12
RICE UNIVERSITY STUDIES
(iii) He was the first to create a precise class of integrable functions,
so as to be able to define the Fourier coefficients (6). Furthermore, in pre-
senting his criterion for (Riemann) integrability, he prominently used,
probably for the first time, the notion of “a necessary and sufficient” con-
dition, literally so.
(iv) He sharply distinguished between a trigonometric series
(7) ⅜⅝+ ∑ (allcosnx + bπsinnx)
л = 1
and a Fourier series. For the latter the coefficients {α,, b,,} are given by the
formulas (6) by means of some function/(x), but in the first case no such
formulas are assumed at all.
(v) For a Fourier series he created the concept of “localization” of
convergence (and he also conceived the Riemann-Lebesgue lemma), thus
creating the concept of a “local” property for mathematics at large.
(vi) For any trigonometric series, with
an,bn→0,
he introduced the sum function
. , 2 ~ uncosnx + bnsin nx
F(x) = -i-⅝x2- Σ --------2-------
n=ι n
and treated it as a present-day Schwartz distribution of level 2. That is,
he introduced “testing” functions” φ(x) and “defined”
f ΦW
J ax
by
ʃ φ"(x)F(x)dx.
(I venture to remark that, long before Schwartz, such “distributions” were
introduced by myself as “generalized” Fourier transforms; see [12, Ch. VI].)
Thirdly, and very decisively, Georg Cantor, after studying closely the
work of Riemann, added the following proposition:
(vii) If the trigonometric series (7) is convergent, and to the limit value O,
at all points of the interval —π ≤ x ≤ π, then the series is identically O,
meaning that an = bn = O, n = 0,1,2,3,..