THE FABER POLYNOMIALS FOR ANNULAR SECTORS

mathematics of computationvolume 64, number 209january 1995, pages 181-203

THE FABER POLYNOMIALS FOR ANNULAR SECTORS

JOHN P. COLEMAN AND NICK J. MYERS

Abstract. A conformai mapping of the exterior of the unit circle to the exteriorof a region of the complex plane determines the Faber polynomials for that

region. These polynomials are of interest in providing near-optimal polynomial

approximations in a variety of contexts, including the construction of semi-

iterative methods for linear equations. The relevant conformai map for an

annular sector {z : R < \z\ < 1, 0 < |argz| < it), with 0 < 9 < it, is

derived here and a recurrence relation is established for the coefficients of its

Laurent expansion about the point at infinity. The recursive evaluation of scaled

Faber polynomials is formulated in such a way that an algebraic manipulation

package may be used to generate explicit expressions for their coefficients, in

terms of two parameters which are determined by the interior angle of the

annular sector and the ratio of its radii. Properties of the coefficients of the

scaled Faber polynomials are established, and those for polynomials of degree

< 15 are tabulated in a Supplement at the end of this issue. A simple closed

form is obtained for the coefficients of the Faber series for 1/z . Known results

for an interval, a circular arc, and a circular sector are reproduced as special

cases.

1. Introduction

The Faber polynomials for particular regions of the complex plane have been

used to provide polynomial and rational approximations in a wide variety ofdifferent contexts. Near-minimax polynomial approximations may be obtained

by truncating Faber series [15, 9, 1], by economization of Faber series [12]and, for solutions of linear differential equations, by the Lanczos r-method [2,3] and by Clenshaw's method [14]. Rational approximations based on Faber

series were discussed in [11] and [13], and applications to iterative methods in

numerical linear algebra appear in [8, 7, 29].The Faber polynomials for any closed bounded continuum D in the complex

plane are associated with a certain exterior conformai map. According to the

Riemann mapping theorem (e.g., [21, p. 380]) there is a unique function <p,

such that

(1.1) 0(oo) = oo, lim^ = l,z->oo Z

Received by the editor June 17, 1993.1991 Mathematics Subject Classification. Primary 30C10, 30C20, 30E10; Secondary 65E05,

65F10.Key words and phrases. Faber polynomials, conformai mapping, annular sector, Faber series,

transfinite diameter.

©1995 American Mathematical Society

0025-5718/95 $1.00+ $.25 per page

181

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use

182 J. P. COLEMAN AND N. J. MYERS

which maps the complement of D in the extended z-plane conformally onto

{w : \w\ > p} , the complement of a closed disc of radius p. The number p is

called the transfinite diameter, or logarithmic capacity, of D. The function <p

has a Laurent expansion

il y Ql<p{z) = Z + CX0+ — + •••

z

about the point at infinity. The Faber polynomial of degree « is obtained by

deleting all negative powers of z from the corresponding Laurent expansion

of [</>(z)]n . For the unit disc, the Faber polynomial of degree « is zn , and

multiples of the Chebyshev polynomials of the first kind are the Faber polyno-mials for an ellipse with foci at (±1,0) and, in particular, for the real interval

[-1,1].Properties of the Faber polynomials and Faber series are described in the

books of Markushevich [24, v. 3, pp. 104-112], Smirnov and Lebedev [28,Chapter 2], Henrici [22, Chapter 18] and Gaier [17, pp. 42-57], and in a sur-vey article by Curtiss [6]. The Faber polynomials are known explicitly only

for a few types of domain. Those for certain lemniscates appear in [24], El-

liott [15] computed the coefficients of some Faber polynomials for the semidisc

{z: \z\ < 1, Rez > 0} and for the square {z: |Rez| < 1, |lmz| < 1}, and

Ellacott [10] expressed the Faber polynomials for a circular arc in terms of

Chebyshev polynomials. Coleman and Smith [4] derived a recurrence relation

for the coefficients of the Faber polynomials for circular sectors, and provided

numerical values of those coefficients for polynomials of degree up to 15 for se-

lected sectors [5]. Gatermann et al. [18] modified the algorithm of [4] to obtaina form involving only rational coefficients and therefore amenable to computer

algebra systems; the algebraic forms in [ 19] allow the computation of the Faberpolynomials of degree up to 20 for an arbitrary circular sector. In other cases,

where explicit formulae were not available, numerical algorithms for conformaimapping have been used to generate Faber polynomials (see Ellacott [9], Starke

and Varga [29] and Papamichael et al. [25]).Virtues of suitably normalized Faber polynomials as residue polynomials for

matrix iterative methods are described in [29]. In that case the Faber poly-nomials are required for some bounded region which contains the estimated

locations of matrix eigenvalues produced by the Arnoldi method or otherwise.

Since one is working with a rough prediction of the eigenvalue spectrum it is not

necessary for the chosen enclosing region to bear any specific relation to the es-

timated eigenvalues. Professor G. Opfer suggested to one of the authors that an

annular sector would be a useful general-purpose region which, by scaling and

rotation, could be adjusted to enclose any estimated eigenvalue cluster boundedaway from the origin. That suggestion motivated the work of the present pa-per, which provides explicit formulae for Faber polynomials for a sector of an

annulus. Results for circular sectors are included as a special case. The maintheoretical results are summarized in the following theorems.

