"–© ¡"

§

1B

T

¬X

fying the constraint can be written as

¡

¡

y T uu

£

H u

F

¬X u

¡ £ uF

H u

¡

T

X

A point on the ellipse is transformed by from a certain in . Similarly, let ¡Y )

¦ ¥ ¥

(

¤

be one of the two inverse transforms of by the mapping, namely, the one with largest ¡

modulus. Then, can be rewritten as T

¡

T uu

u

X

£

r± „ ®

i

“ ¥ w ¥T F

H u

¬X u

Xu

¡ £

“ ¥ w ¥ uF

H u

¤ ¤F ¥

¬H ¬ ¬

uF

Consider the particular polynomial obtained by setting and for ,

Y W

T y

H“ ¥ wH¥

¬X

¨

¡

H“ ¥ wH¥ ¤

¤

which is a scaled Chebyshev polynomial of the ¬rst kind of degree in the variable . It W

is apparent that the maximum modulus of this polynomial is reached in particular when

R

¬ ¬

is real, i.e., when . Thus,

¥ ¥

¡

T

H“ w H

¬„0 X

¨ 0¡ P H 6

7 U 2

8 0 H “ ¥ w H 0¥ ¤

¤

which proves the second inequality.

To prove the left inequality, we rewrite (6.97) as T

T u Xu

£

“ H ¥ w QH ¥ T u F

H u

X H“ ¥ U

¬ T u

Xu

¡ £

“ H ¥ w QH ¥ u F

H“ ¥ H u

U

¤ ¤ ¤

X

and take the modulus of , T

¡

T u

Xu

£

¡ ¡

“ H ¥ w QH ¥ T u F

H u

¡ ¡

q0 X U

H“

¬ ¡ ¡

u

Xu

0¡ £

¡ ¡

“ H ¥ w QH ¥ u F

0¥ H “0 H u

U

¤ ¡ ¡

¤ ¤

The polynomial in of degree inside the large modulus bars in the right-hand side is

¥ §˜

W

T T

such that its value at is one. By Lemma 6.4, the modulus of this polynomial over the ¥

¤ £X