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The standard de¬nition of real Chebyshev polynomials given by equation (6.88) extends 0 0

T T

without dif¬culty to complex variables. First, as was seen before, when is real and ,

X X y

¬ W $ ¡¡ ¡¡

the alternative de¬nition, , can be used. These de¬nitions can

¤ ¤ W“ &

)

HT

be uni¬ed by switching to complex variables and writing T T

¤W ¡¬ X ¤

X X

¬ ¡ ¡¡

where

¤ ¤

(

) H ¡

¬

De¬ning the variable , the above formula is equivalent to

T¥ ¡

r±a „ ®

i

¬X z

¬

where

& H “ ¥ w H $¥ y 8& W “ ¥ w

$¥ y

) H ˜ ˜

T

The above de¬nition for Chebyshev polynomials will be used in . Note that the equation §

X

T

¬

has two solutions which are inverse of each other. As a result, the value

W “ X ¥ w ¥ W£ ¥

of does not depend on which of these solutions is chosen. It can be veri¬ed directly

)

H

that the ™s de¬ned by the above equations are indeed polynomials in the variable and

)

H

that they satisfy the three-term recurrence T T T

± ¯ „ ®

i

X

X X

² ¬

T)

) T H) ` ˜ (

W“H X "UQH

W

X

(!

! )

)

W y

As is now explained, Chebyshev polynomials are intimately related to ellipses in the

¦

complex plane. Let be the circle of radius centered at the origin. Then the so-called

)

Joukowski mapping T

¬X &W “ ¥ w

¥¦ $¥ y

˜

¦ ²

transforms into an ellipse of center the origin, foci , major semi-axis w $ W£

( &W “

) T

yy

z

²

and minor semi-axis . This is illustrated in Figure 6.2. 0 W£ 0W“ X

There are two circles which have the same image by the mapping , one with the ¥¦

radius and the other with the radius . So it is suf¬cient to consider only those circles

W“

T

(& ( Q²$

¬

with radius . Note that the case is a degenerate case in which the ellipse

! X

y y

²9( y Y

reduces to the interval traveled through twice.

% (

T

y yy

An important question is whether or not a generalization of the min-max result of The- X

orem 6.4 holds for the complex case. Here, the maximum of is taken over the ellipse 0¡ 0

boundary and is some point not enclosed by the ellipse. The answer to the question is no;

¡

Chebyshev polynomials are only optimal in some cases. However, Chebyshev polynomials

are asymptotically optimal, which is all that is needed in practice.

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