Mathematical Sciences: Wavelets, Mulivariate Splines, and Radial Functions

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The investigators undertake research in three areas of approximation theory: wavelet analysis, multivariate splines and wavelets, and radial basis functions. In the study of wavelets, beyond the mathematical analysis a major focus is on applications to real-time (or on-line) time-frequency analysis. Hence the investigators study the analysis and algorithmic and computational aspects of spline-wavelets and non-orthogonal wavelet packets. In particular, they will design a predictor-corrector algorithm along the line of the Kalman filter to implement the spline-wavelet decomposition without truncation and to take care of noise processes such as white noise. The research on multivariate splines also extends to the construction of multivariate wavelets using box splines. One of the most difficult problems is the study of minimum supports that give rise to reconstruction sequences with minimum length. Again a predictor-corrector algorithm will be implemented to avoid truncation of the decomposition sequences. "Radial basis functions" is a tool that allows fiting both regular and scattered data by continuously differentiable functions. Certain matrices associated with interpolation and approximation processes arise in connection with the data fitting. The investigators will study the "stability" of these matrices. They also consider various aspects of the question of how rapidly the approximating surfaces approach the underlying surface from which the scattered data are sampled as the number of data points increases. The wavelet transform is a tool that cuts up data or functions as well as their spectra into different frequency components and then studies each component with a resolution matched to the appropriate scale. For example, given a signal, one is interested in "cutting" up its frequency content with local time-information. This is similar to music notation, which tells the player which notes to play at any given moment. Wavelets, wavelet transforms, and wavelet packets help in analyzing signals and data. Wavelet analysis is useful in data reduction, which already has been applied to high-definition T.V. Using wavelets, the investigators aim at decomposing and reconstructing signals as simply and accurately as possible. Another aim is to represent the underlying processes generating scattered data by means of "simple" radial functions. This allows one to analyze and extrapolate data, which is important for artificial learning devices.