Truncated Hilbert Transform: Uniqueness and a Chebyshev series Expansion Approach
This work addresses a mathematical inverse problem in signal processing or imaging, but it appears incremental as it builds on existing theory with specific numerical improvements.
The paper tackles the problem of reconstructing a function from its truncated Hilbert transform by establishing a stronger uniqueness condition using Sokhotski-Plemelj formulas and proposing a Chebyshev series expansion with numerical methods for coefficient estimation, showing through simulations that the extrapolative procedure works well numerically.
We derive a stronger uniqueness result if a function with compact support and its truncated Hilbert transform are known on the same interval by using the Sokhotski-Plemelj formulas. To find a function from its truncated Hilbert transform, we express them in the Chebyshev polynomial series and then suggest two methods to numerically estimate the coefficients. We present computer simulation results to show that the extrapolative procedure numerically works well.