500 17. REAL AND COMPLEX ANALYSISis called the Dirichlet function. For such a function, he thought, no integral could
be defined, and therefore no Fourier series could be defined.^9Fourier integrals. The convergence of the Fourier series of f(x) can be expressed
as the equationf{x) = - I f(y) dy + - V / fiy) cos(ny) cos(na;) dy.
* Jo "^JoThat equation may have led to an analogous formula for Fourier integrals, which
appeared during the early nineteenth century in papers on the wave and heat equa-
tions written by Poisson, Laplace, Fourier, and Cauchy. The central discovery in
this area was the Fourier inversion formula, which we now write as2 f°° ð°
f{x) = - / f(y) cos(zy) cosizx) dy dz.ð Jo Jo
The analogy with the formula for series is clear: The continuous variable æ replaces
the discrete index n, and the integral on æ replaces the sum over n. Once again, the
validity of the representation is much more questionable than the validity of the
formulas of complex analysis, such as the Cauchy integral formula for an analytic
function. The Fourier inversion formula has to be interpreted very carefully, since
the order of integration cannot be reversed. If the integrals make sense in the order
indicated, that happy outcome can only be the result of some special properties of
the function f(x). But what are those properties?
The difficulty was that the integral extended over an infinite interval so that
convergence required the function to have two properties: It needed to be contin-
uous, and it needed to decrease sufficiently rapidly at infinity to make the integral
converge. These properties turned out to be, in a sense, dual to each other. Con-
sidering just the inner integral as a function of z:
Ë f°°
f{z) = / fiy)cosizy)dy,
Jo
it turns out that the more rapidly f(y) decreases at infinity, the more derivatives
f{z) has, and the more derivatives f(y) has, the more rapidly f(z) decreases at
infinity. The converses are also, broadly speaking, true. Could one insist on having
both conditions, so that the representation would be valid? Would these assump-
tions impair the usefulness of these techniques in mathematical physics? Alfred
Pringsheim (1850-1941, father-in-law of the great writer Thomas Mann) studied
the Fourier integral formula {1910), noting especially the two kinds of conditions
that f{x) needed to satisfy, which he called "conditions in the finite region" ("im
Endlichen") and "conditions at infinity" ("im Unendlichen"). Nowadays, they are
called local and global conditions. Pringsheim noted that the local conditions could
be traced all the way back to Dirichlet's work of 1829, but said that "a rather ob-
vious backwardness reveals itself" in regard to the global conditions.
(^9) The increasing latitude allowed in analysis, mentioned above, is illustrated very well by this
example. When the Lebesgue integral is used, this function is regarded as identical with the
constant value it assumes on the irrational numbers.