Theses and Dissertations

Issuing Body

Mississippi State University


Woody, Johnathan R.

Committee Member

Zhang, Haimeng

Committee Member

DuBien, Janice

Committee Member

Sepehrifar, Mohammad

Committee Member

Patil, Prakash N.

Other Advisors or Committee Members

Wu, Tung-Lung

Date of Degree


Document Type

Dissertation - Open Access



Degree Name

Doctor of Philosophy (Ph.D)


College of Arts and Sciences


Department of Mathematics and Statistics


In this dissertation, we utilize the discrete Fourier analysis on axially symmetric data generation and nonparametric estimation. We first represent the axially symmetric process as Fourier series on circles with the Fourier random coefficients expressed as circularlysymmetric complex random vectors. We develop an algorithm to generate the axially symmetric data that follow the given covariance function. Our simulation study demonstrates that our approach performs comparable with the classical approach using the given axially symmetric covariance function directly, while at the same time significantly reducing computational costs. For the second contribution of this dissertation, we apply the discrete Fourier transform to provide the nonparametric estimation on the covariance function of the above circularly-symmetric complex random vectors under gridded data structure. Our results show that these estimates has closely related to the simultaneous diagonalization of circulant matrices. The simulation study shows that our proposed estimates match well with their theoretical counterparts. Finally through the Fourier transform of the original gridded data, the covariance estimator of an axially symmetric process based on the method of moments can be represented as a quadratic form of transformed data that is associated with a rotation matrix.