Trigonometric series

This article is about the mathematical concept. For the book by Zygmund, see Trigonometric Series.

In mathematics, a trigonometric series is a infinite series of the form

${\frac {A_{0}}{2}}+\displaystyle \sum _{n=1}^{\infty }(A_{n}\cos {nx}+B_{n}\sin {nx}),$

an infinite version of a trigonometric polynomial.

It is called the Fourier series of the integrable function ${\textstyle f}$ if the terms $A_{n}$ and $B_{n}$ have the form:

$A_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }\!f(x)\cos {nx}\,dx\qquad (n=0,1,2,3\dots )$

$B_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }\!f(x)\sin {nx}\,dx\qquad (n=1,2,3,\dots )$

Example

The series

${\frac {\sin(2\theta )}{\log(2)}}+{\frac {\sin(3\theta )}{\log(3)}}+{\frac {\sin(4\theta )}{\log(4)}}+\cdots$

is a trigonometric series which converges for all $\theta$ but is not a Fourier series.

The zeros of a 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 $f(x)$ on the interval $[0,2\pi ]$, which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.

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 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 .

See also

• Denjoy–Luzin theorem