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In mathematics, a **trigonometric series** is a series of the form:

It is called a Fourier series if the terms and have the form:

where 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 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]}