} (2πρr) dr = \frac{1}{(2πρ)^{2}} [2πbρJ_{1} (2πbρ) − 2πaρJ_{1} (2πaρ)]

and

2π \frac{1}{b − a} ∫_{a}^{b} g (s) ds × ∫_{a}^{b} rJ_{0} (2πρr) dr = \frac{1}{ρ}B_{1} (2πbρ, 2πaρ) .

Using (2.7) , we deduce the desired representation (2.3) .

In practical applications, we have a = 0 and b = 1. Consequently, we can state the following corollary.

Corollary 1. Let g : [0, 1] → 𝕂 (𝕂 = ℂ, ℝ) be a mapping of bounded variation on [0, 1] . Then we have the representation

(2.10)  G (g) (ρ) = \frac{J_{1} (2πρ)}{ρ} × ∫_{0}^{1} g (s) ds + 2π ∫_{0}^{1} Ã (g) (r) rJ_{0} (2πrρ) dr,

where Ã (g) : [0, 1] → 𝕂 is given by

Ã (g) (r) := ∫_{0}^{r} sdg (s) + ∫_{r}^{1} (s − 1) dg (s) ,

for all ρ ∈ [0, 1] .

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

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

Using the well known property of complex numbers

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

we can state the following corollary:

Corollary 2. With the assumptions from Theorem 1, we have

I (g) (ρ) (2.13)

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

+4\frac{π}{ρ (b − a)} Re [B_{1} (2πbρ, 2πaρ) ∫_{a}^{b} g (s) ds × ∫_{a}^{b} \overline{A (g) rJ_{0} (2πrρ)}dr]

+4\begin{matrix}π & 2 \\ (b − a) & 2 \end{matrix} |∫_{a}^{b} A (g) rJ_{0} (2πρr) dr|^{2} ,

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

I (g) (ρ) (2.14)

= \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} Ã (g) (r) rJ_{0} (2πrρ) dr] .

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 (g) rJ_{0} (2πρr) dr

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

Theorem 2. Let g : [a, b] → 𝕂 (𝕂 = ℝ, ℂ) be of bounded variation on [a, b] . Then we have the inequality

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

≤ 2π ⋁_{a}^{b} (g) ∫_{a}^{b} [\frac{1}{2} + \frac{|r − \frac{a+b}{2} |}{b − a}] |r| dr,

where ⋁_{a}^{b} (g) denotes the total variation of g on the interval [a, b] , for all ρ ∈ [a, b] , ρ ≠ 0.

Proof. Using the identity (2.3) , we obtain

|G (g) (ρ) − \frac{1}{ρ}B_{1} (2πbρ, 2πaρ) × ∫_{a}^{b} g (s) ds| (3.3)

≤ \frac{2π}{b − a} |∫_{a}^{b} (∫_{a}^{r} (s − a) dg (s)) rJ_{0} (2πrρ) dr|

+\frac{2π}{b − a} |∫_{a}^{b} (∫_{r}^{b} (s − b) dg (s)) rJ_{0} (2πrρ) dr|

= : B (ρ) ,

It is obvious that,

|J_{0} (2πrρ)| = |\frac{1}{2π} ∫_{0}^{2π} e^{−i2πrρ cos β}dβ|
≤ \frac{1}{2π} ∫_{0}^{2π} |e^{−i2πrρ cos β}| dβ
= \frac{1}{2π} ∫_{0}^{2π} dβ = 1.

On the other hand, it is well known that if p : [c, d] → 𝕂 is continuous and v : [c, d] → ℝ is of bounded variation on [c, d] , then p is Riemann-Stieltjes integrable with respect to v and

(3.4)  |∫_{c}^{d} p (x) dv (x)| ≤ sup_{x∈[c,d]} |p (x)| ⋁_{c}^{d} (v) .

Using this property, we have that

|∫_{a}^{b} (∫_{a}^{r} (s − a) dg (s)) rJ_{0} (2πrρ) dr| (3.5)

≤ ∫_{a}^{b} |∫_{a}^{r} (s − a) dg (s)| |r| |J_{0} (2πrρ)| dr

≤ ∫_{a}^{b} (r − a) ⋁_{a}^{r} (g) |r| dr,

and, similarly,

(3.6)  |∫_{a}^{b} (∫_{r}^{b} (s − b) dg (s)) rJ_{0} (2πrρ) dr| ≤ ∫_{a}^{b} (b − r) ⋁_{b}^{r} (g) |r| dr.

