Homogeneous approximation property for continuous wavelet transforms (Q999280)

The authors considers the continuous wavelet transform \[ \langle f,\tau(a,b)\psi\rangle = |a|^{-1/2} \int_{-\infty}^\infty f(x) \overline{\psi(a^{-1}(x-b))}\,dx. \] Under certain assumptions one can recover the function \(f\) by an inverse tranform and the question considered in this paper is to what extent this is possible to achieve in a uniform wayfor the time-scale shifts \(\tau(s,t)f(x)= |s|^{-1/2} f(s^{-1}(x-t))\), i.e., the homogeneous approximation property holds. It is shown that this is always the case in the \(L^2\)-sense but not for pointwise convergence and the authors give necessary and sufficient conditions, which depend both on the wavelet and the function \(f\), for the pointwise homogeneous approximation property to hold.