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Description: |
Wavelet theory is applicable to several subjects and fields of science. All wavelet transforms may be considered forms of
time-frequency representation for continuous-time (analog) signals and so are related to harmonic analysis.
A wavelet is a kind of mathematical function used to divide a given function or continuous-time signal into different frequency components and study each component with a resolution that matches
its scale. A wavelet transform is the representation of a function by wavelets. The wavelets are scaled and translated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating
waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional Fourier transforms for representing functions that have discontinuities and sharp peaks, and for
accurately decomposing and reconstructing finite, non-periodic and/or non-stationary signals. In formal terms, this representation is a wavelet series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set of frame of a vector space (also known as a Riesz basis), for the Hilbert space of square integrable functions.
In this talk we will present the mathematical foundations of wavelets analysis on the real line, on the plane and on the unit sphere. The first part is of introductory nature and we will give an
overview of the main mathematical properties of the continuous wavelet transform on the real line and on the plane. Afterwards we will present a more abstract context for the construction of
continuous wavelet transforms based on square integrable representations of a group $G$ on a Hilbert space $\mathcal{H}.$
This allow us to construct continuous wavelet transforms on the unit sphere based on the proper Lorentz group Spin$^+(1,n),$ which is the conformal group on the unit sphere $S^{n-1},$ embedded in Euclidean
space $\mathbb{R}^n.$
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