Theorem 1. The complement of the unit disc {w : \w\ < 1} is mapped confor-mally onto the complement of the annular sector

Q = {z:R< \z\ < 1, 0< |argz| < n}, O<0<n,


THE FABER POLYNOMIALS FOR ANNULAR SECTORS 183

by

where

and

ip(w) - -expJa2 x/-

(x)dx

c =4w L

xA(x)

l)(a-2-a2)-2w(a~2 + a2)

A(x) =■■ [(x - a2)(x - a~2)\2 , B(x) = [(x - b2)(x - b~2)

The parameters a and b satisfy the equations

r*2 \(b2 -x)(b~2 -x)l*

i

Jai [(x-a

and

hM-.r [f£^

)(a-2-x)J

(x - b2)(b~2 - x)

)(a~2-x)

dx

x

* dxx

Theorem 2. The transfinite diameter of the annular sector Q defined in Theorem1 is

(1-a4)p = —-¡— exp

Jo Aa- + a-2 - b2 - b~2

o A(x)[A(x) + B(x)]dx

Theorem 3. The coefficients of the Laurent expansion

ip(w) = p(w + ß0 + ßxw-x + ■■■)

of the function defined in Theorem 1 may be generated recursively. Given aand b, in the notation of Theorem 1, let

2a2(l+b4) 1 +a4

b2(l -a4) ' 1-a4'

Let a¡ — 0 for i < 0, ao = 1 and, for k > 0,

(k + l)ak+x - (2k + l)(s - u)ak-2k(s2 -su- l)ak-X

+ (2k - 1)(5 - u)ak_2 + (1 - k)ak-3

and

ck+x =ak+x -sak + ak_x.

Then ß0 = cx, ßx = \c2 and, for « > 2,

n-l

(n + l)ßn = Cn+l -Y,lcn-lßl-1=1

Theorem 4. The Faber polynomial of degree n for the annular sector Q defined

in Theorem 1 is Fn(z) = pnF„(z), where p is the transfinite diameter of Q.

The scaled Faber polynomial may be written as F„ (pz) — z" + </>„_ x(z), and the



ancillary polynomials {</>„} are generated recursively, in terms of the Laurent

coefficients of Theorem 3, by the formula

n-l n-\

(J)„(Z) = (Z- ßo)cpn-l{z) - £ ßk4>H-k-i{z) - £ ßkZ*~k - (1 + «)/?„.

k=l k=0

Theorem 5. The Faber series for z~x, expressed in terms of the scaled Faber

polynomials and the notation of Theorem 1, is

>+£ [^ &«> ■n=l

Theorem 1, which is proved in §2, provides an expression for the map-

ping function y/ , whose inverse is a multiple of the function cp of (1.1) ; if

z = y/(w), then <j>(z) = pw , where p is given by Theorem 2. Section 3 estab-

lishes the formulae collected in Theorem 3, which allow the recursive evaluation

of the Laurent coefficients essential for the computation of the Faber polynomi-als by the recurrence relation of Theorem 4. In addition to proving Theorem 4,

§4 explores some of the properties of the ancillary polynomials {tpn} and showshow explicit expressions for those polynomials may be obtained with the help

of a computer algebra system such as REDUCE or Mathematica or Maple; the

Supplement provides all that is required to determine the scaled Faber polyno-

mials of degree < 15 . The numerical evaluation of the parameters a, b, and

p, for a given annular sector is discussed in §5.Norms of Faber polynomials are of interest in connection with their use in

approximations. Section 6 summarizes some results which are generalizations

of those of Gatermann et al. [18] for circular sectors. As an example of a Faber

series, the expansion of 1/z is investigated in §7, where we find the simpleformula stated in Theorem 5.

2. The conformal mapping

The aim of this section is to calculate an analytic function, y/(w), whichmaps the complement of the unit disc, A = {u;:|u;|<l}, conformally onto the

exterior of the annular sector Q - {z : R < \z\ < 1, 0 < | argz| < n}, where0 < 0 < n (see Figure 1 ). There is no loss of generality in this choice since

rotations and magnifications allow us to apply the results to any annular sector.

For example, to work with the annular sector {z : R < \z\ < 1, | argz| < 0} we

make the transformations z ^ -z and w —> -w , the latter being required to

maintain the form (1.1).The domain obtained by cutting C\Q along its intersection with the negative

real axis is mapped conformally onto the shaded domain E of Figure 2 by the

function z —► log z, where the principal value of the logarithm is taken. The

domain E is the interior of an infinite polygon with finite vertices at the pointslog/? ± in, logR ± id, ±i0, and ± in . Conversely, the function z -* ez

maps E conformally onto the cut version of <C\Q, and the infinite edges of

the boundary of E are mapped onto the cut.

A Schwarz-Christoffel transformation may be used to map the upper half-

plane n = {v : Im v > 0} conformally onto the domain E in such a way that

the real axis is mapped onto the polygonal boundary in Figure 2 and the finite

1 4pa2

z~ R(l-a4)



Figure 1. The annular sector Q

in *•

-V-^-^ \\\v \ iogy?) \ \

\\ \\\ \

V \'-.

\

\ \ X'\ \ \

-inf

\\\\\\\\ \ \ \ \ \ \ N

\\N\V'\v\ \ E \ \ \ \

\ \ \ \ \ \ \K\W\^\Xx\\\\\\

\ \ \ \ \ "x \ \

Cv\\v\\\4 \ \ \

\\\\\\\V

Figure 2

vertices of that polygon are the images of the points ±a~x, ±b~x, ±b,

± a, with 0 < a < b < 1. The Schwarz-Christoffel map has the form

u(v) = u(a) + KJa

v \(w - b)(w + b)(w - b~x){w + b~x)

(w - a)(w + a)(w - a~x)(w + a-1)

dw

w

where a, b, and K are constants to be determined.

Let

(2.1) A(x) = [(x - a2)(x - úT2)]* , B(x) = [(x-b2)(x-b-2)]2-.



Then

(2.2) u(v) u{a) = K fJa

B(w2)dw

wA(w2,

v B{w2)-A(w2)

(2.3)

where we have written

(2.4)

Ja

=K' L

dw + K\og(-\wA(w2) \al

cosh a - cosh ßdx + K log (-} ,

and b-2 J

To determine the constant K, we note that the positive real axis is mapped

onto the upper boundary of the polygon in Figure 2, and the lower boundary

is the image of the negative real axis. Therefore, for all v e (0, a),

2ni = u(v) - u(-v) - K log Í - j - log (— j

from which K = -2. Furthermore, we require

u(a) = in,

and (2.2) becomes

= -Kni.

u(v) inJa2

B(x)dx.

xA(x)

If we assume that a and b can be determined for any given annular sector

(see §2.1), the rest of the argument is similar to that of Coleman and Smith

[4]. A composition of the Joukowski function wtransformation gives

(w-i

+ w) and a linear

(2.5) ^ 2w[w2 + 1) sinh a - 2w cosh a

which maps C\A, the complement of the unit disc A, conformally onto the slit

plane <C\J, where J is the interval [-a~2, -a2]. Let L denote the interval

(-co, 0] of the real axis. Then the function C —> 'C * maps the cut plane <C\L

conformally onto the upper half-plane U.

