§

¥

¤ ¤

The transformed right-hand side in the proposition is not known explicitly since it is ex-

pressed in terms of the exact solution. However, a procedure can be found to compute

) ˜

c c

it. In other words, it is possible to operate with without invoking . Note that v v

˜ ˜

}

c c c c

. As the next lemma indicates, , as well as , can be

'

D

v v v v

x

W

computed recursively.

™ # 9¥y¤

¢ #%

De¬ne the matrices )

An# H¦ j

˜

''

C

D C

nA¢ H¦ j

˜

˜ c

C

D C v

nA"¦ H¦ j

¤

˜

˜ ˜ c

c

2 DC )# # vC

C C

Dv wC

˜

c c

for . Then , , and the matrices and satisfy the

C C

2i¢i’—D w

thhhte D D

v v

recurrence relations

W W

)

t"2

Dc c

Pn ¦ H¦ j

˜ ’

˜

}

2 tc C

DC xC C

tc d "w

D 2i¢it

thhh

v v

and

¤ )

Dc tc

An ¦ H¦ j

¤

˜ S

˜ }

DC t c vC xC C tc d DVw 2¢iit

thhh h

v

£¤ %

£ c

It is clear by the de¬nitions (13.28) and (13.29) that and that

D Dc

v

, . For the cases , by de¬nition of and

W

2 Dc …' C''

) q)w )

e C

c c c

An¦ ¦ H¦ j

v

˜

}

2 D }c C

2 2

DC xC tC C C c C

Ac t

x

v v v

˜ c