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In this paper, the nonlinear approximation by piecewise constants is investigated, and a partition method for balancing the total variation over the intervals is proposed. Based on this method, the result of Kahane is proved, and the application in signal restoration is given. Finally, the numerical experiments are conducted to assess the proposed method in comparison with different wavelet thresholding methods. One of the experiments is to denoise the real-life precipitation data in Northeast China. The results show that our method performs well in terms of signal to noise ratio (SNR) over many standard wavelet based methods.