1.1. DEFINITIONS AND BASIC PROPERTIES 15

(v) There are polynomials ir^k G ^M-I, (hty G N x

ZN,

such that (1.1.24)

holds with

(1.1.28) 00 = /, 9i= Yl \f - KiMXQiMQ

f°ri^N-

kezN

Furthermore, in each of the cases (ii), (iii), (iv), and (v), (1.1.24) defines a

norm equivalent to \\f\\YL(E)-

The proof of this theorem depends on a number of lemmas, which are given in

Section 1.2. The proof itself is given in Section 1.3.

THEOREM

1.1.15. Let e+, s- G R, r 0, d 0 and let E G S(e+,e_,r,d).

Then, for any nonnegative integers L and M such that L iVmax{- — 1,0} — £+

and M S-, the following conditions on a distribution f G S' are equivalent:

0) / G Y(E).

(ii) The estimate

(1.1.29) IKMSob °°

is satisfied with

h0(x) = sup |(/,/?(• -x))\,

where the supremum is taken over all p G S, normalized so that for a fixed number

X 7Vmax{±,l} + d,

(1.1.30) max(l + \x\)x\Da(p(x)\ 1 for all a with \a\ L ,

X

and with

hi(x) = 2iNsnp\(f,^(2i(- -x)))\ fori€N,

where the supremum is taken over all if G S, satisfying (1.1.30), and such that, in

addition, i\) _ L

*#M-I-

(ii') The estimate (1.1.29) is satisfied with

h0(x) = sup |(/,p(- -x))\,

where the supremum is taken over all Lp G

CQ?(B(0,

1)), such that

\\(f\\cL

1

and

with

hi(x) =

2iN

sup| (/, ^(2( --x)))\ forie N,

where the supremum is taken over all ip G

CQ°(B(0,

1)), such that \\ip\\cL 1? and

such that, in addition, ty _ L

*#M-I-

(iii) / has a representation (1.1.25), converging in S', where the functions a ^

in addition to (1.1.26) satisfy a ^ -L ^PL-I for i G N, and the coefficients Si^ are

such that (1.1.24) is satisfied with gi defined by (1.1.27).

REMARK.

Note that the conditions (ii) and

(ii7)

are analogous to condition (iii)

in Theorem 1.1.14. In fact, the functions hi in the former conditions measure the

degree of local polynomial approximation to the distribution / . We return to this

point in Section 3.2; see in particular Lemma 3.2.1.

The proof of this theorem again depends on a number of lemmas, mainly in

order to achieve the orthogonality conditions. These are given in Section 1.4, and

then the proof of the theorem is given in Section 1.5.