Wavelet Analysis

The following sections are included:

• Wavelet and Wavelet Analysis. Preliminary Notion

  • The space L²(ℝ)

  • The spaces L^p(ℝ)(p ≥ 1)

  • The Hardy spaces H^p(ℝ)(p ≥ 1)

  • The sketch scheme of wavelet analysis

• Rademacher, Walsh and Haar Functions

  • System of Rademacher functions

  • System of Walsh functions

  • System of Haar functions

• Integral Fourier Transform. Heisenberg Uncertainty Principle

• Window Transform. Resolution

  • Examples of window functions

  • Properties of the window Fourier transform

  • Discretization and discrete window Fourier transform

• Bases. Orthogonal Bases. Biorthogonal Bases

• Frames. Conditional and Unconditional Bases

  • Wojtaszczyk's definition of unconditional basis (1997)

  • Meyer's definition of unconditional basis (1997)

  • Donoho's definition of unconditional basis (1993)

  • Definition of conditional basis

• Multiresolution Analysis

• Decomposition of the Space L²(ℝ)

• Discrete Wavelet Transform. Analysis and Synthesis

  • Analysis: transition from the fine scale to the coarse scale

  • Synthesis: transition from the coarse scale to the fine scale

• Wavelet Families

  • Haar wavelet

  • Strömberg wavelet

  • Gabor wavelet

  • Daubechies-Jaffard-Journé wavelet

  • Gabor-Malvar wavelet

  • Daubechies wavelet

  • Grossmann-Morlet wavelet

  • Mexican hat wavelet

  • Coifman wavelet – coiflet

  • Malvar-Meyer-Coifman wavelet

  • Shannon wavelet or sinc-wavelet

  • Cohen-Daubechies-Feauveau wavelet

  • Geronimo-Hardin-Massopust wavelet

  • Battle-Lemarié wavelet

• Integral Wavelet Transform

  • Definition of the wavelet transform

  • Fourier transform of the wavelet

  • The property of resolution

  • Complex-value wavelets and their properties

  • The main properties of wavelet transform

  • Discretization of the wavelet transform

  • Orthogonal wavelets

  • Dyadic wavelets and dyadic wavelet transform

  • Equation of the function (signal) energy balance