On infinitely smooth almost-wavelets with compact support (Q1332986)

It is known that there does not exist infinitely differentiable compactly supported wavelets. We propose a system \(\psi:= \{\varphi_{0k},\psi_{jk},j,k\in \mathbb{Z},j\geq 0\}\) of infinitely smooth functions with the following properties: 1) \(\psi\) is an orthonormal basis in \(L_ 2(\mathbb{R}^ 1)\). 2) \(\varphi_{0k}(t)= \varphi_{00}(t- k)\) and \(\psi_{jk}(t)= \psi_{j0}(t- k2^{-j})\). 3) \(\text{supp }\varphi_{00}= [-3,0]\) and \(\text{supp }\psi_{j0}= [-(j+ 3)2^{-j}, j2^{-j}]\).