Computing the Inversion of the Finite Hilbert Transform with Spline Wavelets

The inversion of the finite Hilbert transform has a wide range of applications in different fields such as medical imaging and signal processing. However, there remains a notable scarcity of effective algorithms for implementing this inversion. Until recently, an interesting fast algorithm for computing the inversion of the finite Hilbert transform by B-splines was developed by B. Yu et al. in 2024. Since B-spline wavelets have many good properties as those of the B-spline itself, for example, the smoothness, the decay, and the vanishing moments, a natural question is: Is it possible to develop a fast algorithm to calculate the inversion of the finite Hilbert transform based on the fast spline wavelet transform? This paper gives a positive answer to this question. Some examples are given in numerical experiments listing L2-norm error and L∞-norm error, and these examples show that the method proposed in this paper has faster computation speed and higher computational accuracy compared to previous results, especially in the L∞-norm error.