20 CHAPTER 2.CHEBYSHEV&FoURIER SERIES are useful to identify both errors and unexpected physics,and also to answer the question: Is a given calculation feasible on my machine? We will return to each of these four key themes in the middle of the chapter,though not in the same order as above.First,though,a brief review of Fourier series. 2.2 Fourier series The Fourier series of a general function f()is fa)=ao+∑acos(n)+∑(n） (2.1) n=l n=1 where the coefficients are a0= (u/2x)["f(a)dr an f(x)cos(nx)dx bm=(1/π) f(x)sin(nx)dx (2.2) First note:because the sines and cosines are periodic with a period of 2 m,we can also compute the Fourier expansion on the interval z E [0,2].The only alteration is that the limits of integration in the coefficient integrals(2.2)are also changed from [-to [0,2]. Second note:the general Fourier series can also be written in the complex form f(x)= Cn exp(inz) (2.3) where the coefficients are cn=(1/2x) f(r)exp(-inx)dx (2.4) The identities cos(r)=(exp(iz)+exp(-ix))/2; sin(x)=(exp(ix)-exp(-ix))/(2i), (2.5) show that(2.3)and(2.1)are completely equivalent,and we shall use whichever is conve- nient.The coefficients of the two forms are related by C0=a0, n=0 (an-ibn)/2 n>0 Cn 1 (an +ibn)/2, n<0 Often,it is unnecessary to use the full Fourier series.In particular,if f(z)is known to have the property of being symmetric about x =0,which means that f(z)=f(-z)for all z,then all the sine coefficients are zero.The series with only the constant and the cosine terms is known as a "Fourier cosine series".(A Chebyshev series is a Fourier cosine series with a change of variable.)If f(z)=-f(-x)for all r,then f(x)is said to be antisymmetric about z =0 and all the an=0.Its Fourier series is a sine series.These special cases are extremely important in applications as discussed in the Chapter 8
20 CHAPTER 2. CHEBYSHEV & FOURIER SERIES are useful to identify both errors and unexpected physics, and also to answer the question: Is a given calculation feasible on my machine? We will return to each of these four key themes in the middle of the chapter, though not in the same order as above. First, though, a brief review of Fourier series. 2.2 Fourier series The Fourier series of a general function f(x) is f(x) = a0 + X∞ n=1 an cos(nx) + X∞ n=1 bn sin(nx) (2.1) where the coefficients are a0 = (1/2π) Z π −π f(x)dx an = (1/π) Z π −π f(x) cos(nx)dx bn = (1/π) Z π −π f(x) sin(nx)dx (2.2) First note: because the sines and cosines are periodic with a period of 2 π, we can also compute the Fourier expansion on the interval x ∈ [0, 2π]. The only alteration is that the limits of integration in the coefficient integrals (2.2) are also changed from [−π, π] to [0, 2π]. Second note: the general Fourier series can also be written in the complex form f(x) = X∞ n=−∞ cn exp(inx) (2.3) where the coefficients are cn = (1/2π) Z π −π f(x) exp(−inx)dx (2.4) The identities cos(x) ≡ (exp(ix) + exp(−ix))/2; sin(x) ≡ (exp(ix) − exp(−ix))/(2i), (2.5) show that (2.3) and (2.1) are completely equivalent, and we shall use whichever is convenient. The coefficients of the two forms are related by c0 = a0, n = 0 cn = ½ (an − ibn)/2, n> 0 (an + ibn)/2, n< 0 Often, it is unnecessary to use the full Fourier series. In particular, if f(x) is known to have the property of being symmetric about x = 0, which means that f(x) = f(−x) for all x, then all the sine coefficients are zero. The series with only the constant and the cosine terms is known as a “Fourier cosine series”. (A Chebyshev series is a Fourier cosine series with a change of variable.) If f(x) = −f(−x) for all x, then f(x) is said to be antisymmetric about x = 0 and all the an = 0. Its Fourier series is a sine series. These special cases are extremely important in applications as discussed in the Chapter 8