6 N.M. DRAGOMIR, S.S. DRAGOMIR, AND G.W. BAXTER

By the estimates (3.5) and (3.6) , we can write down that

B (ρ) ≤ \frac{2π}{b − a} ∫_{a}^{b} (r − a) ⋁_{a}^{r} (g) |r| dr + \frac{2π}{b − a} ∫_{a}^{b} (b − r) ⋁_{b}^{r} (g) |r| dr
= \frac{2π}{b − a} ∫_{a}^{b} [(r − a) ⋁_{a}^{r} (g) + (b − r) ⋁_{b}^{r} (g)] |r| dr
≤ \frac{2π}{b − a} ∫_{a}^{b} max (r − a, b − r) [⋁_{a}^{r} (g) + ⋁_{b}^{r} (g)] |r| dr
= \frac{2π}{b − a} ∫_{a}^{b} [\frac{1}{2} (b − a) + |r − \frac{a + b}{2}|] ⋁_{a}^{b} (g) |r| dr
= \frac{2π}{b − a} ⋁_{a}^{b} (g) ∫_{a}^{b} [\frac{1}{2} (b − a) + |r − \frac{a + b}{2}|] |r| dr,

and the inequality (3.2) is thus achieved. Remark 1. In practical applications, a ≥ 0, so the bound in (3.2) becomes

2π ⋁_{a}^{b} (g) ∫_{a}^{b} [\frac{1}{2} + \frac{|r − \frac{a+b}{2} |}{b − a}] rdr.

As a simple calculation shows that

∫_{a}^{b} [\frac{1}{2} + \frac{|r − \frac{a+b}{2} |}{b − a}] rdr = \frac{a + b}{2} · (b − a) ,

then we get the inequality

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

for all ρ ∈ [a, b] , ρ ≠ 0. Remark 2. If we assume that a = 0 and b > 0 in (3.7) , then we obtain

(3.8)  |G (g) (ρ) − \frac{J_{1} (2πbρ)}{ρ} × ∫_{0}^{b} g (s) ds| ≤ πb^{2} ⋁_{a}^{b} (g) .

The above inequality shows that

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

where b → 0+ and the precision of the approximation is of order 3.

In some practical applications the upper limit of integration is b = 1, therefore the following corollary is required. Corollary 3. Let g : [0, 1] → 𝕂 (𝕂 = ℂ, ℝ) be a mapping of bounded variation on [0, 1] . Then we have the inequality:

(3.9)  |G (g) (ρ) − \frac{J_{1} (2πρ)}{ρ} × ∫_{0}^{1} g (s) ds| ≤ π ⋁_{a}^{b} (g) ,

for all ρ ∈ (0, 1] .

Remark 3. If in Theorem 2, we assume that the mapping g : [0, 1] → ℝ is continuously differentiable on [0, 1] and the integral ∥g′∥_{1} := ∫_{a}^{b} |g′ (s)| ds is finite, then we can put in place of ⋁_{a}^{b} (g) , ∥g′∥_{1}, thus obtaining a result from the paper [3], i.e.,

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

for all ρ ∈ [a, b] , ρ ≠ 0.

Using the inequality (3.2) which provides an upper bound for the remainder R [g] (ρ) , we will point out the following inequality which approximates the mapping I (g) (ρ) .

Corollary 4. Let g : [0, 1] → 𝕂 (𝕂 = ℂ, ℝ) be a mapping of bounded variation on [0, 1] . 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) := 2π ⋁_{a}^{b} (g) ∫_{a}^{b} [\frac{1}{2} + \frac{|r − \frac{a+b}{2} |}{b − a}] |r| dr,

for all ρ ∈ [a, b] , ρ ≠ 0.

4. A Quadrature Formula

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

G (g) (ρ) = 2π ∫_{0}^{1} g (r) rJ_{0} (2πρr) dr,

where g is assumed to be of bounded variation on [a, b] .

