Unconditional bases of wavelets in local fields (Q2204111)

Let \(K\) be a local field, which is a field with a topology such that both the additive and multiplicative groups of \(K\) are locally compact abelian groups. As stated in Definition~4.1, a finite set \(\{\psi^l \; : \; 1\le l\le L\}\subset L^2(K)\) is said to be a set of basic wavelets of \(L^2(K)\) if the system \(\{\psi^l_{j,k} \; : \; 1\le l\le L, j\in \mathbb{Z}, k\in \mathbb{N}_0\}\) forms an orthonormal basis for \(L^2(K)\). Under the decay condition and the Lipschitz condition on a set \(\{\psi^l \; : \; 1\le l\le L\}\) of basis wavelets, the system \(\{\psi^l_{j,k} \; : \; 1\le l\le L, j\in \mathbb{Z}, k\in \mathbb{N}_0\}\) is shown in Theorem 4.3 to be an unconditional basis in the Hardy space \(H^1(K)\) and in Theorem 4.6 to be an unconditional basis in \(L^p(K)\) for \(1<p<\infty\).