2.2.FOURIER SERIES 21 Definition 1(PERIODICITY) A function f(x)is PERIODIC with a period of 2 if f(x)=f(x+2m) (2.6) for all x. To illustrate these abstract concepts,we will look at four explicit examples.These will allow us to develop an important theme:The smoother the function,more rapidly its spec- tral coefficients converge. EXAMPLE ONE:“Piecewise Linear”or“Sawtooth”Function Since the basis functions of the Fourier expansion,{1,cos(nx),sin(nx)},all are peri- odic,it would be reasonable to suppose that the Fourier series would be useful only for expanding functions that have this same property.In fact,this is only half-true.Fourier se- ries work best for periodic functions,and whenever possible,we will use them only when the boundary conditions are that the solution be periodic.(Geophysical example:because the earth is round,atmospheric flows are always periodic in longitude).However,Fourier series will converge,albeit slowly,for quite arbitrary f(). In keeping with our rather low-brow approach,we will prove this by example.Suppose we take f(z)=z,evaluate the integrals(2.2)and sum the series(2.1).What do we get? Because all the basis functions are periodic,their sum must be periodic even if the function f(z)in the integrals is not periodic.The result is that the Fourier series converges to the so-called "saw-tooth"function (Fig.2.1). Since f(x)=x is antisymmetric,all the an are 0.The sine coefficients are bn=(1/x) xsin(nx)dz =(-1)m+1(2/m) (2.70 Since the coefficients are decreasing as O(1/n),the series does not converge with blaz- ing speed;in fact,this is the worst known example for an f(z)which is continuous. Nonetheless,Fig.2.2 shows that adding more and more terms to the sine series does indeed generate a closer and closer approximation to a straight line. The graph of the error shows that the discontinuity has polluted the approximation with small,spurious oscillations everywhere.At any given fixed z,however,the ampli- tude of these oscillations decreases as O(1/N).Near the discontinuity,there is a region where (i)the error is always O(1)and (ii)the Fourier partial sum overshoots f(z)by the same amount,rising to a maximum of about 1.18 instead of 1,independent of N.Collec- tively,these facts are known as "Gibbs'Phenomenon".Fortunately,through "filtering", "sequence acceleration"and "reconstruction",it is possible to ameliorate some of these 2元 3元 Figure 2.1:"Sawtooth"(piecewise linear)function
2.2. FOURIER SERIES 21 Definition 1 (PERIODICITY) A function f(x) is PERIODIC with a period of 2 π if f(x) = f(x + 2π) (2.6) for all x. To illustrate these abstract concepts, we will look at four explicit examples. These will allow us to develop an important theme: The smoother the function, more rapidly its spectral coefficients converge. EXAMPLE ONE: “Piecewise Linear” or “Sawtooth” Function Since the basis functions of the Fourier expansion, {1, cos(nx), sin(nx)}, all are periodic, it would be reasonable to suppose that the Fourier series would be useful only for expanding functions that have this same property. In fact, this is only half-true. Fourier series work best for periodic functions, and whenever possible, we will use them only when the boundary conditions are that the solution be periodic. (Geophysical example: because the earth is round, atmospheric flows are always periodic in longitude). However, Fourier series will converge, albeit slowly, for quite arbitrary f(x). In keeping with our rather low-brow approach, we will prove this by example. Suppose we take f(x) = x, evaluate the integrals (2.2) and sum the series (2.1). What do we get? Because all the basis functions are periodic, their sum must be periodic even if the function f(x) in the integrals is not periodic. The result is that the Fourier series converges to the so-called “saw-tooth” function (Fig. 2.1). Since f(x) ≡ x is antisymmetric, all the an are 0. The sine coefficients are bn = (1/π) Z π −π x sin(nx)dx = (−1)n+1(2/n) (2.7) Since the coefficients are decreasing as