13 Wavelet Methods for Pointwise Regularity
hence the first part of Proposition 1.2. Let us prove the converse part.
Suppose here for simplicity that N = 0 for UJ and 9. We want to bound
|/Cr)-/0ro) | £ , / ( * ) - A . / O r o ) ! .
jez
The sum for j j \ is bounded by
-foo
C^u(2-j) Cu(2-jl)
3i
The sum for j
0
j j \ is bounded by
h
{h-3o)0{\x-x0\) +
Y.e^~J)
^ Ui-Jo)0(\x-xo\) + C9(2-^)
JO
and it is not difficult to see that the sum for j jo is bounded by
jo
\x-x0\^2i[9(\x-x0\) + 9(2-i)] C\x-x0\ 2^(9{\x-x0\)+9(2-^)) .
—oo
Hence Proposition 1.2.
Let us show an example of the application of Propositions 1.1 and 1.2
to lacunary Fourier series. Consider
(1.8) f(x) = Y^rk sm(nkx + (pk)
and suppose that
(1.9) ?±±± qi
a n d
Y\rk\C.
Let 9 be the continuous piecewise linear function such that
\nkJ
Corollary 1.1 If 9 verifies assumption (1.3), 9 is the uniform modulus
of continuity of f and there exists no point xo where f has a modulus
of continuity UJ such that uo(x) o{9{x)) in a neighborhood of x = 0.
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