By composition of the mappings described here we obtain

(2.6) z = - exp L B(x)

xA(xdx

which maps <D\L conformally onto the cut version of (D\ß • At the end of §2.2

it will be shown that the cut introduced in C\ß may be removed and that C\L

is then replaced by €\J .

2.1. Defining equations for a and b. The symmetries inherent in our choice

of the preimages of the vertices of the polygon, under the Schwarz-Christoffel

map, preserve the relationships, which are evident in Figure 2, between the

lengths of the edges. The two distinct lengths which arise are

n-6 = i[u(b)-u(a)]



and

-logR-u(b)-u(b~l]

These equations, which may be written as

(2.7)

and

(2.8)

Jai [{x-a2)(a-2- x)

logR = - [Jb2

(x - b2){b~2 - x)

(x-a2)(a-2-x)\

dx

x

dxx

\

uniquely determine 0 and R for any given a and b such that 0 < a < b < 1.

Furthermore, as the geometrical interpretation in Figure 1 requires, 0 < 0 < n.Since the integrand in (2.7) is nonnegative, the right-hand inequality is true

and, since b2 + b~2 is a decreasing function of b,

(2.9) n-0<Ja

i x dx

2 [(x - a2)(a~2 - x)]2- x= n - 2 sin '

(Ja_).

in particular, 0 > 0. Equation (2.8) may be expressed in a form which is more

useful for numerical computation, by regarding its right-hand side as a sum ofintegrals on [b2, 1] and [1, b~2]. Making the transformation x -» x~x in the

second integral, we obtain

(x - b2)(b~2 - x)]2" dx

)(a~2-x)\ x

The integrals in (2.7) and (2.8) may be expressed in terms of elliptic inte-

grals of the first and third kinds. In a standard notation [27] we find

>{n(ï,-A,*)-n(f,-|,*)};

and

1°«,=i^[Cf(i-l')t't!-1'

x{n('î-k^l")--^u{^-hk2-1")}

where

and

(2.11)

k =b2-a2 (l-fl4)i(l-M)*

l-a2b2' x~ (l-a2b2)

C = 2(cosha - cosh/?).

(1-fe4)

(l-a2b2)'

2.2. Formulae for <p(w). The mapping function y/ may be written in several

different ways, which we list here for later reference.


J. P. COLEMAN AND N. J. MYERS

By using the change of variable x —> x~x in the first formula quoted, bytaking a~x as reference point instead of a as in (2.2), and by using the sub-

stitution x —> x~x again, we obtain

(2.12a)

(2.12b)

(2.12c)

(2.12d)

where

z = ip(w) = -exp - /

\Ja-2 X

r1

= -exp

B(x)

xA(x)

-< B(x)

dx

A(x)

Í R(x)

dx

xA(x)dx

1C = 7T— I (w2 + 1) sinh a - 2w cosh a

In some cases, particularly if the integration interval may include the origin,

it is preferable to remove the logarithmic term from the integral, as in thederivation of (2.3). Corresponding to the formulae (2.12), we have, with C

as in (2.11),

(2.13a)

(2.13b)

(2.13c)

(2.13d)

z = a2Çexp

= a2Çexp

= -2-exp

RC= -rexp

Ja2

C

A(x)[A(x) + B(x)]dx

Ja

í ci A(x)[A(x) + B(x)}

c C

A(x)[A(x) + B(x)]

■f_C

JíyHJÍI A(x)[A(x) + B(x)]dx

Other forms, which do not involve the intermediate variable Ç , may be de-

rived from (2.12). For example, the change of variable

-1

2p(p2 + 1 ) sinh a- 2p cosh a

applied to (2.12b) gives, after some algebra,

(p2-2tp+l)Hß2~2xp + l)2-(2.14)

where

(2.15)

ip(w) exp /u

-1

t =cosh a-b2

sinh a

p(p2 - 2peofna. + 1)

cosh a - b~2

dp

T =sinh a



By combining these definitions with (2.4), it is easily seen that \t\ < 1 and|t| < 1 for 0 < a < b < 1.

To complete the proof that the function y/ provides the required mapping

from C\A to C\7 , it is necessary to show that we may remove the cuts intro-duced in constructing y/ . Let

*(C) = a2exp

= a2exp

A{x)[A(x) + B(x)]

._{- c

dx

A(x)[A(x) + B(x)]dx

The denominator of the integrand is a single-valued analytic function in C\7,the region of the complex plane exterior to the slit J, and it does not vanish in

that region. Integration gives a single-valued analytic function in <C\J and the

identity of the two integrals shows that it remains finite as Ç —► oo . It follows

that x(Q is a single-valued analytic function in €\J and, consequently, Cx(C)is also single-valued in C\7. Since the part of the interval L which lies in

C\7 is mapped onto that part of the negative real axis which lies in C\ß, the

function y/ given by (2.12) and (2.13) maps C\A conformally onto C\ß.

2.3. The transfinite diameter.

(2.16)

Given the mapping z = y/(w), we now define

y/(w)lim

w—>oo W

and (p(z) = py/~x(z), where y/~x denotes the inverse mapping of yi. Then

4>(z) will map C\ß conformally onto the complement of the disc {w: \w\< p},

so the number p is the transfinite diameter of the annular sector ß. As w -*oo,

C = » w sinh a

Therefore, from (2.13a) and (2.16),

(2.17)(1-a4)

p = —-.— exp

(-w

C

A(x)[A(x) + B(x)]dx

2.4. Special cases. There are three special cases which provide useful checks on

the mapping, the formula for the transfinite diameter and the Faber polynomials

themselves.

Case (i): b = a. A real interval

When b = a, the equations (2.7) and (2.8) give 0 = n and R = a4.Consequently, the annular sector ß becomes the interval [-1,axis.