O(1/n), the series does not converge with blazing speed; in fact, this is the worst known example for an f(x) which is continuous. Nonetheless, Fig. 2.2 shows that adding more and more terms to the sine series does indeed generate a closer and closer approximation to a straight line. The graph of the error shows that the discontinuity has polluted the approximation with small, spurious oscillations everywhere. At any given fixed x, however, the amplitude of these oscillations decreases as O(1/N). Near the discontinuity, there is a region where (i) the error is always O(1) and (ii) the Fourier partial sum overshoots f(x) by the same amount, rising to a maximum of about 1.18 instead of 1, independent of N. Collectively, these facts are known as “Gibbs’ Phenomenon”. Fortunately, through “filtering”, “sequence acceleration” and “reconstruction”, it is possible to ameliorate some of these −2π −π 0 π 2π 3π Figure 2.1: “Sawtooth” (piecewise linear) function
22 CHAPTER 2.CHEBYSHEV&FoURIER SERIES 3 1.5 N=18 2.5 N=18 2 1.5 N=3 -0.5 0.5 N=3 w Figure 2.2:Left:partial sums of the Fourier series of the piecewise linear("sawtooth") function (divided by )for N=3,6,9,12,15,18.Right:errors.For clarity,both the partial sums and errors have been shifted with upwards with increasing N. problems.Because shock waves in fluids are discontinuities,shocks produce Gibbs'Phe- nomenon,too,and demand the same remedies. EXAMPLE TWO:“Half-Wave Rectifier”Function This is defined on t E [0,2x]by sin(t), 0<t<π f(t)三 0 π<t<2m and is extended to all t by assuming that this pattern repeats with a period of 2 m.[Geo- physical note:this approximately describes the time dependence of thermal tides in the earth's atmosphere:the solar heating rises and falls during the day but is zero at night.] Integration gives the Fourier coefficients as a0=(1/πj: a2m=-2/儿π(4n2-1)】 (n>0): a2m+1=0(n≥1) (2.8) b1=1/2;b2m=0(m>1) 2.9) Fig.2.3 shows the sum of the first four terms of the series,fa(x)=0.318+0.5 sin(t)- 0.212 cos(2t)-0.042 cos(4t).The graph shows that the series is converging much faster than that for the saw-tooth function.At t =/2,where f(t)=1.000,the first four terms sum to 0.988,an error of only 1.2 % This series converges more rapidly than that for the "saw-tooth"because the "half- wave rectifier"function is smoother than the "saw-tooth"function.The latter is discontin- uous and its coefficients decrease as O(1/n)in the limit n-oo;the "half-wave rectifier"is continuous but its first derivative is discontinous,so its coefficients decrease as O(1/n2).This is a general property:the smoother a function is,the more rapidly its Fourier coefficients will decrease,and we can explicitly derive the appropriate power of 1/n
22 CHAPTER 2. CHEBYSHEV & FOURIER SERIES 0 1 2 3 0 0.5 1 1.5 2 2.5 3 N=18 N=3 0 1 2 3 -1 -0.5 0 0.5 1 1.5 N=18 N=3 Figure 2.2: Left: partial sums of the Fourier series of the piecewise linear (“sawtooth”) function (divided by π) for N=3 , 6, 9, 12, 15, 18. Right: errors. For clarity, both the partial sums and errors have been shifted with upwards with increasing N. problems. Because shock waves in fluids are discontinuities, shocks produce Gibbs’ Phenomenon, too, and demand the same remedies. EXAMPLE TWO: “Half-Wave Rectifier” Function This is defined on t ∈ [0, 2π] by f(t) ≡ sin(t), 0 <t<π 0, π<t< 2π and is extended to all t by assuming that this pattern repeats with a period of 2 π. [Geophysical note: this approximately describes the time dependence of thermal tides in the earth’s atmosphere: the solar heating rises and falls during the day but is zero at night.] Integration gives the Fourier coefficients as a0 = (1/π); a2n = −2/[π(4n2 − 1)] (n > 0); a2n+1 = 0(n ≥ 1) (2.8) b1 = 1/2; b2n =0 (n > 1) (2.9) Fig. 2.3 shows the sum of the first four terms of the series, f4(x)=0.318 + 0.5 sin(t) − 0.212 cos(2t)−0.042 cos(4t). The graph shows that the series is converging much