Spectral expansions of laplace-beltrami operator on unit sphere - Rasedee, Ahmad Fadly Nurullah (2015)
The reconstruction of functions from its expansions is a prominent problem in harmonic analysis. These type of problems are not always solvable with the definition of the sum of a Fourier series as the limit of its partial sums. Functions that are not very smooth (such functions are the most interesting and from a pratical point of view have important expansions), have successive strongly oscillating terms of the partial sums which does not correspond with the characterization of the to be reconstructed function. This causes the sequence of partial sums to oscillate around the function rather than approach it. When attempting to overcome this issue, it is interesting to consider some arithmetic means. For purpose of this research, the Riesz means is taken into consideration. Due to the consistent behavior of these oscillations, the Riesz means acts as a regularization of the partial sums to better approximate the function. In the present research we investigated convergence and summability problems of the spectral expansions of differential operators. The most specific properties of the decompositions are established. The investigations of spectral expansions of the differential operator in modern methods of harmonic analysis, incorporates the wide use of methods from functional analysis, modern operator theory and spectral decomposition. New methods for approximating the functions from different spaces (Nikolskii,Sobolev, Liouville) are constructed using asymptotic behavior of the spectral function of the differential operators. We consider the summation, conditions and principles for the localization of Riesz means of the Fourier-Laplace series of distributions. When working in the field of spectral theory of differential operators, specifically on localization of spectral decompositions by Riesz means, it is important to consider how the dependence on convergence of spectral decompositions of their Riesz means affects the behavior of the function in a small neighborhood of the given point. This research focuses on convergence and summability problems of the spectral expansions of differential operators related to the Fourier-Laplace series. The Fourier-Laplace series can be interpreted as eigenfunction expansions of the Laplace-Beltrami operator, which is a symmetric and nonnegative elliptic operator on the unit sphere. Application of Neuman’s Theorem allows us to present the Laplace-Beltrami through partial sums of the Fourier-Laplace series, which is referred as spherical expansions related to the Laplace-Beltrami operator.