Trigonometric series

It is an infinite version of a trigonometric polynomial.

be extended periodically (see sawtooth wave).

Then its Fourier coefficients are: Which gives an example of a trigonometric series: However, the converse is false.

[3][4] 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

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.

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 .