On the analog telephone and the mobile telephone system, the bandwidth of the speech is limited from 300 Hz to 3.4 kHz. This paper proposes a new recovery method of wideband speech (50 Hz--7.4 kHz) from narrow-band speech based on piecewise linear mapping. In this method, first, the narrow-band spectrum envelope of input speech is transformed to a wideband spectrum envelope using linearly transformed matrices which are associated with several spectral spaces. These matrices were estimated by speech training data, so as to minimize the mean-square error between the transformed and the original spectra. Second, the residual wave with a wideband spectrum is generated from the residual wave of input speech by the nonlinear computation. Finally, the reconstructed wideband speech is synthesized from the transformed spectrum envelope and the generated residual wave. This algorithm is compared to the following other methods: (1) the codebook mapping [Y. Yoshida and M. Abe, Proc. of ICSLP94, 1591--1594 (1994)]; and (2) the neural network. Through the evaluation by the spectral distance measure, it was found that the proposed method achieved a 0.67-dB maximum lower spectral distortion than the other methods. Perceptual experiments indicate a good performance for the reconstructed wideband speech.