Periodic wavelets on the \(p\)-adic Vilenkin group
Let \((G,\oplus)\) be the locally compact \(p\)-adic Vilenkin group, \(U_0\) the unit ball, \(A\) the dilation operator and \(U_n=A^{-n}U_0\), \((w_l)_{l=0}^\infty\) Vilenkin functions. Using the modified Vilenkin-Dirichlet kernel \[ D_n^*(x):=\alpha+\sum_{k=1}^{N-l}w_k(x)+\alpha w_{N-1}(x), \] the author introduces the functions \[ \begin{multlined}\varphi_{n,k}(x)=\Phi_n(x\ominus x_{n,k}), \psi_{n,k}^{(j)}(x):=\\ \sum_{s=0}^{p-1}\varepsilon_p^{js}\varphi_{n+1,pk+s}(x), 0\leq k\leq N-1, 1\leq j\leq p-1, N=p^n,\end{multlined} \] where \(\Phi_n(x):=N^{-1}D_N^\alpha(x)\) and shows that \(\{\varphi_{n,k}\}_{k=0}^{N-1}\) and \(\{\psi_{n,k}^{(l)}\}_{k=0}^{N-1}\) are bases for the spaces \(V_n:=\text{span}\{1,w_1(x),\dots,w_{N-1}(x)\}\), \(W_n^{(j)}:=\text{span}\{w_{jN}(x),w_{jN+1}(x),\dots,w_{(j+1)N-1}(x)\}\) respectively. The corresponding fast algorithms for the decomposition and reconstruction are discussed.
scientific article; zbMATH DE number 6105086
scientific article