From (2.3) it is evident that

R] of the real

when b = a. Therefore,

z = eu = a2C

u{v) = in - 2 log Í - J



which correctly maps <E\A onto the complement of the real interval [-1, -R],

When b = a, the integrand in (2.17) vanishes, so

(I-*4) 1P = -4O-Ä).

Case (ii): b = 1. A circular arc

When b = 1, the integral in (2.8) vanishes to give R = 1, and (2.7) gives

(see (2.9))

-, / 2a(2.18) 0 = 2 sin

a2 + l

The annular sector ß then degenerates to an arc of the unit circle \z\ - 1, ofhalf-angle n - 0 .

In this case, x = t in (2.14), and

p2 -2tp+ly/(w) — - exp

= tu exp

[/

/w

.x p2 - 2/¿cotha+ 1

The remaining integral is elementary, and the result is

! p(p2 - 2/zcotha+ 1)

2 rw dp

dp

(2.19) z = y/(w) =tü(it;tanhf - 1)

w -tanh I

which correctly maps C\A onto the complement of the circular arc

{z : \z\ = 1, 0 < |argz| < n} ,

where 0 is given by equation (2.18).

From the expression (2.19) for y/(w), the transfinite diameter of the arc is,

p = tanh2 - \l+a2)

cos0_2'

which, with allowance for the difference in notation, agrees with Ellacott [10].

The formula (2.17) similarly gives

,2-(1-a4)p = . exp H^)\<m

Case (iii): a->0 and b -> 0. A circular sector

In the integrand of (2.7), x « b~2 < a~2 when 0 < a < b « 1. Expand-

ing (b~2 - x)h(a~2 - x)'2' in terms of a and b, and evaluating the integrals,we obtain

n - 0 = (n - ^) [l + 0(a2) + 0(b2)

so when a -► 0 and b -» 0, we have

(2.20)

Similarly, from (2.10),

ñ^na

logR2a

' b/' {X " b2)\ dx + 0(a2) + 0(b2

Jb2 x(x - a2)2



log\1-W a

log

The substitution y2 = (x - b2)/(x - a2) allows us to evaluate the integral as

'b-ay0\

J + ayo)

with y0 — (1 - b2)ï(l - a2)'2'. Since further expansion shows that R -> 0 as

a and b both tend to zero, the annular sector ß tends to the circular sector

{z : \z\ < 1, 0 < |argz| < n}

where 0 = na/b ; our results in that limit should agree with those of Colemanand Smith [4] and of Gatermann et al. [18].

To find the transfinite diameter in the required limit, we write a = lb inequation (2.17) and make the substitution x = tb2 to obtain

(1-X4b4)

where

I{X,b)=fJo

exp[I(X,b)] ,

[bA(X2-l)+X-2-l] dt

lo {X2 - i)ï(A-2 - rè4)î [(A2 - t)\{X~2 - tb4)i + (1 - í)*(l - ¿40*

In the limit as b —► 0 this reduces to

1P = 4 exp

The substitution y2 = (1 - t)/{X2 - t) and use of partial fractions converts theintegral to a sum of elementary integrals, and we find

_1_P~ (\-A)l-X(l+X)M'

and in this limit I = 0/n . Theorem 2 of Coleman and Smith [4] shows thatthis is the transfinite diameter for a sector of the unit disc of half-angle n - 0 .

Similar reasoning applied to the integral in (2.13a) allows us to compute y/<¿ ,

the limit of the mapping function y/ as a and b both tend to 0. Noting that

[(u;-l)2 + cV)l ,C4a2w

as a —» 0, we find

y/0(w) =(w - I)2

4wexp

J-Xq {

¿(1-0*(X2-t)\

-1

where xrj = 4X2w(w - 1)~2 .

To obtain the configuration chosen by Coleman and Smith [4], it is necessaryto carry out the transformations z —► — z and w —> -w . In the notation of

[4], modified to avoid conflicting uses of a ,

w + 1u(-w) =

2arwh 'aLc = 1 ■X1

when the correct branch of the square root function is chosen. Straightforward

algebraic manipulation then shows that -y/0(-w) is the mapping function in



Theorem 1 of Coleman and Smith [4], thus confirming that the map derived

here correctly reduces to that for a circular sector in the limit as a -* 0 and

6^0.

3. The Laurent expansion of y/(w) about the point at infinity

The function y/(w) has a Laurent expansion of the form

(3.1) y/(w) = p(w + ß0 + ßxw-x +■■■)

about the point at infinity. The coefficients of this expansion are required in a

recurrence relation used to generate the Faber polynomials.

Differentiation of (2.14) gives

(3.2) £«r<«0- M^M^dw w(w2-2w coin a+1)'

where

M(w, y) = (w2 - 2yw + l)i.

If we now let w = £_1 and *¥(£) = y/(w), equation (3.2) may be rewritten as

n,s m*y C(t2-2c:cotha+l)dV(c:)

{ ' {q) M(t,t)M(i,T) d£ ■

Since |i| < 1 and |t| < 1, there is a convergent expansion of the form

oo

(3.4) [M(i,t)M{Z,i)]-l=Ylaktkfc=0

for |i| < 1 . Substitution in (3.3) gives

(3.5) ^)=(E^)(-^)'

where

(3.6) ck — ak_2-2cothaak_x + ak, k>0,

with a-2 = a_i =0. Also, from (3.1),

W) p(^ + ßo + ßii + ---)

for \Ç\ < 1. We may substitute this in (3.5) and equate coefficients of the

powers of £, on both sides to obtain ßo = cx, ßx = {c2 and, for « > 2,

n-l

(3.7) (l+n)ßn = c„+x-Y,iCn-ißi.1=1

The recurrence relation in (3.7) and the definition (3.6) allow us to generate

ßo, ßx, ... , ß„, for a given positive integer «, when ak is known for k —

0, 1,...,« + 1. In the case of circular sectors, Coleman and Smith [4 ] found

a very simple expression for the corresponding coefficients in terms of Legendre


the faber polynomials for annular SECTORS 193

polynomials. A similar approach can be used here; noting that M(w, y) is the

reciprocal of a generating function for the Legendre polynomials, we find

k

ak = y£Pi(t)Pk-,(r),

1=0

where P„ is the Legendre polynomial of degree « . This formula has the com-

putational disadvantage that as k increases an increasing number of Legendre

polynomials must be evaluated and stored. For this reason we have derived a

five-term recurrence relation from which {ak} may be computed directly. Letf(t, x, Ç) be the function in (3.4). Then, by differentiation and rearrangement,

[t + T - 2f (1 + 2tx) + 3(t + x)i2 - 2?]f(t, x, 0

= (\-2tt:+e){\-2x^+e)dñt,d¡A).

