Document Type

Dissertation

Degree

Doctor of Philosophy

Major

Applied Mathematics

Date of Defense

4-20-2021

Graduate Advisor

Wenjie He, Ph.D.

Committee

Qingtang Jiang, Ph.D.

Adrian Clingher, Ph.D.

Haiyan Cai, Ph.D.

Abstract

In solving the data interpolation problem, which is fundamental in data analysis, we typically deal with the data samples spread in a finite interval [a, b], which results in the operations involving finite-dimensional matrices. There are many interesting results developed under this framework. However, when the data samples are given from an infinite interval [a, ∞) (for certain special types of real-world applications), many existing results would not work anymore due to the special properties of the infinite data samples. A new framework should be established to support the infinite data samples.

In this dissertation, we develop a special tool called local linear quasi-interpolant for an infinite interval with the following properties: 1) Each linear functional of the quasi-interpolant is determined by at most three data samples, so that the spline coefficients can be calculated in real-time; 2) The quasi-interpolant preserves all the linear polynomials; 3) Our framework does not impose any restriction on the relationship between the sample locations and the spline knots, which provides us the necessary flexibility in the real-world applications.

Our construction is based on a matrix factorization method with respect to infinite-dimensional matrices. In order to ensure that the infinite version of the Shoenberg-Whitney matrices are invertible, we take the constructive approach that results in both the left-inverses and the right-inverses. Furthermore, since the associative law of the matrix multiplication does not work for the infinite matrices, we verify all the formulas derived from the infinite matrix operations. Finally, our local method allows us to calculate the spline interpolating coefficients in real-time on the fly for the infinite data samples.

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Additional Files

Joharadis-PhD-May-2021.pdf (443 kB)
Construct Linear Quasi-Interpolants on Infinite Intervals

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