INTRODUCTION. NOTATION 3
Lp,\oc the space of measurable functions on R^, belonging to LP(A) for all
compact A c
RiV.
V CQ°, V', P
/ m
the usual spaces of test functions, distributions, and
distributions of order at most m.
S, S' the Schwartz spaces of test functions and temperate distributions.
T,
T~x
the Fourier transformation, and its inverse.
Mr,df the maximal function, defined for measurable functions / , and r 0,
d 0 by
Mr4f(x)=sup(a-N f \f(x + y)\r(l + \y\)-rddy) .
a0\ JB(0,a) /
Mr$f is denoted Mrf.
The action of a distribution / on a test function p is denoted (/, (f).
The convolution of a distribution / and a test function ip is denoted / * (p and
defined by / * ip(x) = {f,ip(x- •)).
^Pm5 TU —1, 0, 1, 2, ..., is the set of polynomials of degree at most m. (We
define ^3_i = {0}). The notation / _ L ^3
m
for a function or distribution / means
that JnN f-K dx = 0, or (/, 7r) = 0 , for all polynomials ir G £pm.
Let A C R ^ be bounded, let m = 0, 1, 2, ..., and let X be a normed or
quasi-normed space of functions defined on R^ . The degree of local approximation
is
£m(f,A,X)= mf \\f-ir\\x{A),
where X(A) is the restriction of X to A.
We adopt the convention that the letter C (without any index) is used to denote
a constant, whose exact value is irrelevant and can change even within a chain of
inequalities.
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