faster than that for the saw-tooth function. At t = π/2, where f(t)=1.000, the first four terms sum to 0.988, an error of only 1.2 %. This series converges more rapidly than that for the “saw-tooth” because the “halfwave rectifier” function is smoother than the “saw-tooth” function. The latter is discontinuous and its coefficients decrease as O(1/n) in the limit n → ∞; the “half-wave rectifier” is continuous but its first derivative is discontinous, so its coefficients decrease as O(1/n2). This is a general property: the smoother a function is, the more rapidly its Fourier coefficients will decrease, and we can explicitly derive the appropriate power of 1/n
2.2.FOURIER SERIES 23 0.5 10 12 0 Figure 2.3:Top:graph of the"half-wave rectifier"function.Bottom:A comparison of the "half-wave rectifier"function [dashed]with the sum of the first four Fourier terms [solid]. fa(x)=0.318 +0.5 sin(t)-0.212 cos(2 t)-0.042 cos(4 t).The two curves are almost indistinguishable. Although spectral methods(and all other algorithms!)work best when the solution is smooth and infinitely differentiable,the "half-wave rectifier"shows that this is not always possible. EXAMPLE THREE:Infinitely Differentiable but Singular for Real x f(r)=exp{-cos2(r)/sin2(x)} (2.10) This function has an essential singularity of the form exp(-1/z2)at z=0.The power series about x =0 is meaningless because all the derivatives of(2.10)tend to 0 as x0. However,the derivatives exist because their limit as x-0 is well-defined and bounded. The exponential decay of exp(-1/z2)is sufficient to overcome the negative powers of z that appear when we differentiate so that none of the derivatives are infinite.Boyd(1982a) shows that the Fourier coefficients of(2.10)are asymptotically of the form an~[]exp(-1.5n2/3)cos(2.60n2/3+π/4） (2.11) where denotes an algebraic factor of n irrelevant for present purposes.Fast convergence, even though the power series about x=0 is useless,is a clear signal that spectral expan- sions are more potent than Taylor series(Fig.2.4). This example may seem rather contrived.However,"singular-but-infinitely-differentiable" is actually the most common case for functions on an infinite or semi-infinite interval.Most functions have such bounded singularities at infinity,that is,at one or both endpoints of the expansion interval. EXAMPLE FOUR:"Symmetric,Imbricated-Lorentzian"(SIP)Function f(x)≡(1-p2)/{(1+p2)-2pcos(x)} (2.12)
2.2. FOURIER SERIES 23 0 1 2 3 4 5 6 0 0.5 1 t 0 2 4 6 8 10 12 0 0.5 1 t Figure 2.3: Top: graph of the “half-wave rectifier” function. Bottom: A comparison of the “half-wave rectifier” function [dashed] with the sum of the first four Fourier terms [solid]. f4(x)=0.318 + 0.5 sin(t) − 0.212 cos(2 t) − 0.042 cos(4 t). The two curves are almost indistinguishable. Although spectral methods (and all other algorithms!) work best when the solution is smooth and infinitely differentiable, the “half-wave rectifier” shows that this is not always possible. EXAMPLE THREE: Infinitely Differentiable but Singular for Real x f(x) ≡ exp{− cos2(x)/ sin2(x)} (2.10) This function has an essential singularity of the form exp(−1/x2) at x = 0. The power series about x = 0 is meaningless because all the derivatives of (2.10) tend to 0 as x → 0. However, the derivatives exist because their limit as x → 0 is well-defined and bounded. The exponential decay of exp(−1/x2) is sufficient to overcome the negative powers of x that appear when we differentiate so that none of the derivatives are infinite. Boyd (1982a) shows that the Fourier coefficients of (2.10) are asymptotically of the form an ∼ [ ] exp(−1.5n2/3) cos(2.60n2/3 + π/4) (2.11) where [] denotes an algebraic factor of n irrelevant for present purposes. Fast convergence, even though the power series about x = 0 is useless, is a clear signal that spectral expansions are more potent than Taylor series (Fig. 2.4). This example may seem rather contrived. However, “singular-but-infinitely-differentiable” is actually the most common case for functions on an infinite or semi-infinite interval. Most functions have such bounded singularities at infinity, that is, at one or both endpoints of the expansion interval. EXAMPLE FOUR: “Symmetric, Imbricated-Lorentzian” (SIP) Function f(x) ≡ (1 − p2)/ © (1 + p2) − 2p cos(x) ª (2.12)
