D. Gabor, Progr. Opt. 1, 109 (1961).
H. Gamo, J. Appl. Phys. Japan 25, 431 (1956) 26, 102 (1957); 26, 414 (1957); 27, 577 (1958).
T. di Francia, J. Opt. Soc. Am. 45, 497 (1955).
M.Born and E. Wolf, Principles of Optics (Pergamon Press Inc., New York, 1959), P). 483.
R. Paley and N. Wiener, The Fourier Transform in the Complex Domain (American Mathematical Society, Colloquium Publications, New York, 1934), Vol. XIX.
E. T. Whittaker, Proc. Roy. Soc. (Edinburgh) 35, 181 (1915).
E. H. Linfoot, J. Opt. Soc. Am. 16, 740 (1956).
R. Barakat and A. Houston, J. Opt. Soc. Am. 53, 1244 (1963).
E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals (Oxford University Press, London, 1948), 2nd ed.
The theory of conjugate Fourier series and finite Hilbert transforms is employed extensively in aerodynamics and just about the only readable accounts are given in aerodynamic treatise. An excellent discussion is contained in: A. Robinson and J. Laurman, Wing Theory (Cambridge University Press, Cambridge, England, 1956), p. 101.
E. L. O'Neill and A. Walther, Opt. Acta 10, 33 (1962).
No Hilbert transform equivalent to conjugate series in two dimensions is known to the author. Howvever, some progress along these lines is contained in the recent paper: S. Goldman, J. Opt. Soc. Am. 52, 1131 (1962).
The cardinal series (6.10) is quite old and in fact is implicitly contained in: J. M. Whittaker, Interpolatory Function Theory (Cambridge University Press, Cambridge, England, 1935), p. 7
Footnote added in proof: The problem of determining t in terms of the sampled value of τ has been solved and will appear in a sequel to the present paper.
R. Barakat and A. Houston, J. Opt. Soc. Am. 54, 768 1964).