2 INTRODUCTION. NOTATION

In the case of spaces of Besov and Lizorkin-Triebel type the results of Sections

2.2 and 2.4 were announced by Netrusov in the short note [35]. In the special

case of L*, 1 p oo, proofs were published in [4], Chapter 10, but proofs of

the general results have not been previously available. Section 2.3 is devoted to an

extension of the spectral synthesis theorem to Besov spaces of distributions.

Chapter 3 is devoted to approximation theorems of Luzin type for the spaces

studied in Chapter 1. In Section 3.1, which is independent of the results of Sections

2.2 - 2.4, we prove a theorem which improves earlier such results even in the case

of Sobolev spaces. In part this result was announced by Netrusov in [32], but the

proof has remained unpublished.

This section is continued in Section 3.2 with an analogous theorem for spaces

of distributions, which seems to be a question that has not been previously studied.

Finally, in order to make the paper more self-contained, we give as an appendix

a complete proof of the Whitney theorem on polynomial approximation in

LP(HN)

for 0 p oo which plays a crucial role in the theory.

The purpose of this joint paper is to give a detailed and accessible exposition

of the results sketched above. Since the theory is developed from the beginning,

the reader is not required to be familiar with the theory of B- and F-spaces.

NOTATION.

Throughout the paper we will use the following notation:

Z, R — the integers, the real line.

ZN

— iV-tiples of integers, k = (fci, &2, • • •, &AT), h G Z, i = 1, 2, ..., N.

R ^ — the real iV-dimensional Euclidean space, equipped with the norm \x\ =

\{xux2,-..,xN)\ = (r^ + ^ + .-. + x^)

1

/

2

.

N = {1, 2,... }, the natural numbers. N

0

= {0} U N.

N — the dimension of the space R^ .

B(x,a) — the open ball in R ^ with radius a 0 and center at x.

Qo,o — the unit cell [0,

l)N

in R^ .

{Qi,k}(i,k)eZxZN — the family of dyadic cells

Qhk = 2-\k +

[0,1)N)

= 2~\k + Q0,o), i e Z , k G

ZN.

Qi,k{a), a 0 — the closed cube concentric to Q;^ and with sidelength a2~\

x(A) or XA— the characteristic function for the set A.

Xi,k — the characteristic function for Qi,k'

Da

with a multi-index a = (ai, c*2,..., &N) — & partial differentiation operator

of order |a| = ^2i=1 &i

Th and A^ = Th — I with h G R ^ — translation and difference operators;

Thf(x) = f(x + ft), Ahf(x) = f[x + ft) - f{x).

AJJ2 = (Th — I)m — difference operator of order m.

C m , m G NU{oo} — the m times continuously differentiate functions, not

necessarily bounded. In situations where we use the norm

ll/||c-(A) = max sup

\Daf(x)\,

\x\mx£A

it is understood that we are dealing with bounded functions.

CQ1 — the subspace of C m of compactly supported functions.

\A\ = fA dx, A C HN — the Lebesgue measure.

LP(A), 0 p oo, A C R ^ — the Lebesgue spaces, equipped with the

(quasi)-norm defined by

\\f\\pL

,A\ — fA

\f(x)\pdxJ

modified in the usual way for

p — oo.