Equating the coefficients of the powers of £ on both sides of the equation, we

obtain(k + l)ak+x =(2k + l)(t + x)ak -2k(l + 2tx)ak_x

+ (2k-l)(t + x)ak-2 + (l-k)ak_3

for k > 0, with a0 = 1 and a, = 0 for /' < 0. With

. cosh a cosh ßs — 2-T—,— and u = 2^——

sinh a sinh a

this becomes

(3 8) ^ + 1^+1 = ^2k +l^s~ u^Uk ~ 2k^2 ~su~ l)ak-i

+ (2k-l)(s-u)ak_2 + (l-k)ak_3.

4. The Faber polynomials

The Faber polynomials, F„(z), satisfy the recurrence relation

n-l

(4.1) Fn+x(z) = (z-b0)Fn(z)-Y,bkFn-k(z)-(l + n)bn, «>0,A:=l

where bk = ßkpk+x, p is the transfinite diameter of the region, and the ßk are

generated from the recurrence relations above. Following Gatermann et al. [18],

we introduce the scaled Faber polynomials

(4.2) F„(z) = Fn(z)p-"=<l>n(j\

and let

(4.3) d>„(z) = z" + 0„_1(z),

where (f)„-X is a polynomial of degree n-l for « > 1 , and </>-i(z) = 0.

Substitution in (4.1) gives the recurrence relation

n-l n-l

(4.4) cpn(z) = (z - ßo)4„-i{z) - £ßk<P„-k-i(z) - ¿2ßkzn~k - (1 + n)ß„.

k=l k=0



Our notation differs slightly from that used by Gatermann et al. [18], in their

work on circular sectors, because no factor analogous to their 1 - c is evident,

except in the limit as a —> 0 and b —► 0, when the annular sector tends to

a sector of the unit disc. Given the Schwarz-Christoffel parameters a and bcorresponding to a particular annular sector, the Faber polynomials of degree

up to «max may be computed by the following algorithm.

Algorithm.

5 = 2(1 + a4)/(l - a4) ; u = 2a2b~2(l + b4)/(l - a4) ;

a_3 = a-2 = a_i = 0; a0 = 1 ; ax=s-u;

cx=ax-s; ßo = cx; (f>0 = -ß0;F0(z) = 1; Fx(z) = z + p</>0.

For « = 1, «max - 1

an+x = [(2« + 1)(5 - u)an - 2n(s2 -su- l)an-X

+(2« - 1)(5 - w)a„_2 + (1 - n)an-3]/(n + 1) ;

cn+x =an+x -sa„ + an-X;

ßn = (Cn+i-Z":1lc„-lßi)/{n + l);

(ßn = (z- ßoWn-l - E"k~Jl ßk(4>n-k-l + Zn~k) - (1 + n)ßn - ß0Zn ;

<D„+1 = z»+' + <f>H ; Fn+X(z) = />"+1On+1 (f ) .

end.

Example. With «max = 2 we obtain

flo=l; ûi=5-m; cx=-u; ß0 = -u; (f>o = u;a2 = (s2 - 4su + 3u2 + 2)/2 ; c2 = (-s2 - 2su + 3u2 + 4)/2 ;

ßx = (-s2 - 2su + 3u2 + 4)/4 ; (px = 2hz + (s2 + 2su - u2 - 4)/2 ;

a3 = (-s3 - 3s2u + 9su2 - 5w3 + 85 - 8w)/2 ;

Ci = (-253 + s2u + 6su2 - 5w3 + 85 - 10w)/2 ;

ß2 = (-4si+s2u + lOsu2 - 7M3 + ißs _ 16m)/12;

02 = 3mz2 + (352 + 65« + 3m2 - 12)z/4 + (253 + s2u - 2su2 + u* + 2u- Ss)/2.

Then Q>x(z) = z + u, 02(z) = z2 + (f>x(z), 03(z) = z3 + (j)2(z).

In the limit as a —> 0 and b —> 0, when the annular sector becomes a sector

of the unit disc, we have that i-»2 and, from (2.20),

M^2(£)2 = 2(l-c)

where the parameter c is as used by Gatermann et al. [18]. Making the changes

described at the end of §2.4 to obtain the appropriate sector, we find, for exam-ple,

<D,(z) = z-2(l-c),

<D2(z) = z2 + ( 1 - c)[-4z + 2 + 2c],

<D3(z) = z3 + ( 1 - c)[-6z2 + (9 - 3c)z + -2 - 4c],

which agree with Gatermann et al. [18].


the faber polynomials for annular sectors 195

4.1. The coefficients of (j>n(z). Letting

(4.5) 0„(z) = ¿^;Jz"->

;=0

in the recurrence relation (4.4), we obtain, after some algebra,

(4.6a) Pn,o=Pn-i,o- ßo = (n+l)u,

n

(4.6b) pn,n = -J2ß"-kPk-l,k-l ~(l+n)ßnk=\

and, for /= 1,...,«- 1,

n

(4.6c) Pn,n-i=Pn-l,n-i- Yl ßn-kPk-l ,k-l-i ~ ßn-i-k=i+l

In the interest of brevity we shall use the term ^-polynomial to describe a

polynomial in two variables, s and u, which is invariant or changes sign, underthe planar antipodal map (s, u) -> (-s, -u), according as the degree of the

polynomial is even or odd; in other words, such a polynomial of even (odd)

degree contains only terms of the form s'u-> where i + j is even (odd).

