Discrete Fourier Analysis and Wavelets

Harmonic Shape Analysis: From Fourier to Wavelets

Harmonic analysis studies the representation of functions as the linear combination of basic wave-like functions. It plays a fundamental role in the processing of time-series signals and images. Recent years have witnessed many efforts to adapt the Fourier and wavelet analysis to the domain of 3D shapes. The manifold Fourier analysis relies on the eigenfunctions of the Laplace-Beltrami operator...

Wavelets, Fourier Transform, and Fractals

Wavelets and Fourier analysis in digital signal processing are comparatively discussed. In data processing, the fundamental idea behind wavelets is to analyze according to scale, with the advantage over Fourier methods in terms of optimally processing signals that contain discontinuities and sharp spikes. Apart from the two major characteristics of wavelet analysis multiresolution and adaptivit...

From Fourier Transform to Wavelets

Quantitative Fourier Analysis of Approximation Techniques: Part II—Wavelets

In a previous paper, we proposed a general Fourier method that provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions. Here, we apply our results when these functions satisfy the usual two-scale relation encountered in dyadic multiresolution analysis. As a consequence of this additional constraint, the quantities introdu...

Topics in Fourier Analysis: DFT&FFT,Wavelets, Laplace Transform

In addition to their inestimable importance in mathematics and its applications, Fourier series also serve as the entry point into the wonderful world of Fourier analysis and its wide-ranging extensions and generalizations. An entire industry is devoted to further developing the theory and enlarging the scope of applications of Fourier–inspired methods. New directions in Fourier analysis contin...