| The researches presented in this thesis belong to the field of numerical approximation. We mainly focus on the following topics:Chebyshev approximation by multivariate spline functions, the Haar and Walsh functions over triangular domains, numerical integration over the n-dimensional simplex and sparse solutions of elliptic differential equations. The contents are summarized as follows:In Chapter 1, we present the background of the researches in this thesis. Some prelim-inaries are also given.In Chapter 2, we study Chebyshev approximation by multivariate spline functions, in-cluding the uniqueness of the best approximation spline and the geometry characterization of the critical point set of S10(Δ) over some triangulations. Generally, the best approx-imation in several variables isn’t unique. Since the function values of any two distinct best approximations are the same, we get a sufficient condition for the uniqueness of the best approximation. Based on the concept of the properly posed knot set, the geometry characterization of the critical point set of S10(Δ) over some triangulations is given.In Chapter 3, the Haar and Walsh functions over triangular domains are constructed. The uniform convergence of the Haar-Fourier series is proved. Moreover, based on the relation between the Walsh and Haar functions, the convergence of the Walsh-Fourier series is studied. Additionally, the relation between the Walsh functions in the two different orders, i.e. Paley order and Hadamard order, is pointed out.In Chapter 4, a fourth degree integration formula is given for the n-dimensional simplex Tn for all n≥2, which is invariant under the group G of all affine transformations of Tn onto itself. The formula contains (n+2)(n+3)/2 nodes.In Chapter 5, based on the theory of compressed sensing, a rapid and exact algorithm for sparse solutions of elliptic differential equations is proposed. The algorithm transforms the problem of elliptic differential equations to the problem of a linear programming. |
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