Using Wavelets Based on B-splines for Calculation of the Hankel Transform E.B. POSTNIKOV Theoretical Physics Department, Kursk State University Radishcheva st, 33, Kursk, 305000 RUSSIA
[email protected] Abstract: The purpose of this paper is to represent the integral Hankel transform as a series. If one uses Bspline wavelet this series is a linear combination of the hypergeometrical functions. Numerical evaluation of the test function with known analytical Hankel transform of the null kind illustrates the proposed results. Key-Words: the Hankel transform, hypergeometrical function, B-spline wavelet.
1 Introduction The Hankel transform [1] is a very useful tool of mathematical physics. Its application area is the axial symmetrical problems, especially seismology, electromagnetic methods in geophysical prospecting [2, 3], optical beam propagation [4] and other. There are two types of the Hankel transform. The first one is defined on the semiinfinite interval. In this case the direct and inverse transforms of the n-th kind are represented as an symmetric pair ∞
Fn ( p ) = ∫ f (r ) J n ( pr )rdr , 0
∞
(1)
f ( p ) = ∫ Fn ( p ) J n ( pr ) pdp. 0
In the case of the finite Hankel transform only a direct transform has an integral form. Without loss of generality its expression is 1
Fn ( p ) = ∫ f (r ) J n ( pr )rdr.
(2)
0
Unfortunately, comparatively little functions have a closed form solution of the Hankel integrals. But, there are several techniques for numerical evaluating of the Hankel transform. They are based on digital filtering, see a review in [2]; Discrere Hankel Transform (this is similar to the Fast Fourier Transform, see [5] and reference therein, and methods based on the Filon Quadratures. Filon’s approach suggests that it is possible to fit only f (r ) (or Fn ( p ) ) by a quadratic function instead of the entire integrand. In this paper the generalization of this idea is considered. First note that one can use a more flexible basis as the intermediate agent in the
calculation. The role of this intermediate agent can be played by the wavelet basis. This paper is organized as follows. In Section 2 we represent the integral Hankel transform as an exact series using B-spline wavelets of the arbitrary order. Section 3 consists of the numerical example of evaluation of the test function.
2 Outline of algorithm Suppose there exist an exact analytical representation of the local weighted average of the function, and the weight is a power functions. Then there exist a representation of the integral Hankel transform as a series. The wavelet basis can play this role. Suppose there exist a wavelet expansion of the function f ( x) [6]: ∞
∞
∞
f ( x) = ∑ cokϕ k (r ) + ∑∑ d jkψ k (r ), k =0
j =0 k =0
where ϕ k (r ) = ϕ (r − k ) , ψ jk (r ) = 2 j 2ψ (2 j r − r ) are a scaling function and a wavelet respectively. Let us consider the semi-orthogonal wavelet basis based on the B-spline N n ( x) [6]:
ψ (r ) = ψ m (r ) = This
wavelet
3m − 2
∑qN n=0
has
n
a
n
(2r − n) .
compact
(3)
support
suppψ m = [0,2m − 1] , where m is an order of the spline. The coefficients in (3) are
qn =
(−1) n 2 m −1
m
m
l =0
∑ l N
2m
(n + 1 − l ) .
