Haar wavelets on the Lebesgue spaces of local fields of positive characteristic (Q2879354)

The author studies Haar wavelet systems on the space \(L^p(K)\) where \(1<p <\infty\) and \(K\) is a local field, that is, a field which is locally compact, nondiscrete and totally disconnected. The scaling function associated with the Haar wavelets is the characteristic function of the set \(\mathfrak {D}=\{x\in K : |x|\leq 1\}\) where the valuation \(|\alpha|\) is defined with the aid of the Haar measures \(d(\alpha x)= |\alpha|d x\). It is shown that the Haar wavelet system defined using coset representatives in \(K^+\) of \(\mathfrak{D}\) and a prime element \(\mathfrak p\) of \(K\) is an unconditional basis in \(L^p(K)\) and when properly normalized a democratic and a greedy basis as well.