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Applied Mathematical Modelling 33 (2009) 812–821 www.elsevier.com/locate/apm
Fourier series of half-range functions by smooth extension Jeremy Morton 1, Larry Silverberg *,2 North Carolina State University, Raleigh, NC 27695-7910, United States Received 1 December 2005; received in revised form 1 November 2007; accepted 5 December 2007 Available online 15 December 2007
Abstract This paper considers Fourier series approximations of one- and two-dimensional functions over the half-range, that is, over the sub-interval [0, L] of the interval [L, L] in one-dimensional problems and over the sub-domain [0, Lx] [0, Ly] of the domain [Lx, Lx] [Ly, Ly] in two-dimensional problems. It is shown how to represent these functions using a Fourier series that employs a smooth extension. The purpose of the smooth extension is to improve the convergence characteristics otherwise obtained using the even and odd extensions. Significantly improved convergence characteristics are illustrated in one-dimensional and two-dimensional problems. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Fourier series; Smooth extension; Half-range problems
1. Introduction The Fourier series is the most widely used series expansion in mathematical modeling of engineering systems. It serves as the basis for the Fourier integral, the Laplace transform, the solution of autonomous linear differential equations, frequency response methods and many engineering applications. There are many good treatments on the subject; too many to mention in a comprehensive manner. However, the treatment by Tolstov [1] is classical. Jerri [2] provides an excellent overview on convergence of the Fourier series and discusses Gibbs-like phenomena in continuous and discrete wavelet representations. Sidi [3] reviews the state of the art of extrapolation methods giving applied scientists and engineers a practical guide to accelerating convergence in difficult computational problems. Also, accelerated convergence by means of periodic bridge functions was developed by Anguelov [4]. Convergence improvements in the Fourier transform are realized by the Cooley– Tukey FFT algorithm [5] and by the efforts of Brenner and Rader [6]. Direct evaluation of terms in a Fourier transform requires an order of N2 arithmetical operations while the fast Fourier transform requires an order of N log N operations. Other efforts include the short time Fourier transform (STFT) and its variations [7,8] and
*
1 2
Corresponding author. E-mail address:
[email protected] (L. Silverberg). Ph.D. candidate, Mechanical Engineering. Professor, Mechanical and Aerospace Engineering.
0307-904X/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2007.12.009
J. Morton, L. Silverberg / Applied Mathematical Modelling 33 (2009) 812–821
813
the recent and popular technique of wavelets [9,10] in which individual wavelet functions are localized in time, unlike with Fourier transforms. Problems in which Fourier series are used to represent periodic functions are known as full-range problems. In the half-range problem, a non-periodic function is defined over the interval [0, L]. The half-range function is extended to the interval [L, L] using a bridge function that is defined over the interval [L, 0]. The bridge function can be an even image or an odd image of the function. The use of the even image is called the even extension and it results in a cosine expansion; the use of the odd image is called the odd extension and it results in the sine expansion. This paper introduces another type of extension, called the smooth extension. The smooth extension uses a different type of bridge function, as described in the paper. The purpose of the smooth extension is to enhance the convergence characteristics of the Fourier series of half-range functions. This paper begins by developing the Fourier series by smooth extension and then examples are presented that compare its convergence rates with Fourier series that use the even and odd extensions. The results demonstrate significant improvements in convergence in both one-dimensional and two-dimensional problems. 