C. J. Bouwkamp, Rept. Progr. Phys. 17, 39 (1954); and other references in this report.
Mathematical terms which are used but not defined in this section may be found in, for instance, E. C. Titchmarsh, The Theory of Functions, 2nd ed. (Oxford University Press, 1939).
R. R. Goldberg, Fourier Transforms (Cambridge University Press, 1961) p. 51. This book provides a very readable summary of the results of Fourier-transform theory. It will be noticed, however, that it treats only the one-dimensional case, whereas this paper deals with functions of two variables. The extension of the theory to n dimensions is, in fact, quite straightforward and is discussed in, for instance, S. Bochner, Lectures on Fourier Integrals (Princeton University Press, 1959) Ch. IX and particularly Theorem 63.
Reference 3, Theorem 2K, p. 4. 1235
E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed. (Oxford University Press, 1948), p. 83.
Reference 3, Theorem 133E, p. 48.
R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1966), p. 311. Note however, that the conditions satisfied by U(x,y,z) in this reference are weaker than those in the present paper. Our requirement b(iii) is an important additional condition.
It should be noted that the right-hand side of Eq. (2) was derived as an improper Riemann integral, whereas the L2 Fourier-transform theory and in particular Theorem IV which is used in Eq. (3) makes use necessarily of the Lebesgue definition of the integral. This leads to no difficulty here, however, since it may be shown that both definitions of the integral lead to the same result in the present case. See, for instance, E. Asplund and L. Bungart, A First Course in Integration (Holt, Rinehart and Winston, Inc., New York, 1966) p. 75.
H. Weyl, Ann. Physik 60, 481 (1919). Equation (5) of the present paper is a straightforward modification of the result given in this reference.
E. Asplund and L. Bungart (see Ref. 8) p. 163.
This statement is true only when U(x,y,O) is continuous. For a discussion of the values assumed at a point where U(x,y,O) is discontinuous see: N. Mukunda, J. Opt. Soc. Am. 52, 336 (1962).
G. C. Sherman, J. Opt, Soc. Am. 57, 546 (1967).