On perturbations of the Haar system (Q1889626)

This paper discusses the basis properties in \(L_p[0,1]\), \(1 \leq p < \infty\), of sequences of the form \[ \{e,\phi_{k,j};\;k=0, 1\dots,\;j= 0,\dots, 2^k -1\} \tag{1} \] which are close, in a certain sense, to the classical system of Haar functions \[ \{e, \chi_{k,j};\;k=0, 1, \dots,\;j= 0, \dots, 2^k -1\}, \] where \(e = \chi_{[0,1)}\), \(h_{k,j}(t) = 2^{k/2} h(2^kt-j)\), and \(h = \chi_{[0,1)} -\chi_{[1/2,1)}\). The paper contains four theorems. The principal result is the following Theorem. {Suppose that \(1\leq p<\infty\) and \(1/p + 1/q =1\). If the sequence (1) is \(\omega\)-linearly independent in \(L_p[0,1]\) and \[ \sum_{k=0}^{\infty} 2^{k(1/2-1/q)} \left ( \sum_{j=0}^{2^k-1} | | h_{j,k}-\phi_{j,k} | | _p^q \right )^{1/q} < \infty, \] then it is a basis of \(L_p[0,1]\)}. It is also shown that if (1) is not necessarily \(\omega\)-linearly independent then it is still a basis of \(L_p[0,1]\), provided that the left-hand side of the preceding displayed equation is strictly less than \(1\). As an application, a sufficient condition for the density of a nonstationary wavelet is also given.