Brief Historical Introduction -- Hilbert Spaces and Orthonormal Systems -- Fourier Transformations and Their Applictions -- The Gabor Transform and Time-Frequency Signal Analysis -- The Wigner-Ville Distribution and Time-Frequency Signal Analysis -- The Wavelet Transforms and Their Basic Properties -- Multiresolution Analysis and Construction of Wavelets -- Extensions of Multiresolution Analysis -- Newland's Harmonic Wavelets -- Wavelet Transform Analysis of Turbulence.