Abstract:
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.
Keywords:Haar wavelets, Fourier–Haar series, continuously differentiable homeomorphism, changes of variable.
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