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DISCRETIZATION AND APPROXIMATION ON HIGH DIMENSIONAL DOMAINS
- Author / Creator
- Niu, Yeli
This thesis includes 2 parts. Part 1 is: Discretization on High Dimensional Compact Domains. Part 2 is: Polynomial Approximation on High Dimensional Spheres. A special example for part 1 is the result on the unit sphere of high-dimensional Euclidean spaces. In chapter 2, we obtained general results about discretization of integration on compact metric domains. Related results on spheres, closed balls and simplexed could be as special examples of our results. The first main result is about regular partitions on compact path-connected metric space equipped with non-atomic Borel probability measure. An example of this result is the regular partition on the unit sphere, which improves the previous results by the uniform absolute constant in the diameter of each partition. Many similar results were obtained with constants depending on the dimension of the Euclidean space (they are an exponential form of the dimension). It is a pity that our method here is not constructable. Resting main results
are about numerical integration. Numerical integration plays an important role in approximation theory. To integrate a given function, we sometimes do not know its original function,
sometimes it is too complicated to find its original function. Thus in many applied problems, we need to use numerical integration (discrete weighted summation) to asymptotically express it. A main topic is finding fixed nodes and weights to approximately express integration for a class of function, modifying weights and notes to improve the uniform approximation error for the class of functions. We here used methods from  to prove the existence of nodes and
weights for numerical integration which result in a better approximation error. One result is about discretization of integration on compact matric spaces that equipped with certain measures. Results here is better than previous results under the following points. First, we reduced the smoothness requirements. Functions here do not need to be differentiable, satisfying Lipschitz condition is enough. Second, example of our result about discretization of integration for piecewise polynomials on the unit sphere gives a better approximation error, somewhat overcome the curse of dimensionality. The last main result is about discretization of integration on
finite-dimensional compact domains.
Chapter 3 mainly discusses the Jackson type’s inequality and its matching inverse inequality, equivalence of K-functional and modulus of smoothness on the unit sphere Sd−1. There
are many definitions for K-functional and modulus of smoothness. Here we use the modulus of smoothness defined by Z. Ditzian via rotation operator on the unit sphere. K-functional here we defined through partial derivative in Euler angles. In 1964, D. J. Newman, and H. S. Shapiro  proved that for f ∈ C(Sd−1), the constant appear in Jackson type’s inequality for r = 1, p = ∞ can be a dimension-free constant. Results in this chapter show that this result can be extended to all cases of positive integer r and p ≥ 1. We also obtain the matching inverse Jackson’s inequality, equivalence of K−functional and modulus of smoothness with constant
in equivalence independent of dimension d. In this sense, we improved Ditzian’s results on Jackon’s inequality. Jackson type’s inequality and its matching inverse inequality connect the rate of polynomial approximation to the smoothness properties of functions on the sphere. Equivalence of K-functional and modulus of smoothness builds the relation between difference with differentiation.
I will describe in detail my current research projects in these two directions and the progress I’ve made.
- Graduation date
- Fall 2021
- Type of Item
- Doctor of Philosophy
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