Multiresolution analysis on local fields and characterization of scaling functions (Q2882907)

Let \(K\) be a local field of positive characteristic and \(K=\bigcup_{j\in{\mathbb Z}} {\mathfrak p}^{-j}{\mathfrak D}\), where \(\mathfrak p\) is a fixed element of maximum absolute value in the prime ideal in \(K\) and \({\mathfrak D}\) is the ring of integers in \(K\). Suppose that \(\{u(n)\mid n\in \{0\}\cup{\mathbb N}\}\) is a complete list of discrete coset representatives of \({\mathfrak D}\) in \(K\). We can introduce analogous notions of translation and dilation as NEWLINE\[NEWLINE q^{j/2}f({\mathfrak p}^{-j}x-u(k)), NEWLINE\]NEWLINE where \(q=|{\mathfrak p}^{-1}|\). Then, similarly as in the Euclidean case, we can develop the theory of wavelets, especially, multiresolution analysis (\text{MRA}) on \(K\), that is a sequence \(\{V_j\mid j\in{\mathbb Z}\}\) of closed subspaces of \(L^2(K)\) satisfying certain conditions. In this paper the authors find some crucial properties of \text{MRA} on \(K\): In the definition of \text{MRA} on \(K\) it is enough that the translations of the scaling function form a Riesz basis of \(V_0\) instead of an orthogonal basis and the triviality condition \(\bigcap_{j\in{\mathbb Z}}V_j=\{0\}\) follows from other conditions. Finally, they characterize the scaling functions by their Fourier transforms.