About the book
In the last years, different versions of wavelet transforms (continuous, discrete, real, and complex) have revolutionized the managing of spectral data and computing data related to the multi-resolution analysis, graph representation in trees to sub-representation of spectral bands in wavelets decomposition to an image, time-series data to localization of time meaning signals, spectrometry and spectrograms data, etc. The applications are diverse and are focused on the complementation of the Fourier transforms data in its spectral image as well as slight benefits over Fourier transforms in reducing computations when examining specific frequencies. Despite its major advantages, we can interpret a two-dimensional spectral image related to Fourier analysis and dynamical systems in search for signals of a known, non-sinusoidal shape.
Such is the case, for example, with wavelets that have an important role in standard STFT/Morlet analyses. From a purely mathematical point, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by certain orthonormal series generated by wavelets. An important fact to consider with these representations and their consistencies is the Riesz theorem and aspects of functional analysis related to the convergence of the different wavelet series that can be constructed and generated to start an appropriate wavelet. Also, related problems with wavelets and other functional transforms include, for example, the Mexican hat wavelet, Haar Wavelet, Daubechies wavelet, triangular wavelet, and more.