On infinitely smooth compactly supported almost-wavelets. (English. Russian original) Zbl 0843.42017
Math. Notes 56, No. 3, 877-883 (1994); translation from Mat. Zametki 56, No. 3, 3-12 (1994).
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) \(\Psi\text{ is an orthonormal basis in } L_2(\mathbb{R}^1);\)
(2) \(\varphi_{0k}(t)= \varphi_{00}(t- k),\quad \psi_{j, k}(t)= \psi_{j, 0}(t- k2^{- j});\)
(3) \(\text{supp } \varphi_{00}= [- 3,0],\quad \text{supp } \psi_{j, 0}= [-(j+ 3) 2^{- j}, j2^{- j}].\)
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 \(\psi_{j, 0}\).
(1) \(\Psi\text{ is an orthonormal basis in } L_2(\mathbb{R}^1);\)
(2) \(\varphi_{0k}(t)= \varphi_{00}(t- k),\quad \psi_{j, k}(t)= \psi_{j, 0}(t- k2^{- j});\)
(3) \(\text{supp } \varphi_{00}= [- 3,0],\quad \text{supp } \psi_{j, 0}= [-(j+ 3) 2^{- j}, j2^{- j}].\)
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 \(\psi_{j, 0}\).
Reviewer: B.Rubin (Jerusalem)
MSC:
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
Keywords:
waveletsReferences:
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