On Some Double Orthogonality Properties of the Spheroidal and Mathieu Functions

Abstract

1.Introduction. The existence of complete sets of functions that are orthogonal over two different regions simultaneously, one contained within the other, appears to have been discovered by Bergman [1,2,3]. They were applied later by Davis [7] to the problem of finding the best mean-square approximation to a function given over the smaller region when a constraint is imposed upon the mean-square value of the approximating function over the larger region. In the theory of doubly orthogonal functions as developed by Bergman and Davis the regions of orthogonality are areas in the complex plane. More recently Sleplan and Pollak [13] have shown that the zero-order prolate spheroidal wave functions are doubly orthogonal over two dIfferent intervals along the real axis. These functions have been found to be of considerable importance in certain physical problems, namely in the representation of signals [13] and in the synthesis of line source antennas [12]. Further investigation into the antenna synthesis problem has led to the existance of a broader class of doubly orthogonal functions, a class that includes all of the first kind of spheroidal wave functions and periodic Mathieu functions.

Finite Fourier self-transforms

“…Their double orthogonality has been demonstrated by Rhodes [5,21] who used a Sturmian proof which is considerably simpler than the commutation of the operators. Another interesting case was considered by Chalk [11], The search for optimum pulse shapes for communication channels led him to consider the integral equation ( where a" is a solution of:…”

Untitled

“…and which for special values of the parameter a reduces to spheroidal wave functions and Mathieu functions. Their double orthogonality has been demonstrated by Rhodes [5,21] who used a Sturmian proof which is considerably simpler than the commutation of the operators. Another interesting case was considered by Chalk [11], The search for optimum pulse shapes for communication channels led him to consider the integral equation (2.8) with w -1) "y = (1 + w2/c2)~1/2, T = (-1, 1) and 0 = (-°°, «>).…”