Preface

Over the last few years, the wavelet transforms in their multiple forms—continuous, discrete, real, and complex—have revolutionized the way we manage spectral and computing data for multi-resolution analysis, graph representation in trees, sub-representation of spectral bands in wavelet decomposition to an image, time series data to localization of time-meaning signals, spectrometry, and spectrograms data, and more. It can be used in numerous ways, which is attributable to its ability to complement Fourier transforms data in its spectral image while also providing a few advantages over Fourier transforms in terms of reduced calculations when analyzing particular frequencies. While it has many applications in Fourier analysis and dynamical systems, its main strengths lie in the interpretation of spectra in two-dimensional spectral images—searching for signals of a known, nonsinusoidal shape. The wavelets play a significant role in the typical STFT/Morlet analysis. From a purely mathematical perspective, a wavelet series is a representation of a squareintegrable (real- or complex-valued) function by a certain orthonormal series created by wavelets. Notably, The Riesz theorem and other features of functional analysis about the convergence of various wavelet series that can be created after the generation of an appropriate wavelet are crucial to the consistency of these representations. However, this does not come without its fair share of challenges. We have observed a few problems with the wavelet transform, along with other functional transforms: the Mexican hat wavelet, Haar Wavelet, Daubechies wavelet, triangular wavelet, and many more.

> **Dr. Francisco Bulnes** Professor, IINAMEI Director, Head of Research Department in Mathematics and Engineering, Tescha, Mexico

Section 1

Introduction to Analyisis

and Signal Processing

**1**

Section 1
