Trigonometric Series - The Zeros of A Trigonometric Series

The Zeros of A Trigonometric Series

The uniqueness and the zeros of trigonometric series was an active area of research in the 19th century. First, Georg Cantor proved that if a trigonometric series is convergent to a function on the interval, which has at least one zero, but only finitely many, then the coefficients of the series are all zero. Contrast this situation with a polynomial p(x) of degree d. If it has more than d zeros, then its coefficients are all zero.

Later Cantor proved that if the set of zeros S 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 .