(pp. 858-860)

For ease of notation, we will limit the discussion to the one-dimensional case; however, the statements and proofs extend immediately to the general case.

Let$\psi (x)$be a Lipschitz function of the real variable$x$that also satisfies the following conditions:

$|\,\psi (x)\,|\; \leqslant \;C{(1\; + |\,x\,|)^{ - 3}},{\text{ |}}\,\psi '{\text{(}}x)\,|\; \leqslant \;C{(1\; + |\,x\,|)^{ - 3}}$; (1)

the set${\psi _{(j,k}}) = {2^{j/2}}\psi ({2^j}x - k),\;j,\;k \in \mathbb{Z}$, is an orthonormal basis for${L^2}(\mathbb{R})$. (2)

If

$\int_{ - \infty }^\infty {|\,f(x)\,|\,{{(1\; + \;|\,x\,|)}^{ - 3}}dx\; < \;\infty } $,

then the wavelet coefficients

$c(j,\;k) = \int_{ - \infty }^\infty {f(x)} {{\bar \psi }_{(j,k)}}(x)\;dx$

make sense. If, in addition,$f(x)$satisfies$\,f(x)\; - \;f({x_0})\,|\; \leqslant \;C|\,x\; - \;{x_0}\,{|^\alpha }$for some exponent$\[\alpha \in (0,1]\]$, then this implies immediately that the condition

$|\,c(j,\,k)\,|\; \leqslant \;C'{2^{ - j(\alpha + 1/2)}}(1\; + |\,{2^j}{x_0} - \;k\,{|^\alpha })$(3)

is satisfied for all$j\; \in \;\mathbb{R}$and all$k\; \in \;\mathbb{Z}$.

We propose to examine, conversely, if (3) implies...