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.