A Discrete Fractional Hankel Transform Based on the Eigen Decomposition of a Symmetric Kernel Matrix of the Discrete Hankel Transform

摘要

Recently a discrete Hankel transform (DHT) has been introduced using a symmetric involutory kernel matrix T. Although Namias contributed the fractional Hankel transform (FRHT) in 1980, no discrete counterpart has appeared till now. Here a definition is proposed for a discrete fractional Hankel transform (DFRHT) based on the eigen decomposition of the diagonalizable matrix T. Being a real symmetric involutory matrix, T has two orthogonal eigen spaces corresponding to its two distinct eigenvalues. Simple explicit expressions are derived for the orthogonal projection matrices of T on its eigen spaces. Expressions are derived for the dimensions of the two eigen spaces in terms of the trace of matrix T. Initial orthonormal bases are generated for the two eigen spaces by the singular value decomposition of the orthogonal projection matrices. Final superior orthonormal bases - which better approximate samples of the eigen functions of the FRHT - are generated by either the Gram Schmidt algorithm or the orthogonal procrustes algorithm.