On the Hilbert transform of wavelets

Details On the Hilbert transform of wavelets On the Hilbert transform of wavelets

2011-07-22

arxiv.org/abs/1107.4619v1

Mathematics

Functional Analysis

Appears in IEEE Transactions on Signal Processing, vol. 59, no. 4, pp. 1890-1894, 2011

Scientific paper

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.

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