Denote the restriction of the B-spline to the unit interval by N mα . Then, there exist an expansion of
N mα into a set of Bernstein polynomials: m−1 m−l −1 l Nmα ( y) = ∑alm−1(α) (1−[ y −α +1]) ( y −α +1) l =0 α m −1 where al (α ) are known Bernstein coefficients. m−1
m 3 m − 2 m −1 m − l −1 β + l
ψ mjk (r ) = ∑ ∑ ∑
∑ ∑ (−1)
β
α =1 n = 0 l = 0 β = 0 γ = 0
Take B-spline as a series m
m
α =1
α =1
N m ( y) = ∑ N mα ( y ) = ∑ N m ( y ) χ ([α − 1, α ), y ) , 1, if y ∈ [α − 1, α ) . 0 , f y [ α 1 , α ) ∉ −
where χ ([α − 1, α ), y ) =
Therefore, using Bernstein polynomials and a binomial expansion we get
m − 1 m − l − 1 β + l β + l −γ γ m −1 r × qn al (α ) (1 − 2k − n − α ) α β γ
χ ([α + 2k − n − 1,α + 2k − n), r ) For any r γ there exist a definite integral [7] ζ
γ ∫ r Jν ( pr )rdr == 0
12 (γ + 2 + ν ) pν ζ γ + 2+γ p 2ζ 2 F − 1 2 1 (γ + 4 + ν ),ν + 1 2ν (γ + 2 + ν )Γ(ν + 1) 4 2
Here 1 F2 is a hyperheometrical function. Finally, the result of the Hankel transform of B-spline wavelet is the following analytical expression: m 3 m − 2m −1 m − l −1 β + l
ψ mjk (p) = ∑
∑ ∑ ∑ ∑ (−1)
α =1 n = 0 l = 0 β = 0 λ = 0
β
m − 1 m − l − 1 β + l β + l −γ q n a lm −1 (α )(1 − 2k − n − α ) × γ α β
pν × 2 ( j + 2 )ν + 2 ( j +1) (γ + 2 + ν )Γ(ν + 1) 1 (γ + 2 + ν ) p 2 (α + 2k − n) 2 − (α + 2k − n) γ + 2 +ν 1 F2 1 2 2 2( j + 2) 2 (γ + 4 + ν ),ν + 1
1 (γ + 2 + ν ) p 2 (α + 2k − n − 1) 2 − (α + 2k − n − 1) γ + 2 +ν 1 F2 1 2 (γ + 4 + ν ),ν + 1 2 2( j + 2) 2 Note that c0k , d jk can be found by the convolution f (r ) with the dual basis functions. But these functions are also linear combinations of the power functions. This concludes the proof.
3 Numerical example Consider a simple example [8]. The B-spline wavelet of the first kind is the Haar function:
ψ 1jk ( p ) =
2− j p
(4)
−
1, r ∈ (0, 12 ) ψ 1 (r ) = −1, r ∈ ( 12 ,1) 0, r ∉ (0,1) The corresponding scaling function is
1, t ∈ (0,1) ϕ1 (r ) = 0, t ∉ (0,1) In this case (5) take a form
1 −j −j −j 1 2 k + 2 J1 ( 2 p (k + 2 ) ) − (k + 1) J1 ( 2 p(k + 1) ) − kJ1 ( 2 pk ) ,
and the Hankel transform of the scaling function as
1 J1 ( p(k + 1) ) − J1 ( p ) . p
ϕ1jk ( p) =
The absolute error of this approximation is shown in Figure 3.
The average and detail coefficients are k +1
c0 k =
∫
f (r )dr ,
k
2− j ( k +1/ 2)
d jk = 2
∫
−j
2− j ( k +1)
f (r )dr − k
2
−j
∫
f (r )dr .
( k +1/ 2)
We can compare he accuracy of the proposed algorithm. There is the known exact analytical Hankel transform: ∞
∫e 0
r − a
2
p
2
p − a . J 0 ( pr )rdr = e 2a
Let us truncate this transforming Gaussian function in the segment [ 0, R ] . This function is represented in the Fig. 1 (the replacement r to x = r / h is used).
Fig .1 The exact transform (solid line) and approximate transform at the level of expansion J = 3 (dotted line) are plotted in Figure 2.
Fig. 2
Fig. 3 One can see that the proposed method can be used for computational purposes.
References: [1] C.J. Tranter, Integral Transforms In Mathematical Physics, Methuen & Co, 1956. [2] M.N. Nabighian (Ed), Electromagnetic methods in applied Geophysics – Theory. Vol. 1, Society of Exploration Geophysics, 1988. [3] S.H. Ward (Ed.), Geotechnical and Environmental Geophysics, Society of Exploration Geophysics, 1990. [4] M. Guizar-Sicairos, J.C. Guitierres-Vega, Computation of quasi discrete Hankel transforms of integer order for propagating optical wave fields, Journal of Optical Society of America A, Vol. 21, No. 1, 2004, p. 53-58. [5] J.D. Secada, Numerical evaluation of Hankel transform, Computer Physics Communications, Vol. 116, No. 2-3, 1999, p. 278-294. [6] C.K. Chui, An Introduction to Wavelets, Academic Press, 1992. [7] Y.L. Luke, The Special Functions and Their Approximations, Academic Press, 1969. [8] E. B. Postnikov, About calculation of the Hankel transform using preliminary wavelet transform, Journal of Applied Mathematics, No. 6, 2003, p. 319-325.
