The optimum line source for the best mean-square approximation to a given radiation pattern

An optimum aperture distribution for pattern shaping by a continuous line source of arbitrary length is derived in terms of the functions most natural to a least-squares fit: the eigenfunctions of the finite Fourier transform. It is expressed as an explicit function of Taylor\´s superdirective ratio . The new distribution produces the best mean-square approximation to a specified radiation pattern that is possible to obtain from an aperture of a given length for a given value of the superdirective ratio. The best mean-square pattern approximation is shown to be represented exactly by the orthogonal expansion , and the resulting optimum aperture distribution by , where the eigenfunctions of the finite Fourier transform are the angular prolate spheroidal wave functions, are the radial prolate spheroidal wave functions evaluated at unity, is the aperture length, and the expansion coefficients are ; are the expansion coefficients of the given radiation pattern, the eigenvalues are , is a unique positive number satisfying , and is the normalization factor for the eigenfunctions on (-1, 1). The pattern approximation is determined largely by the first terms of its expansion, beyond which the expansion converges quickly for practical values of the superdirective ratio.