On infinitely smooth compactly supported almost-wavelets (Q1905276)

The authors construct the system \[ \Psi= \{\varphi_{0, k}, \psi_{j, k},\;j= 0, 1,\dots;\;k\in \mathbb{Z}\} \] of infinitely smooth functions with the properties (1) \quad \(\Psi\text{ is an orthonormal basis in } L_2(\mathbb{R}^1);\) (2) \quad \(\varphi_{0k}(t)= \varphi_{00}(t- k),\quad \psi_{j, k}(t)= \psi_{j, 0}(t- k2^{- j});\) (3) \quad \(\text{supp } \varphi_{00}= [- 3,0],\quad \text{supp } \psi_{j, 0}= [-(j+ 3) 2^{- j}, j2^{- j}].\) Unlike wavelets, the system \(\Psi\) is not generated by contractions and translations of one function. However, the length of the support of the functions \(\psi_{j, k}\) tends to 0 as \(j\to \infty\), and for each fixed \(j\) the functions \(\psi_{j, k}\), \(k\neq 0\), are obtained by translations of \(\psi_{j, 0}\).