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.
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