Let \(S\) be the Strömberg wavelet [J.-O. Strömberg, Harmonic analysis, Conf. in Honor of A. Zygmunt, Chicago 1981, Vol. 2, 475-494 (1983; Zbl 0521.46011)], then \((S_{r,m})_{r,m\in \mathbb Z}\) is an orthonormal basis in \(L^2(\mathbb R),\) where \(S_{r,m}(t)=2^{r/2}S(2^rt-m)\), \(r,m\in\mathbb Z.\) Furthermore, let \(P_{r,m}\) denotes the extension onto \(L^{\infty}(\mathbb R)\) of the partial sum operators for the system \((S_{r,m})_{r,m\in \mathbb Z},\) i.e., \[ P_{s,n}f=\sum \langle f,S_{r,m}\rangle S_{r,m}, \] where \(\langle \cdot,\cdot\rangle\) stands for the scalar product in the Hilbert space \(L^2(\mathbb R)\) and the sum extends over all pairs \((r,m)\) such that \((r,m)\leq(s,n)\) if \(n<\infty\) and \((r,m)<(s,\infty)\) if \(n=\infty.\) Here the notation \((r,m)<(s,n)\) means that either \(r<s\) or \(r=s\) and \(m<n.\)
The main result is the following expression for the Lebesgue constant \(\|P_{r,m}\|_{\infty}:\) \[ \|P_{r,m}\|_{\infty}=2+(2-\sqrt{3})^2 \text{ if } r,m\in\mathbb Z, \]
\[ \|P_{r,\infty}\|_{\infty}=2 \text{ if } r\in\mathbb Z. \]