Trigonometric series explained
In mathematics, trigonometric series are a special class of orthogonal 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:
\sin{(nx)}=2\sin{(x)}-
+ …
However, the converse is false. For example,
} = \frac + \frac + \frac+\cdotsis a trigonometric series which converges for all
but is not a
Fourier series.
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.
[1] 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
.[2] See also
References
- Book: Bari, Nina Karlovna . Nina Bari . 1964 . A Treatise on Trigonometric Series . 1 . Pergamon . Mullins . Margaret F. . limited .
- Book: Edwards, R. E. . Fourier Series . Springer New York . New York, NY . 64 . 1979 . 978-1-4612-6210-7 . 10.1007/978-1-4612-6208-4.
- Book: Edwards, R. E. . Fourier Series . Springer New York . New York, NY . 85 . 1982 . 978-1-4613-8158-7 . 10.1007/978-1-4613-8156-3.
- Book: Hardy, G. H. . Rogosinski . Werner . G.H. Hardy. Werner_Wolfgang_Rogosinski. Fourier series . Dover Publications . Mineola, N.Y . 1999 . 978-0-486-40681-7 . limited .
- Book: Zygmund, Antoni . Antoni Zygmund . Trigonometric Series . 1 and 2 . Cambridge University Press . 2nd, reprinted . 0236587 . 1968 . limited .
Notes and References
- 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 . ..