Nonhomogeneous Dual Wavelet Frames with the $${p}$$-Refinable Structure in $${L}^{\mathbf{2}}{({\mathbb{R}}^{+})}$$

In recent years, nonhomogeneous wavelet frames have been widely studied by many researchers, while the ones in $$L^{2}(\mathbb{R}^{+})$$ have not. Some practical applications indicate that it is desirable to have a nonhomogeneous dual wavelet frame in $$L^{2}(\mathbb{R}^{+})$$ , because the time variable can not take negative values in signal sampling. In addition, similar to the homogeneous dual wavelet frames, the nonhomogeneous ones derived from refinable functions have fast wavelet algorithms. In view of this, under the setting of $$L^{2}(\mathbb{R}^{+})$$ , we study the properties of nonhomogeneous dual wavelet frames and obtain a construction of nonhomogeneous dual wavelet frames from a pair of $$p$$ -refinable functions.