The name is absent

(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+ ∑ (a_llcosnx + b_πsinnx)

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

a_n,b_n→0,

he introduced the sum function

.    ,    ₂    ~ u_ncosnx + b_nsin 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 a_n = b_n = O, n = 0,1,2,3,..

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