Reproducing kernel of image space of Haar wavelet transform

In this paper, we first show that the reproducing kernel Hilbert space H[0, +∞) is the solution space of the wave equation. Second, using the reproducing kernel of the Hilbert space H[0, +∞) we obtain the concrete expressions of the reproducing kernel function of the image space of Haar wavelet transform. Based on the reproducing kernel function of the image space of Haar wavelet transform, we give the sampling theorem. The sampling formula makes the numerical computation easier than before, and it also provides theoretical base for us to further study the image space of the general wavelet transform.