Document Type

Dissertation

Degree

Doctor of Philosophy

Major

Applied Mathematics

Date of Defense

5-5-2020

Graduate Advisor

Wenjie He

Committee

Qingtang Jiang

Haiyan Cai

Yuefeng Wu

Abstract

The data interpolating problem is a fundamental problem in data analysis, and B-splines are frequently used as the basis functions for data interpolation. In the real-world applications, the real-time processing is very important. To achieve that, we cannot use any matrix inversion for large amount of data, and we also need to avoid using any global operator. To solve this problem, we develop a new method based on a local quasi-interpolation operator. To construct the local quasi-interpolation operator, we need to factorize the Shoenberg-Whitney matri- ces for the given data samples. Furthermore, our local quasi-interpolation operator should correspond to a band matrix with the minimum bandwidth, which is criti- cal for the real-time data processing. Finally, we bridge the gap between our local quasi-interpolation operator and a local spline interpolation operator through an impulse interpolation operator using a “blending” method.

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