This study deduces a general inversion of continuous wavelet transform (CWT) with timescale being real rather than positive. In conventional CWT inversion, wavelet’s dual is assumed to be a reconstruction wavelet or a localized function. This study finds that wavelet’s dual can be a harmonic which is not local. This finding leads to new CWT inversion formulas. It also justifies the concept of normal wavelet transform which is useful in time-frequency analysis and time-frequency filtering. This study also proves a law for CWT inversion: either wavelet or its dual must integrate to zero.
Su, X.Q., Liu, L.T., Hsu, H. and, Wang, G.C. (2014) Long-Term Polar Motion Prediction Using Normal Time-Frequency Transform. Journal of Geodesy, 88, 145-155. http://dx.doi.org/10.1007/s00190-013-0675-7