an odd number of integrations.

Third, if the function is periodic and differentiable to all orders, we can integrate“by“

parts an arbitrary number of times. In that case, the theorem implies that for large n, the

Fourier coef¬cients are decreasing faster than any ¬nite power of n. This is the property of

“in¬nite order” or “exponential” convergence de¬ned above.

Fourth, since a Chebyshev series in x becomes a Fourier series in y only after we

make the substitution x = cos(y), a Chebyshev series “ after the change of variable “

is automatically periodic in y. Therefore, a CHEBYSHEV SERIES ALWAYS HAS THE

PROPERTY OF INFINITE ORDER CONVERGENCE EVEN FOR FUNCTIONS THAT ARE

NON-PERIODIC (in x, the Chebyshev argument) if all its derivatives are bounded on the

expansion interval, x ∈ [’1, 1].

Fifth, the constant F increases with k, the number of times we integrate-by-parts. Con-

sequently, we cannot take the limit of k ’ ∞ for ¬xed n. (Taking this limit with the false

assumption of k-independent F would imply that all the coef¬cients are zero!) Rather, the

theorem tells us how rapidly the coef¬cients decrease for large n as n ’ ∞ for ¬xed k.

Sixth, this theorem can be strengthened to the statement that if f (x) is in the Lipschitz

space L» “on the circle”, then an , bn ∼ O(n’» ). A function is in the Lipschitz space L» (for

0 < » < 1) if |f (x) ’ f (y)| = O(|x ’ y|» ) for all x, y on the interval; Lipschitz spaces for

» > 1 are de¬ned by taking higher order differences of f . The phrase “on the circle” means

that continuity is analyzed as if the interval x ∈ [’π, π] is bent into a circle so that x = π is

continuous with x = ’π; for example, f (’π) = f (π) implies that the function is not in any

Lipschitz space with » > 0.

EXAMPLE: Let us calculate the index of convergence of the Fourier series solution to

(2.59)

uxx + Q(x) u = f (x); u(0) = u(π) = 0

Since the boundary conditions do not impose periodicity, the solution to this problem is

usually not periodic even if the functions Q(x) and f (x) are periodic themselves.

Because of the lack of periodicity, we really ought to use Chebyshev polynomials for

this problem. Nonetheless, many old papers have applied Fourier series to non-periodic

CHAPTER 2. CHEBYSHEV & FOURIER SERIES

44

problems; certain analytical techniques, such as the method of separation-of-variables, fail

otherwise. It therefore is useful to see just how bad this choice of basis functions is.

Because the boundary conditions demand u(0) = 0, we must exclude the cosine terms

(and the constant) from the Fourier series and use only the sine functions:

∞

(2.60)

u(x) = bn sin(nx)

n=1

Since the sum of the sine series and the individual sine functions are both antisymmetric

with respect to x = 0, we halve the interval of integration to [0, π] and double the result.

This is natural for this problem since [0, π] is the interval between the boundaries. Thus,

π

(2.61)

bn = (2/π) u(x) sin(nx)dx

0

One integration-by-parts gives

π

π

bn = ’(2/[nπ]) u(x) cos(nx) + (2/[nπ]) u(1) (x) cos(nx) (2.62)

0 0

The boundary conditions u(0) = u(π) guarantee that the boundary term in (2.62) is zero

independent of Q(x) and f (x). Integrating“by“parts a second time gives

π

π

(x) sin(nx) ’ (2/[n π])

2 (1) 2

u(2) (x) sin(nx)dx (2.63)

bn = (2/[n π]) u

0 0

Since sin(0) = sin(nπ) = 0, it does not matter whether u(1) (0) = u(1) (π); the boundary term

in (2.63) is 0 anyway.

