].If {d r lr o is a convex null sequence,

We finally are prepared to prove the main theorem of this section due to

THEOREM 6.3d. [ 96 ] Let 1 == Bo $ B1 $ · · · $ Bn be a finite non-

decreasing sequence and for f E ..J./ let E8 be defined by (6.3-5) as

where

(6.3-15)

and

then

(6.3-16)

and { Cv } is a positive increasing sequence. With this new notation, we see

that

and that (6.3-14) is equivalent to

I EsUl(z) I ::;; I z lk => or

(6.3-17)

In order to prove this last result, we consider the convolution inverse of C given by

conditions for which