On the Hilbert transform of wavelets

TL;DR

This paper proves that the Hilbert transform of a wavelet retains key properties such as smoothness and vanishing moments, with localization influenced by the original wavelet's vanishing moments, extending understanding in wavelet analysis.

Contribution

It provides rigorous proofs and sharp estimates showing the Hilbert transform of a wavelet preserves smoothness and vanishing moments, even for non-compactly supported wavelets.

Findings

Hilbert transform of a wavelet is again a wavelet with similar smoothness.

Localization of the transformed wavelet depends on the original wavelet's vanishing moments.

Results apply to non-compactly supported wavelets with minimal smoothness and decay.

Abstract

A wavelet is a localized function having a prescribed number of vanishing moments. In this correspondence, we provide precise arguments as to why the Hilbert transform of a wavelet is again a wavelet. In particular, we provide sharp estimates of the localization, vanishing moments, and smoothness of the transformed wavelet. We work in the general setting of non-compactly supported wavelets. Our main result is that, in the presence of some minimal smoothness and decay, the Hilbert transform of a wavelet is again as smooth and oscillating as the original wavelet, whereas its localization is controlled by the number of vanishing moments of the original wavelet. We motivate our results using concrete examples.