Theorem 6. The coefficient p„j, for « = 0,1,... and j — 0, ... , n, is an

A-polynomial of degree j +1 in s and u.

Proof. An induction argument applied to the recurrence relation (3.8) shows

that ak is a polynomial in 5 and u of degree k. Furthermore, since s - u

and s2 - su - 1 are ,4-polynomials of degree 1 and 2, respectively, the hy-

pothesis that ak is an ^-polynomial of degree k for k — 0, ... , n leads to

the conclusion that the same property holds for k = « + 1 ; the induction hy-

pothesis is readily confirmed for « = 1. It then follows from (3.6) that cn is

an /1-polynomial of degree n , and an induction argument applied to equation(3.7) shows that ßn is an ^-polynomial of degree n + 1 .

Turning now to the equations (4.6), we assume, as an induction hypothe-

sis, that for each « the coefficient pnj is an ,4-polynomial of degree j + 1.

Then each term on the right-hand side of (4.6c) is an /1-polynomial of degree

n-i+1, and (4.6b) gives the corresponding result for / = 0 ; finally (4.6a) shows

that pn _ o is an /i-polynomial of degree 1. Clearly the hypothesis is true for

« = 1 . ' D

In view of Theorem 6, we may write

Pn-l,0 = YnlU,

Pn-l,l= 7n2 + Yn3S2 + y„4SU + y„5U2 ,

Pn-i,2 = ïnes + 7niu + yn853 + y„9s2u + y„xosu2 + ynXXu3,

etc. In keeping with the notation of Gatermann et al. [18], we regard the coef-

ficients ynk as the elements of the «th row of a matrix

r = {y„k), «=1,2,...; k = 1,2, ... , m(n),

where «i(«) is the number of terms in the «th row of Y.



Theorem 7. The elements of the matrix Y are rational numbers, and the number

of elements in the nth row is

m(n)24(2«3 + 15«2 + 37«-30)--^

«-1

where [x] denotes the integer part of x .

Proof. It is clear from the various defining equations that the coefficients of the

polynomials pnj are rational numbers.

A homogeneous polynomial of degree j has j + 1 terms. An ^-polynomial

of degree 2r is a sum of homogeneous polynomials of even degree from 0 to

2r inclusive; it consists of

¿(2/+l) = (r+l)2

/=o

terms. Similarly, an ^-polynomial of degree 2r + 1 has

r+l

Y,2l = r2 + 3r + 2l=i

terms.Equation (4.6a) shows that p„:o consists of a single nonzero term. For /' > 2,

the polynomial pn,i-i, being an A -polynomial of degree ¿,has \(i + 2)2 terms

if i is even, and 3(/+2)2-^ terms if /' is odd. For a given «, the total number

of terms is

, , 1 ^—^ /. ^ ~> l it — 1

-Ta12"'(2«3+15«2 + 37«-30)-i«-1

D

A computer algebra system may be used to compute the polynomials p„jfrom (4.6), as polynomials in s and u. We have used Mathematica and

REDUCE for this purpose. The polynomials pn-i,j, for j = 0(1)« - 1 andn = 0(1)15, are listed in the Supplement section of this issue, where pn-ij is

denoted by p{j), for each value of « . The results given there may be used with

(4.2), (4.3) and (4.5), to construct the Faber polynomials of degree < 15.

The first ten rows of the matrix Y have the form

/ i2

34

567

89

\10

-2 1/2 1 -1/2-3 3/4 3/2 3/4-4123

-5 5/4 5/2 25/4-6 3/2 3 21/2-7 7/4 7/2 63/4-8 2 4 22

-9 9/4 9/2 117/4

\

-4

-16/3-20/3

-8

-28/3-32/3-12

1

-8/3-25/3-16

-77/3

1

4/35/32

7/3

1/25/3

-1

2/31/21/35/311/2

■112/3 8/3-51 3

10/3 10/311/2 749/6 35/3 77/634/3 52/3 74/3

15 24 42

-10 5/2 5 75/2 -40/3 -200/3 10/3 115/6 95/3 395/6



and other entries may be read from the Supplement. As Gatermann et al. [18]

found for circular sectors, all entries in a column of Y are expressible in terms

of a polynomial in the row index. For example, we find

Yni = n, ynl = —n-n ¿,

1Yn2 = ~n, 7118=3«»

10-« - «

n

1 17*3=4«, 7n9 = -T2" + 4"2>

1 11 1 ,7«4=2"> ymo = —g-"+2W .

1 2 5 29 5 2 1 37«5 = 2« 4n> 7nXX = T2n~4 6

4yn6 = ~jn,

5. Numerical evaluation of a, b, and p

To compute the Faber polynomials for a particular sector of an annulus,

we need to evaluate the Schwarz-Christoffel parameters ( a and b ) and the

transfinite diameter p. Given R and 0 defining a particular sector, a and b

are found by solving the pair of nonlinear equations (2.7) and (2.8). We used

a modified Newton iteration in which the partial derivatives in the Jacobian

were approximated by central difference formulae of the form

df(a,b) _f(a + D,b)-f(a-D,b)da 2D

The convergence of the Newton iteration depends on having sufficiently good

initial estimates of a and b . Table 1 gives suitable values for certain ranges

of R and 0. As 0 tends to 0 or i, and as R tends to 0 or 1, the con-

vergence becomes much more sensitive to the choice of starting value, but thecorresponding regions, which are close to known limits, are less likely to be of

practical interest than those covered by Table 1.

At each step of the Newton iteration it is necessary to evaluate the integrals in(2.7) and (2.8) numerically to an accuracy consistent with that required in theNewton iteration. Despite the square root singularities at the endpoints of the

integration intervals, the NAG routine D01AHF, which is based on the Gauss-

Kronrod-Patterson family of formulae, works satisfactorily. Figures 3 and 4

show a and b, respectively, as functions of R and 0, for 0.001 < R < 0.999

and 0.3 < Ö < 3.1414.The integral in equation (2.17), which defines the transfinite diameter, must

also be evaluated numerically to whatever accuracy we require. Again, the NAG

routine D01AHF is appropriate.



