Geodesic
On Changes of Variable Preserving the Convergence and Absolute Convergence of Fourier Series in the Haar Wavelet System
Matematičeskie zametki, Tome 108 (2020) no. 2, pp. 179-189

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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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     author = {K. Bitsadze},
     title = {On {Changes} of {Variable} {Preserving} the {Convergence} and {Absolute} {Convergence} of {Fourier} {Series} in the {Haar} {Wavelet} {System}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {179--189},
     publisher = {mathdoc},
     volume = {108},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2020_108_2_a2/}
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K. Bitsadze. On Changes of Variable Preserving the Convergence and Absolute Convergence of Fourier Series in the Haar Wavelet System. Matematičeskie zametki, Tome 108 (2020) no. 2, pp. 179-189. http://geodesic.mathdoc.fr/item/MZM_2020_108_2_a2/