Integrating-by-parts a third time gives

π

π

(x) cos(nx) ’ (2/[n π])

3 (2) 3

u(3) (x) cos(nx)dx (2.64)

bn = (2/[n π]) u

0 0

Since cos(nπ) = cos(0) for odd n, we are stuck with k = 3 as the algebraic index of conver-

gence unless u(2) (0) = u(2) (π). Since one can only impose one pair of boundary conditions

on the solution of a second order differential equation, we would seem to be out of luck.

Evaluating the differential equation at x = 0, however, gives

= ’Q(0) u(0) + f (0)

u(2) (0)

(2.65)

= f (0)

since the boundary conditions on the differential equation require u(0) = 0. Similarly,

u(2) (π) = f (π). We see that if

(2.66)

f (0) = f (π) = 0,

then the combination of the differential equation with its associated boundary conditions

forces the boundary term in (2.64) to vanish. We can integrate twice more before being

defeated by the fact that u(4) (0) = u(4) (π). Thus, whether the index of convergence is k

= 3 or 5 depends entirely upon whether f (x) satis¬es condition (2.66) (as is often true in

practical applications). A k = 5 geophysical example is Eliasen (1954), who used a Fourier

sine series to solve the “barotropic instability problem” for a cosine jet.

The important point is that we deduced the index of convergence simply from the form

of the differential equation and the boundary conditions. We did not have to specify Q(x)

2.10. ASYMPTOTIC CALCULATION OF FOURIER COEFFICIENTS 45

at all, and f (x) only at two points to compute k. The integration“by“parts theorem is a

very powerful tool because we can apply it before we solve the differential equation.

This theorem is also important because it generalizes to all other spectral basis sets. That

is to say, integration“by“parts can be used to provide similar coef¬cient bounds for any

of the standard basis functions including Chebyshev and Legendre polynomials, Hermite

functions and spherical harmonics.

2.10 Asymptotic Calculation of Fourier Coef¬cients

Theorem 5 (STRIP OF CONVERGENCE FOR FOURIER SERIES) Let z = x + iy and let

ρ denote the absolute value of the imaginary part of the location of that singularity of f (z) which is

nearest the real z-axis. In other words, if zj , j = 1, 2, . . . denotes the location of the singularities of

f (z), then

ρ = min | (zj )| . (2.67)

j

Then the Fourier series converges uniformly and absolutely within the STRIP in the complex z-

plane, centered on the real axis, which is de¬ned by

| y |< ρ, (2.68)

x = arbitrary [convergence]

and diverges outside the strip,

| y |> ρ, (2.69)

x = arbitrary [divergence]

If the limit exists, then the asymptotic coef¬cients of the Fourier series

∞ ∞

(2.70)

f (z) = a0 + an cos(nz) + bn sin(nz)

n=1 n=1

are related to ρ, the half-width of the strip of convergence, by

lim sup log | an /an+1 |= ρ (2.71)

n’∞

and similarly for the sine coef¬cients bn .

The asymptotic rate of convergence for real z is given by µ = ρ.

If the series has a ¬nite algebraic index of convergence, then ρ = 0 and the series converges only

on the real axis.

If the function is periodic and entire, i. e., has no poles or branch points except at in¬nity, then

usually ρ = ∞ and the coef¬cients are O[exp(’(n/k) log(n) + O(n))] for some constant k. (The

exceptions with ¬nite ρ are entire functions that grow very rapidly with | z | such as exponentials-

of-exponentials, as discussed in Boyd (1994b).)

Note: In this context, “singularity” includes not only poles and branch points, but also discon-

tinuities caused by a lack of periodicity, as for the saw-tooth function.

PROOF: We omit a rigorous proof, but offer the following argument. Let z = x + iy so

that | exp(i n z) | = exp(’ n y). Thus, in the complex form of the Fourier series

∞ ∞

(2.72)

f (z) = cn exp(inz) = cn exp(inx) exp(’ny)

n=’∞ n=’∞

CHAPTER 2. CHEBYSHEV & FOURIER SERIES

46

all terms with negative n grow exponentially as exp(| n |y) in the upper half-plane while

those with n > 0 grow in the lower half-plane. It follows that the series can converge at

| y | = ρ if and only if | c±n | decay at least as fast as exp(’ n ρ).

