On progressive functions
The authors investigate functions \(\phi:\mathbb R\to\mathbb R\) that are solutions of functional equations of the form NEWLINE\[NEWLINE \phi = C s + \sum_{k=1}^{N-1} \alpha_k \phi (\cdot - k), NEWLINE\]NEWLINE where \(s\in L^2(\mathbb R)\) with \(\mathrm{supp} s \subset [0, \infty)\), \(1 < N\in \mathbb N\), \(C\in \mathbb R\setminus\{0\}\), and \(\alpha_k\in \mathbb R\), \(k = 1, \dots, N-1\). A function \(g\in L^2(\mathbb R)\) is called \textit{progressive} iff the support of its Fourier-Plancherel transform \(\widehat{g}\) is \([0, \infty)\). The space of progressive functions is identical to the Hardy space \(H^2 (\mathbb R)\).NEWLINENEWLINEThe authors derive sufficient conditions and provide explicit formulas for the construction of progressive functions with prescribed exponential decay. Moreover, conditions for the positive integer translates of a progressive function to form an orthonormal or Riesz basis are derived. It is further shown that progressive functions may be employed to construct wavelets of arbitrary smoothness on \([0,\infty)\) provided that they are generated by a smooth compactly supported refinable function.
scientific article; zbMATH DE number 6363002
scientific article