Representation of measurable functions by absolutely convergent series of translates and dilates of one function (Q2785847)

Let \(\delta\) be a positive function on \((0,\infty)\) which satisfies \(\delta(0+)= 0\). A system of functions \(\{\sigma_k\}\) on \(\mathbb{R}\) is call a \(\delta\)-\textit{representation} if given any almost everywhere finite, measurable function \(f\) on \(\mathbb{R}\) there are coefficients (not necessarily unique) \(\{a_k\}\) such that \(\sum a_k\sigma_k\) converges almost everywhere on \(\mathbb{R}\) to \(f\) and \(\sum\delta(|a_kf_k|)\) is almost everywhere finite. When \(\delta(t)=t\), a \(\delta\)-representation is called an \textit{absolute representation}.NEWLINENEWLINENEWLINEThe author identifies conditions under which the systems \(\psi_{jk}:= \sqrt{2^j}\psi(2^jx-k)\), \(j,k\in\mathbb{Z}\), and \(\varphi_{jk}:= \varphi(2^jx-k)\), \(0\leq k<2^j\), \(j\in\mathbb{N}\), are representations.NEWLINENEWLINENEWLINEFor the first type of system, if \(\psi\) is \(C^1\) and of mean zero on \(\mathbb{R}\), if both \(\psi(x)\) and \(\psi'(x)\) are dominated by the reciprocal of the cubic \((1+|x|)^3\), and if \(\psi_{jk}\) is an orthonormal basis in \(L^2(\mathbb{R})\) (this includes regular wavelets), then \(\psi_{jk}\) is an absolute representation.NEWLINENEWLINENEWLINEFor the second type of system, if \(\varphi\) is integrable, supported in \([0,1]\), and \textit{not} of mean zero on \(\mathbb{R}\), then \(\varphi_{jk}\) is an absolute representation. If, in addition, \(\varphi\) is piecewise constant on \([0,1]\), then \(\varphi_{jk}\) is a \(\delta\)-representation for all positive functions \(\delta\) which satisfy \(\delta(0+)=0\).