On the other hand, if f (z) has a pole at y = ρ, the series must converge to ever-larger

values as (z) ’ ρ. This implies that the coef¬cients cannot decrease faster than exp(’ nρ)

(modulo the usual algebraic factor of n). (Note that if the coef¬cients could be bounded by

exp(’n(ρ ’ )) for some small positive , for example, then one could bound the Fourier

series, term-by-term, by a convergent geometric series to show it was ¬nite even at the

pole.)

Thus, the series coef¬cients must decrease as exp(’ n ρ) when f (z) has a singularity at

| (z) | = µ. Q. E. D.

The theorem shows that “geometric” convergence is normal for a Fourier series. It is only

when the function has a discontinuity or other singularity for real z that the series will have

sub-geometric or algebraic convergence.

2.11 Convergence Theory for Chebyshev Polynomials

Since a Chebyshev polynomial expansion is merely a Fourier cosine series in disguise, the

convergence theory is very similar to that for Fourier series. Every theorem, every identity,

of Chebyshev polynomials has its Fourier counterpart. Nonetheless, the mapping does

produce some noteworthy consequences.

The mapping is

(2.73)

z = cos(θ)

and then

Tn (z) ≡ cos(nθ) (2.74)

The following two series are then equivalent under the transformation:

∞

(2.75)

f (z) = an Tn (z)

n=0

∞

(2.76)

f (cos θ) = an cos(nθ)

n=0

In other words, the coef¬cients of f (z) as a Chebyshev series are identical with the Fourier

cosine coef¬cients of f (cos(θ)).

FIRST IMPLICATION: EXPONENTIAL CHEBYSHEV CONVERGENCE for

NON-PERIODIC FUNCTIONS.

Even if f (z) itself is not periodic in z, the function f (cos θ) must inevitably be periodic

in θ with period 2π. As we vary θ over all real θ, the periodicity of cos(θ) implies that z (=

cos[θ]) merely oscillates between -1 to 1 as shown in Fig. 2.15. Since f (cos θ) is periodic, its

Fourier series must have exponential convergence unless f (z) is singular for z ∈ [’1, 1]. It

does not matter if f (z) is periodic in z nor does it matter if f (z) has singularities for real z

outside the interval [-1, 1]. The Fourier cosine series in (2.76) sees f (cos θ) only as a periodic

function. For real θ, the Fourier series sees only those variations in f (z) that occur for

z ∈ [’1, 1]. The exponential convergence of the Fourier series (2.76) then implies equally

fast convergence of the Chebyshev series since the sums are term“by“term identical.

2.11. CONVERGENCE THEORY: CHEBYSHEV POLYNOMIALS 47

1

0.5

0

z

’0.5

’1

’6 ’4 ’2 0 2 4 6

θ

Figure 2.15: The Chebyshev polynomials, Tn (z), are related to the terms of a Fourier co-

sine series through the identity Tn (cos[θ]) = cos( n θ). The graph shows the relationship

between z and θ, z = cos(θ).

SECOND IMPLICATION: the FOURIER EQUIVALENT of a CHEBYSHEV SERIES is a

FOURIER COSINE SERIES.

This is implicit in (2.76). Since cos(θ) is symmetric about θ = 0, f (cos θ) is forced to be

symmetric in θ, too, even if f (z) has no symmetry whatsoever with respect to z. Conse-

quently, we need only the cosines.1

THIRD IMPLICATION:

Theorem 6 (CHEBYSHEV TRUNCATION THEOREM) The error in approximating f (z) by

the sum of its ¬rst N terms is bounded by the sum of the absolute values of all the neglected coef¬-

cients. If

N

fN (z) ≡ (2.77)

an Tn (z)

n=0

then

∞

ET (N ) ≡| f (z) ’ fN (z) |¤ | an | (2.78)

n=N +1