Complete analysis of the Eigen functions of Fourier transform

TLDR: The extended theory of systematic construction of Eigen function concept for sine and cosine functions was applied and the necessary expressions for complex Eigen functions when function f1 is even and function f2 is odd vice versa were derived.

Abstract: In this paper, we proposed an analysis of Eigen functions of the n-Dimensional Fourier transform. We extended the theory developed in our previous publication [1]. We applied our extended theory of systematic construction of Eigen function concept for sine and cosine functions. We explained the concept in a clear way through example construction of Eigen functions of 1-D and 2-D Fourier transform for real even case with negative Eigen value. PP Vaidyanathan discussed in his paper [2] that construction of complex Eigen function f1+ jf2 is possible when both functions f1 and f2 are either real even or real odd. But he didn't discuss what will happen when f1 and f2 are different. We subjected this concept and derived the necessary expressions for complex Eigen functions when function f1 is even and function f2 is odd vice versa.