On Changes of Variable Preserving the Convergence and Absolute Convergence of Fourier Series in the Haar Wavelet System

It is established that, among all continuously differentiable homeomorphic changes of variable, the absolute convergence of Fourier series in the Haar wavelet system is preserved by only those for which $\varphi^{-1}(0)$ is binary-rational and $\varphi'(x)=\pm 2^m$, where $m$ is an integer and $x\in\mathbb R$. It is also established that this condition is necessary for a continuously differentiable homeomorphic change of variable to preserve the convergence of Fourier series in the Haar wavelet system.