Characterization of the stability range of the Hilbert transform with applications to spectral factorization

The Hilbert transform plays an important role in many different applications. Especially in the area of detection and estimation it is closely related to the calculation of the spectral factorization. Generally, it is not possible to calculate the Hilbert transform in closed form. Therefore approximation methods are applied. This paper studies the stability of a general class of approximation algorithms for the Hilbert transform which contains all traditional numerical integration methods. To this end, the paper introduces a scale of signal spaces with finite energy in which a factor (log n)^β measures the concentration of the signal energy in its Fourier coefficients c_n. It will be shown that if the energy concentration is too weak, i.e. if 0 ≤ β ≤ 1, then every approximation method diverges. Conversely, if the energy concentration is sufficiently good, i.e. if β > 1, convergent approximation methods do exist and we give a natural characterization of all convergent methods.