Truncated Hilbert Transform: Uniqueness and a Chebyshev series Expansion Approach

TL;DR

This paper establishes a stronger uniqueness theorem for functions with compact support based on their truncated Hilbert transform and introduces Chebyshev series methods for numerical reconstruction, supported by simulation results.

Contribution

It provides a new uniqueness result using Sokhotski-Plemelj formulas and develops two Chebyshev-based numerical methods for reconstructing functions from their truncated Hilbert transform.

Findings

Stronger uniqueness result for functions with compact support

Chebyshev series expansion effectively estimates the function from its truncated Hilbert transform

Numerical simulations demonstrate the method's practical effectiveness

Abstract

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.