Table 1. Starting values for modified Newton iteration

0.999 - 0.1 0.1-0.02 0.02-0.01 0.01 - 0.005 0.005-0.001

3.1414-3.0 o = 0.4894 = 0.513

D = 1.0 x 10-«

a = 0.37604 = 0.3761

D = 1.0 x 10"6

a = 0.3107b = 0.3108

D = 1.0x 10-

a = 0.265916 = 0.26592

D = 1.0 x 10-»

a = 0.17774b = 0.177751.0 x 10"'

3.0-1.9 a = 0.49¡> = 0.52

D = 0.001

o = 0.12b = 0.24

D = 0.001

a = 0.09b = 0.12

D = 0.001

a » 0.07

4 = 0.12D = 0.001

a = 0.07

6 = 0.10D = 0.001

1.9-1.57 a = 0.124 = 0.24

D = 0.001

o = 0.126 = 0.24

D = 0.001

o = 0.064 = 0.10

D = 0.001

a = 0.06

4 = 0.10Z> = 0.0001

a = 0.014 = 0.02

D = 0.0001

1.57-1.00 a = 0.054 = 0.16

D = 0.001

a = 0.034 = 0.08

D = 0.001

a = 0.014 = 0.03

D = 0.001

a = 0.014 = 0.03

D = 0.0001

a = 0.00056 = 0.001

D = 0.0001

1.00-0.75 a = 0.054 = 0.07

D = 1.0 x 10"s

a = 0.03

4 = 0.080 = 0.001

a = 6.0x 10-5

4 = 6.0 x 10-4D = 1.0 x 10-'

a = 4.0 x 10"5

4 = 3.0 x lO-4D = 1.0x10-»

a = 4.0 x 10-6

6 = 3.0 x lO-5D = 1.0 x 10"6

0.75 - 0.50

0.50 - 0.30

a = 0.01

4 = 0.06D = 1.0 x 10"5

a = 0.034 = 0.08

D = 0.001

a = 6.0 x lO-8

4 = 6.0 x 10"4D = 1.0 x 10"3

a = 4.0 x 10"5

4 = 3.0 x lO"4£> = 1.0xlQ-5

a = 4.0 x 10"64 = 3.0 x 10"s

D = 1.0 x IP"6

a = 1.1 x 10-*4 = 1.6 x 10"3D = 1.0 x 1Q-'

a = 5.0 x lO-84 = 5.0 x 10-*.P =1.0x10-"

a = 1.0 x 10"64 = 1.0x10"'

D = 1.0x 10"7

a = 1.0x10-'

4 = 1.0 x lO"8D = 1.0 x 10"*

a = 1.0 x 10"9

4 = 8.0 x lO"9D = 1.0x 10 10

Figure 3 . A graph of a(R, 0)



Figure 4. A graph of b(R, 0)

6. Norms of Faber polynomials on annular sectors

Any assessment of the accuracy of an approximation based on Faber poly-

nomials requires some knowledge of a relevant norm. Both the area and line

versions of the 2-norm may be computed explicitly by a slight modification

of the work of Gatermann et al. [18] for circular sectors. An upper bound is

available for Halloo •

6.1. The area 2-norm. The square of the relevant norm of the scaled Faber

polynomial F„ is

(6.1) = f <&n(-) dxdy = p2[ \0>n(u)\2 dvdw.JQy,RA V^/ JQy.K.P

Here, z = x + iy, u = v + iw, z = pu and Qy,R,p is an annular sector of

half-angle y, outer radius 1/p, and inner radius R/p. Using the approach of

[18], we obtain

n-l n-l

(6.2)

where

= In.n + 2 y ^pn-\jIn-\-jt„ + y jPn-X jIn-\-j,n-l-j

7=0 j=0

n-l

+ 2 / ^Pn-l,jPn-l,kIn-\-i.n-l-k ,j>k

( 2 ,--/Sin(t-7)a1 P (i-J)

U,j=<

1 - Ri+J+2

i+j+2J

p-2i(l-R2i+2)a

i+1

ifiïj,

if/= ;.



As R -* 0 these expressions reduce to those of §3.1 of [18], when account istaken of the difference in notation mentioned after equation (4.4) above.

6.2. The line 2-norm. In defining the line 2-norm, ||F„||, the surface integral

of (6.1) is replaced by a line integral around the boundary of the annular sector.

The square of that norm is obtained by replacing each I¡j in (6.2) by

h,j =2p-'-J [(1 - Ä»*») (%^) + S$£$2 (Ri+J+X + 1)] if i* j,

6.3. The maximum norm. Upper and lower bounds may be obtained for the

norm

= maxzee

F«{z)

of a scaled Faber polynomial; by the maximum principle, that maximum value

occurs on the boundary of the domain ß. Let

Tn{2) z" + a„-xz"-x + + a0

be the Chebyshev polynomial of degree « for the annular sector ß, the monic

polynomial of smallest maximum modulus on ß. It is known (Walsh [30,p. 320]) that || 7), ||„j > pn and therefore, since no monic polynomial of degree

« can have smaller norm, Halloo > 1 ■ An upper bound, independent of the

degree « , comes from the inequality

If, v< —n

(see Ellacott [9]), where V is the total rotation of the boundary of the domainQ . For an annular sector of interior angle 2y,

V=l2n + 4y for 0 < y <

2'

6n - 4y for — < y < n.

Combining these bounds, we have

(6.3) 1 <V

< —.oo n

Starke and Varga [29], who used a different normalization for the Faber poly-

nomials, provided bounds in terms of the norm of the corresponding Chebyshev

polynomials. Their Theorem 3.4, for nonconvex regions, is applicable to annu-

lar sectors, and 0(0), which appears in their bounds, is given in closed form

by equation (7.3) below.

