An asymptotic formula for some wavelet series (Q1356643)

This paper describes the pointwise behavior of a wavelet series in the neighborhood of a point of divergence. The main result is as follows. Consider the wavelet series \(F(x)=\sum_{j=1}^\infty c_j\psi(2^{n_j}x-k_j)\). The \(n_j\) form a sequence of integers which is relatively dense in \(\mathbb{N}\); the \(c_j\geq 0\) tend to zero but not fast enough such that \(\sum c_j=\infty\). The wavelet \(\psi\) has a bounded derivative and there exist \(C\) and \(N\) such that \(\psi(x)\leq C/(1+|x|^N)\). Moreover, for \(x=x_0\) there holds \(2^{n_j}x-k_j\to\theta^*\) as \(j\to \infty\). Then it is shown that for a specific integer \(r\), depending on \(\delta\), \(\lim_{\delta\to0} F(x_0+\delta)/(\sum_{j=1}^r c_j)=\psi(\theta^*)\).