... TO 2-D FMM. The principal tool in the theory of the fast multipole method for 2-D scattering problems is the addition theorem (Gradshteyn and Ryzhik,. 1965).
TRUNCATION BOUNDS FOR THE FAST MULTIPOLE METHOD BASED ON ANALYTIC FOURIER SERIES
Luc Knockaert and Daniel De Zutter Department of Information Technology University of Ghent Sint-Pietersnieuwstraat 41, B-9000 Ghent, Belgium.
ABSTRACT Analytic Fourier series admit a uniform exponential truncation bound. Applied to the fast multipole method this leads to pertinent truncation bounds for the multipole potential expansion. 1. INTRODUCTION The problem of the truncation of infinite Fourier sums is of utmost importance in computational electromagnetics, especially since the advent of the fast multipole method (FMM) (Coifman et al., 1993), (Song et al., 1997), (Bindiganavale and Volakis, 1997). For a general Fourier series it is a priori difficult to know how or when to truncate, i.e. to estimate the number of principal harmonics, without making use of statistical information theoretic criteria such as MDL (Rissanen, 1978) or MAPME (Knockaert, 1997). On the other hand, in important electrical engineering areas such as frequency modulation (Gibson, 1989) and electromagnetic scattering theory (Bowman et al., 1969), (Coifman et al., 1993) one often deals with analytic Fourier series, i.e. Fourier series
2 associated with periodic analytic functions. In this paper we exploit a recent result on analytic Fourier series (Knockaert and De Zutter, 1998) to obtain pertinent truncation bounds for the multipole potential expansion. 2. PROBLEM STATEMENT AND MAIN RESULT Let f (x) be a complex analytic periodic function with period 2π defined over the real axis. Then f (x) admits the Fourier series ∞ X
f (x) =
an e−inx
(1)
n=−∞
which converges absolutely and uniformly. By the principle of analytic continuation (Davis, 1975) p. 306, the series (1) can be continued in the complex plane to yield absolute and uniform convergence within a strip |=(z)| < t. The above Fourier series can be written as fN (x) + SN (x), where N X
fN (x) =
an e−inx
(2)
n=−N
is the truncated series and SN (x) =
∞ ³ X
an e−inx + a−n einx
n=N +1
´
(3)
is the residual. Clearly, on the real axis we have |SN (x)| ≤ RN =
∞ X
n=N +1
(|an | + |a−n |)
(4)
Although we know that limN →∞ RN = 0, it is important to have more explicit information concerning the convergence rate. This is the purpose of the following Proposition 1: RN ≤ 4eM (δ)
2+δ √ N δ e−N δ δ2
for
1 ≤δ 0 and N be such that N ≥ min Λ(δ)
(7)
0 r and defined by
P∞
l=0+
∞ X
(2)
Jl (kr)Hl (kR) cos lθ
xl = 12 x0 +
|R − r| =
q
(12)
l=0+
P∞
l=1
xl . The angle θ is of course
R2 + r2 − 2rR cos θ.
(13)
Since the Hankel function is singular at the origin only, it is an easy matter to find the strip of convergence of the Fourier series (12) . It is determined by max | cos(x + it)| = cosh t =
0≤x≤2π
and hence
R2 + r 2 2Rr
(14)
R t = ln r √ Next we need M (δ). Denoting b = k 2Rr we have µ
¶
(15)
¯ ¶¯ µ ¯ ¯ (2) q ¯ M (δ, b, t) = max ¯H0 b cosh t − cos(x + iδ) ¯¯ 0≤x≤2π
(16)
and following the corollary of Proposition 1 &
Ã
1 4e(2 + δ)M (δ, b, t) Nmin (², b, t) = min φ 0
