Trigonometric series explained
In mathematics, a trigonometric series is an infinite series of the form
A0+
An\cos{nx}+Bn\sin{nx},
where
is the variable and
and
are
coefficients. It is an infinite version of a
trigonometric polynomial.
A trigonometric series is called the Fourier series of the integrable function if the coefficients have the form:
Examples
Every Fourier series gives an example of a trigonometric series.Let the function
on
be extended periodically (see
sawtooth wave). Then its Fourier coefficients are:
\begin{align}
An&=
\pix\cos{nx}dx=0, n\ge0.\\[4pt]
Bn&=
\pix\sin{nx}dx\\[4pt]
&=-
\cos{nx}+
\\[5mu]
&=
, n\ge1.\end{align}
Which gives an example of a trigonometric series:
The converse is false however, not every trigonometric series is a Fourier series. The series
} = \frac + \frac + \frac+\cdotsis a trigonometric series which converges for all
but is not a
Fourier series.
[1] Here
for
and all other coefficients are zero.
Uniqueness of Trigonometric series
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
on the interval
, which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.
[2] Later Cantor proved that even if the set S on which
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 S
0
= S
and let S
k+1
be the derived set of S
k
. If there is a finite number n
for which S
n
is finite, then all the coefficients are zero. Later, Lebesgue proved that if there is a countably infinite ordinal α
such that S
α
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 S
α
.[3] References
- Book: Bari, Nina Karlovna . Nina Bari . 1964 . A Treatise on Trigonometric Series . 1 . Pergamon . Mullins . Margaret F. . limited .
- Book: Zygmund, Antoni . Antoni Zygmund . Trigonometric Series . 1 and 2 . Cambridge University Press . 2nd, reprinted . 0236587 . 1968 . limited .
See also
Notes and References
- Book: Hardy . Godfrey Harold . G.H. Hardy . Rogosinski . Werner Wolfgang . Werner Wolfgang Rogosinski . 1956 . 1st ed. 1944 . Fourier Series . 3rd . Cambridge University Press . 4–5 . limited .
- Web site: Set theory and uniqueness for trigonometric series. Alexander S.. Kechris. Caltech. 1997.
- Cooke. Roger. Uniqueness of trigonometric series and descriptive set theory, 1870–1985. Archive for History of Exact Sciences. 45. 281–334. 1993. 10.1007/BF01886630. 4. 122744778 . ..