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
\gamma
. 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
P(\frac{\pi a}{\lambda} \sin \theta) = \sum \min{n=0} \max{\infty} a_{n}S_{0n}(c,\sin \theta)
, and the resulting optimum aperture distribution by
a(\frac{2x}{a}) = \sum \min{n=0} \max{infty} \frac {\pi i^{-n}} {R_{0n}^{(1)}(c,1)} a_{n}S_{0n}(c,\frac{2x}{a})
, where the eigenfunctions
S_{0n}(c, \eta)
of the finite Fourier transform are the angular prolate spheroidal wave functions,
R_{0n}^{(1)}(c, 1)
are the radial prolate spheroidal wave functions evaluated at unity,
a
is the aperture length,
c=\pi a/\lambda
and the expansion coefficients
a_{n}
are
a_{n} = \frac{b_{n}}{1+\mu(\lambda_{n}^{-1}-\gamma)}
;
b_{n}
are the expansion coefficients of the given radiation pattern, the eigenvalues
\lambda_{n}
are
(2c/\pi) [R_{0n}^{(1)} (c, 1)]^{2}
,
\mu
is a unique positive number satisfying
\sum \min{n=0}\max{\infty}\frac{\frac{k_n}|b_{n}|^{2}} {\lambda_{n}^{-1}-\gamma}} {(\mu + \frac{1}{\gamma_{n}^{-1}-\gamma)^{2}}=0
, and
k_{n}
is the normalization factor for the eigenfunctions on (-1, 1). The pattern approximation is determined largely by the first
(2a/\lambda)+1
terms of its expansion, beyond which the expansion converges quickly for practical values of the superdirective ratio.