A trigonometric series is a series of the form:
A
0
2
+
∑
n
=
1
∞
(
A
n
cos
n
x
+
B
n
sin
n
x
)
.
It is called a Fourier series if the terms
A
n
and
B
n
have the form:
A
n
=
1
π
∫
0
2
π
f
(
x
)
cos
n
x
d
x
(
n
=
0
,
1
,
2
,
3
…
)
B
n
=
1
π
∫
0
2
π
f
(
x
)
sin
n
x
d
x
(
n
=
1
,
2
,
3
,
…
)
where
f
is an integrable function.
The uniqueness and the zeros of trigonometric series was an active area of research in 19th century Europe. First, Georg Cantor proved that if a trigonometric series is convergent to a function
f
(
x
)
on the interval
[
0
,
2
π
]
, which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.[1] But almost half a millennium back the Indian Mathematicians, notably from Kerala school of astronomy and mathematics like Madhava of Sangamagrama and Neelakanta Somayaji had already created the whole basis of the same theory. Due to the imperialism that occurred in India most of the information was hidden from the outside world.
Later Cantor proved that even if the set S on which
f
is nonzero is infinite, but the derived set S' of S is finite, then the coefficients are all zero. In fact, he proved a more general result. Let S0 = S and let Sk+1 be the derived set of Sk. If there is a finite number n for which Sn is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α such that Sα is finite, then the coefficients of the series are all zero. Cantor's work on the uniqueness problem famously led him to invent transfinite ordinal numbers, which appeared as the subscripts α in Sα .
Antoni Zygmund wrote a classic two-volume set of books entitled Trigonometric Series, which discusses many different aspects of these series.The first edition was a single volume, published in 1935 (under the slightly different title "trigonometrical series"). The second edition of 1959 was greatly expanded, taking up two volumes, though it was later reprinted as a single volume paperback. The third edition of 2002 is similar to the second edition, with the addition of a preface by Robert A. Fefferman on more recent developments, in particular Carleson's theorem about almost everywhere pointwise convergence for square integrable functions.