24 CHAPTER 2.CHEBYSHEV&FoURIER SERIES f Fourier coeffs. 100 0.8 102 0.6 10￥ 0.4 106 0.2 108 1010 -2 0 10 203040 degree n Figure 2.4:Left:graph of f(x)=exp(-cos2(x)/sin2(x)).Right:Fourier cosine coeffi- cients of this function.The sine coefficients are all zero because this function is symmetric with respect to z =0. where p 1 is a constant.This f()is a periodic function which is infinitely differentiable and continuous in all its derivatives.Its Fourier series is f(x)=1+2∑pcos(nx) (2.13) n=1 This example illustrates the "exponential"and "geometric"convergence which is typ- ical of solutions to differential equations in the absence of shocks,corner singularities,or discontinuities. We may describe (2.13)as a "geometrically-converging"series because at z =0,this is a geometric series.Since cos(nz)<1 for all n and z,each term in the Fourier series is bounded by the corresponding term in the geometric power series in p for all z.Because this rate of convergence is generic and typical,it is important to understand that it is qual- itatively different from the rate of the convergence of series whose terms are proportional to some inverse power of n. Note that each coefficient in(2.13)is smaller than its predecessor by a factor of p where p <1.However,if the coefficients were decreasing as O(1/n)for some finite k where k=1 for the“saw-tooth”andk=2 for the“half-wave rectifier”,then an+1/an~n/(n+1) 1-k/n for n>>k (2.14) ~1 [Non-exponential Convergence] Thus,even if k is a very large number,the ratio of an+1/an tends to 1 from below for large n.This never happens for a series with "exponential"convergence;the ratio of Ian+1/an is always bounded away from one-by p in(2.13),for example
24 CHAPTER 2. CHEBYSHEV & FOURIER SERIES 10 20 30 40 10-10 10-8 10-6 10 -4 10 -2 100 degree n Fourier coeffs. -2 0 2 0 0.2 0.4 0.6 0.8 1 x f Figure 2.4: Left: graph of f(x) ≡ exp(− cos2(x) / sin2(x) ). Right: Fourier cosine coeffi- cients of this function. The sine coefficients are all zero because this function is symmetric with respect to x = 0. where p < 1 is a constant. This f(x) is a periodic function which is infinitely differentiable and continuous in all its derivatives. Its Fourier series is f(x)=1+2X∞ n=1 pn cos(nx) (2.13) This example illustrates the “exponential” and “geometric” convergence which is typical of solutions to differential equations in the absence of shocks, corner singularities, or discontinuities. We may describe (2.13) as a “geometrically-converging” series because at x = 0, this is a geometric series. Since | cos(nx) | ≤ 1 for all n and x, each term in the Fourier series is bounded by the corresponding term in the geometric power series in p for all x. Because this rate of convergence is generic and typical, it is important to understand that it is qualitatively different from the rate of the convergence of series whose terms are proportional to some inverse power of n. Note that each coefficient in (2.13) is smaller than its predecessor by a factor of p where p < 1. However, if the coefficients were decreasing as O(1/nk) for some finite k where k = 1 for the “saw-tooth” and k = 2 for the “half-wave rectifier”, then an+1/an ∼ nk/(n + 1)k ∼ 1 − k/n for n >> k (2.14) ∼ 1 [Non − exponential Convergence] Thus, even if k is a very large number, the ratio of an+1/an tends to 1 from below for large n. This never happens for a series with “exponential” convergence; the ratio of | an+1/an | is always bounded away from one — by p in (2.13), for example