2. Standard Fourier series Let f(x) be p-times differentiable over the interval [L, L]. The standard Fourier series approximation of f(x) over [L, L] is N a0 X npx npx þ bn sin an cos fN ðxÞ ¼ þ 2 L L n¼1 Z L 1 npx ð1a–cÞ f ðxÞ cos dx ðn ¼ 0; 1; . . . ; N Þ an ¼ L L L Z 1 L npx bn ¼ f ðxÞ sin dx ðn ¼ 1; 2; . . . ; N Þ L L L The Fourier series produces a periodic function over (-1, 1) with period 2L. When the interest lies in using the Fourier series to represent a half-range function f(x) over the interval [0, L] a bridge function g(x) is defined over the interval [L, 0]. The function f(x) in Eq. (1) becomes the composite of f(x) over [0, L] and g(x) over [L, 0]. The even extension defined by the bridge function g(x) = f(x), yields the cosine expansion Z N a0 X npx 2 L npx an ¼ dx ðn ¼ 0; 1; . . . ; N Þ ð2a; bÞ an cos f ðxÞ cos fN ðxÞ ¼ þ 2 L L L 0 n¼1 and the odd extension defined by the bridge function g(x) = f(x) yields the sine expansion Z N X npx 2 L npx fN ðxÞ ¼ bn sin f ðxÞ sin bn ¼ dx ðn ¼ 1; 2; . . . ; N Þ L L 0 L n¼1
ð3a; bÞ
Next, consider the convergence of the Fourier series. Eqs. (1b) and (1c) are integrated by parts p times, to yield 8 p=2 nþm p=2 Z > X > ð1Þ L d2m1 f ðLÞ d2m1 f ðLÞ ð1Þ L L dpþ1 f ðxÞ npx > > pþ1 dx ðp ¼ 0; 2; 4; 6. ..Þ sin > 2m 2m1 2m1 pþ1 < n p dx dx n p L dx L m¼1 an ¼ > ðpþ1Þ=2 X ð1Þnþm L d2m1 f ðLÞ d2m1 f ðLÞ ð1Þðpþ1Þ=2 L Z L dpþ1 f ðxÞ > npx > > þ cos dx ðp ¼ 1; 1;3;5; ...Þ > : 2m p 2m1 2m1 pþ1 p pþ1 n dx dx n dx L L m¼1 ð4aÞ 8 p=2þ1 > X ð1Þnþm L d2m2 f ðLÞ d2m2 f ðLÞ ð1Þp=2 L Z L dpþ1 f ðxÞ > npx > > pþ1 cos dx ðp ¼ 0; 2; 4;6 ...Þ > 2m1 p 2m2 2m2 pþ1 < p n dx dx n dx L L m¼1 bn ¼ ðpþ1Þ=2 > X ð1Þnþm L d2m2 f ðLÞ d2m2 f ðLÞ ð1Þðpþ1Þ=2 L Z L dpþ1 f ðxÞ > npx > > þ dx ðp ¼ 1; 1; 3;5;.. .Þ sin > : 2m1 p 2m2 2m2 pþ1 p pþ1 n dx dx n dx L L m¼1 ð4bÞ
814
J. Morton, L. Silverberg / Applied Mathematical Modelling 33 (2009) 812–821
in which the function f(x) and its first p derivatives are assumed to exist in the open interval (L, L). Notice in Eqs. (4a, b), if the derivatives of the function f(x) of orders 0, 1, 2, . . ., p at x = L are further set equal to the corresponding derivatives at x = L, i.e., if the conditions dm f ðLÞ dm f ðLÞ ¼ ðm ¼ 0; 1; 2; . . . ; pÞ dxm dxm are imposed, then the Fourier coefficients reduce to 8 p=2 Z > ð1Þ L L dpþ1 f ðxÞ npx > > dx ðp ¼ 0; 2; 4; 6 . . .Þ sin < npþ1 p pþ1 dx L L an ¼ ðpþ1Þ=2 Z L pþ1 > > ð1Þ L d f ðxÞ npx > :þ dx ðp ¼ 1; 1; 3; 5; . . .Þ cos pþ1 pþ1 n p dx L L
ð5aÞ
ð5bÞ
and 8 Z ð1Þp=2 L L dpþ1 f ðxÞ npx > > > cos dx ðp ¼ 0; 2; 4; 6 . . .Þ < pþ1 npþ1 p dx L L bn ¼ ðpþ1Þ=2 Z L pþ1 > > ð1Þ L d f ðxÞ npx > :þ sin dx ðp ¼ 1; 1; 3; 5; . . .Þ pþ1 npþ1 p dx L L
ð5cÞ
The convergence rate in Eq. (5) is of the order of np+1. The question arises whether there is a practical means of achieving this convergence rate. 3. The smooth extension In light of the preceding discussion, return to the function f(x) defined over the interval [0, L] and consider the bridge function g(x) that satisfies the two sets of conditions dm g dm f dm g dm f ¼ for ðm ¼ 0; 1; . . . ; pÞ ¼ for ðm ¼ 0; 1; . . . ; pÞ ð6Þ dxm dxm dxm dxm 0
0þ
Lþ
L
A bridge function that satisfies Eq. (6) is named a smooth bridge. The composite function f(x) in Eq. (1) that is constructed from f(x) over [0, L] and the smooth bridge g(x) over [L, 0] is named a smooth extension. The smooth extension is p-times differentiable over the domain [L, L] and, when replicated, is p-times differentiable over the interval (1, 1), too. Substituting the smooth extension into Eq. (1) yields the Fourier coefficients Z 0 Z L 1 npx npx gðxÞ cos f ðxÞ cos dx þ dx ðn ¼ 0; 1; . . . ; N Þ an ¼ L L L L 0 ð7a; bÞ Z 0 Z L 1 npx npx dx þ dx ðn ¼ 1; 2; . . . ; N Þ bn ¼ gðxÞ sin f ðxÞ sin L L L L 0 4. Constructing a smooth bridge from a spline: One-dimensional problems A smooth bridge can be constructed in a variety of ways; the smooth bridge is constructed here from a spline of order 2p + 1, written gðxÞ ¼
2pþ1 X
sn xn
ð8Þ
n¼0
Substituting Eq. (8) into Eq. (6) yields a set of linear algebraic equations, written in the partitioned form " # " # gð1Þ C ð1Þ m mn ¼ ½sn 2pþ2 ð9aÞ gð2Þ C ð2Þ m mn ð2pþ2Þð2pþ2Þ 2pþ2
J. Morton, L. Silverberg / Applied Mathematical Modelling 33 (2009) 812–821
in which gð1Þ m
dm f ¼ m ; dx 0þ
C ð1Þ mn ¼ m!dmn ;
gð2Þ m
dm f ¼ m dx L
C ð2Þ mn ¼
815
ðm ¼ 0; 1; 2; . . . ; pÞ
n! Lmn ðn mÞ!
ðm ¼ 0; 1; 2; . . . ; p; n ¼ 0; 1; 2; . . . ; 2p þ 1Þ ðn mÞ! ¼ 1 for n < m
ð9bÞ
4.1. Linear extension When p = 0, the bridge function is a first-order polynomial. Solving Eq. (9a) for p = 0 yields the linear extension having the linear bridge function x L
