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.

中文翻译：

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Haar小波系统中傅立叶级数收敛与绝对收敛的保变量变化

已经确定，在变量的所有连续可微同胚变化中，Haar 小波系统中傅立叶级数的绝对收敛性仅由 $$\varphi^{-1}(0)$$ 是二元有理的那些保持和 $$\varphi'(x)=\pm 2^m$$ ，其中 $$m$$ 是一个整数， $$x\in\mathbb R$$ 。还确定了这个条件对于变量的连续可微同胚变化是必要的，以保持 Haar 小波系统中傅立叶级数的收敛性。