Continuous and localized Riesz bases for \(L^2\) spaces defined by Muckenhoupt weights (Q2348426)

The authors construct absolutely continuous Riesz bases in \(L^2(\mathbb R, wdx)\) for \(A_\infty\)-weights using a regularisation procedure of an unbalanced Haar system in \(L^2(\mathbb R, wdx)\). These bases can be chosen to have supports as close to the dyadic intervals and with Riesz bounds as close to \(1\) as desired. Starting with \(h_I^w(x)=\frac{1}{\sqrt{w(I)}}(\sqrt{\frac{w(I_r)}{w(I_l)}}\chi_{I_l}- \sqrt{\frac{w(I_l)}{w(I_r)}}\chi_{I_r})\) where \(I\) stands for a dyadic interval and \(I_l\) and \(I_r\) are the left and right half of it, they first use a change of variables defined by \(W^{-1}\) where \(W(x)=\int_0^x\), then regularise by convolution with a smooth compactly supported function and finally reverse the change of variables. This procedure allows them to generate a Riesz basis with the desired properties.