Firstly, let us assume that in formula (3.7) we have 0 ≤ a ≤ b ≤ 1. Then we have the coarser upper bound

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

Now, assume I_{n} : 0 = x_{0} < x_{1} < ... < x_{n−1} < x_{n} to be a division of the interval [a, b] . Put h_{i} := x_{i+1} − x_{i} (i = 0, ..., n − 1) and ν (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} ∑_{k=0}^{n−1} B_{1} (2πx_{i+1}ρ, 2πx_{i}ρ) × ∫_{x_{i}}^{x_{i+1}} g (s) ds

= \frac{1}{ρ} ∑_{k=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 3. Let g : [0, 1] → 𝕂 (𝕂 = ℂ, ℝ) be a mapping of bounded variation 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)| ≤ 2πν (h) ⋁_{a}^{b} (g) ,

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

≤ 2πh_{i} ⋁_{x_{i}}^{x_{i+1}} (g) .

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

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

≤ 2π ∑_{k=0}^{n−1} h_{i} ⋁_{x_{i}}^{x_{i+1}} (g) ≤ 2πν (h) ∑_{k=0}^{n−1} ⋁_{x_{i}}^{x_{i+1}} (g)

= 2πν (h) ⋁_{a}^{b} (g) ,

and the theorem is proved.

Remark 4. Observe that if ν (h) → 0, then, by (4.4) , R (g, I_{n}, ρ) → 0 uniformly on ρ ∈ (0, 1] , which proves that H (g, I_{n}, ρ) can approximate the Hankel transform with any accuracy.

In practical problems, the interval [0, 1] is divided into equidistant subintervals by the division I_{n} : x_{i} = \frac{i}{n}, i = 0, ..., n. If we consider the sum

H_{n} (g, ρ) (4.5)

: = \frac{1}{ρ} ∑_{k=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} ∑_{k=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 5. Let g be as in Theorem 3. Then we have

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

where H_{n} (g, ρ) is as given in (4.5) and the remainder R_{n} (g, ρ) satisfies the estimate

(4.7)  |R_{n} (g, ρ)| ≤ \frac{2π}{n} ⋁_{0}^{1} (g) .

Remark 5. Supposing that we know the total variation of g on the interval [0, 1] and would like to approximate G (g) (ρ) with a given error ε > 0. Then we have to divided the interval [0, 1] into at least n_{ε} ∈ ℕ^{∗} points, where

n_{ε} := [\frac{2π}{ε} ⋁_{0}^{1} (g)] + 1,

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

Remark 6. Using the results and techniques outlined in papers [1], [4]-[7], the authors intend to investigate the error estimates in approximating Hankel transforms of r−Hölder continuous, Lipschitzian or monotonic nondecreasing functions. The corresponding results will be comunicated in another paper that is in preparation.

References

[1] N.S. Barnett and S.S. Dragomir, An inequality of Ostrowski’s type for cumulative distribution functions, Kyungpook Mathematical Journal, 39(2)(1999), 303-311

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

[3] N.M. Dragomir, S.S. Dragomir, M. Gu, X. Gan and R. White, An approximation of the Hankel transform for absolutely continuous mappings, J. KSIAM, 6(1)(2002), 17-31.

[4] S.S. Dragomir, The Ostrowski’s integral inequality for Lipschitzian mappings and applications, Computers and Mathematics with Applications, 38(1999), 33-37.

[5] S.S. Dragomir, Ostrowski’s inequality for montonous mappings and applications, J. KSIAM, 3(1)(1999), 127-135.

[6] S.S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Mathematical Inequalities and Applications, 4(1)(2001), 59-66.

[7] 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.

School of Electrical Engineering, Victoria University of Technology, PO Box 14428, Melbourne City, MC 8001 Australia

   E-mail address: nicoleta.dragomir@vu.edu.au

School of Computer Science & Mathematics, Victoria University of Technology, PO Box 14428, Melbourne City, MC 8001 Australia

   E-mail address: sever@matilda.vu.edu.au

School of Electrical Engineering, Victoria University of Technology, PO Box 14428, Melbourne City, MC 8001 Australia

   E-mail address: gregory.baxter@vu.edu.au