7. The Faber series for z"x on an annular sector

The Faber series for a function /, analytic in the annular sector ß, is an

expression of the formoo

j=0


the faber polynomials for annular sectors 201

See Curtiss [6], Markushevich [24, v. 3, p. 109], or Gaier [17, p. 44]. Thecoefficients are

(7.1)1 f f(y(w)) ,

a¡ = =-^-^ / \, dw,1 2mpJ JM=Ri wJ+x

where Rx > 1 is sufficiently small that / can be extended analytically to the

closed region bounded by the image under y/ of the unit circle \w\ = Rx. In

particular, when f(z) = z~x, the Faber series is

(7.2)1z

1

w*y/'(w*)'+£

n=l

Fn{z)

(w*)n

where F„(z) is the scaled Faber polynomial introduced in (4.2), and w* is the

root of magnitude greater than 1 of the equation y/(w) = 0; in other words,w* is the point which y/ maps to the origin in the z-plane. Equation (7.2)

may be established either by applying Cauchy's residue theorem to (7.1) or,

as in Chui et al. [1], by using a generating function for Faber polynomials andthe uniqueness of the Faber series. It is clear from (2.13) that z = 0 implies

Ç = 0, and therefore

(7.3) -• 1+fl2w

1-a2'

From (2.13c) and (3.2) we obtain

,... „ > R(l-a4) f-{

C dx

A(x)[A(x) + B(x)]

Setting C = 0 in this expression, and noting that

4a2

(1-a2)2

C dx

M(w , t)M(w, x).

and

we obtain

(7.5)

M(w\ t)M(w*, x)

fJo

(1-a4)p = . exp

w*y/ (w*) =

o A(x)[A(x) + B(x)]

R(l-a4)

4a2p

Then, from equation (7.2), the Faber series for z ' is

1 _ 4pa2

z(7.6) R(l-a4)

'+£

n=l

V ~) FniÎtI)

As a check on (7.6), it can be shown that it correctly gives a known Cheby-

shev expansion when b = a (case (i) of §2.4). The Faber polynomial of degree

« > 1 for the interval [-1,1] is 2x~"Tn(x), where Tn is the Chebyshev

polynomial of degree « , and the corresponding polynomial for z e [-1, -R]

is

^)-(^H^)License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use


With the help of results from §2.4 for this particular case, the expansion (7.6)

becomes

(7.7) M1+2lfeS) T"nz+,+R'l-R

The Chebyshev expansion

1 1

X-Ô y/S2^!

oo n 1

l+2Y/(S-y/ô2-l) Tn(x)\,n=l J

for ô > 1 and x e [-1, 1], may be established by a technique used, for exam-

ple, by Fox and Parker [16, p. 85]. By transforming to the interval [-1, -R]

and letting S = (1 + R)(l - R)~x, we again obtain (7.7).The maximum norm of the error in approximating z~x on the domain ß

by a truncated Faber series

Qn(z) = -w ^(l+tw-^)j

is easily bounded. From (6.3) we obtain

-Qn{z) <V\w*\-"

(see also [1]), and from (7.3)

1 < y<

n\w*y/'(w*)\(\w*\ - 1)

2Vp (1-a2

nR(\+a2) \l+a2

Acknowledgments

We are very grateful to Dr. R. A. Smith for his invaluable help in constructing

the conformai map, and to Professor G. Opfer for a discussion which prompted

the research reported here. N. J. Myers is supported by a research studentship

awarded by the Science and Engineering Research Council.

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Department of Mathematical Sciences, University of Durham, South Road, Durham,

DH1 3LE, England

E-mail address : j ohn. colemanSdurham .ac.uk

Department of Mathematical Sciences, University of Durham, South Road, Durham,

DH1 3LE, EnglandE-mail address : n. j . myer sfidurham .ac.uk


mathematics of computationvolume 64, number 209january 1995, pages s1-s6

Supplement to


JOHN P. COLEMAN AND NICK J. MYERS

Appendix

Table 2. Faber polynomials up to degree 15

!>-l

Polynomials <t>n-\ = V^z"-7-1 for 1 < n < 15.

Degree of Faber Polynomial: 1

p(0): U

Degree of Faber Polynomial : 2

p(0) : 2*Up(l) : (SA2+2*S*U-UA2-4)/2


p(0): 3*Up(l): (3*(SA2+2»S*U+UA2-4)1/4

p(2): (2*SA3+SA2*U-2*S*UA2-8*S+UA3+2*U)/2


p(0): 4*Up(l) : SA2+2*S*U+3*UA2-4

p(2) : (4*SA3+5*SA2*U+2*S*UA2-16«S+UA3-8*U)/3

p(3): (11*SA4-4*SA3*U-6*SA2*UA2-48»SA2+12*S*UA3

+32*S*U-5*UA4-16*UA2+16)/8


p(0): 5*Up(l): (5*<SA2+2*S*U+5*UA2-4))/4

p(2): (5*(SA3+2*SA2*U+2*S*UA2-4*S+UA3-5*U))/3

p(3): (5*(9*SA4+8*SA3*U+4*SA2*UA2-42*SA2+4*S*UA3-20*S*U-UA4-10*UA2+24))/24

p{4): (12«SA5-13*SA4*U+8*SA3*UA2-56*SA3+14*SA2*UA3

+60*SA2*U-2 0*S*UA4-72*S*UA2+32*S+7*UA5+28*UA3+8*U)/{


p(0): 6*Up(l): (3*(SA2+2*S*U+7«UA2-4))12

p(2): (4*SA3+11*SA2*U+14*S*UA2-16*S+11*UA3-32*U)12

p(3): (39*SA4+76*SA3*U+74*SA2*UA2-192*SA2+44*S*UA3

-256*S*U+7*UA4-12 8*UA2+144)/16

p(4): (46*SA5+31*SA4*U+34*SA3*UA2-248-SA3+22*SA2*UA3-12 0*SA2-U-20*S*UA4-13 6*S*UA2+2 56*S+7*UA5+2 4*UA3

i-1044U)/20

p(5): (21*S"6-34*SA5*U+65*SA4*UA2-100*SA4-20*SA3'UA3+112*SA3*U-65*SA2*UA4-312*SA2*UA2+72*SA2+70*S*UA5

+304*S*UA3+144*S*U-21*UA6-100*UA4-72*UA2-321/16

SI

© 1995 American Mathematical Society

0025-5718/94 $1.00 + $.25 per page


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