Trigonometric series

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 zeros of a trigonometric seriesEdit

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 S0 = S and let Sk+1 be the derived set of Sk. If there is a finite number n for which Sn is finite, then all the coefficients are zero. 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 .[2]

ReferencesEdit

  1. ^ [1]
  2. ^ Cooke, Roger (1993), "Uniqueness of trigonometric series and descriptive set theory, 1870–1985", Archive for History of Exact Sciences, 45 (4): 281–334, doi:10.1007/BF01886630.