In mathematics, a trigonometric series is a

infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, ma ...

of the form :

\frac+\displaystyle\sum_^(A_ \cos + B_ \sin),

an infinite version of a

trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...

. It is called the

Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...

of the

integrable function In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

f

if the terms

A_

and

B_

have the form: :

A_=\frac\displaystyle\int^_0\! f(x) \cos \,dx\qquad (n=0,1,2,3 \dots)

B_=\frac\displaystyle\int^_0\! f(x) \sin\, dx\qquad (n=1,2,3, \dots)

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 Georg Ferdinand Ludwig Philipp Cantor ( , ; – January 6, 1918) was a German mathematician. He played a pivotal role in the creation of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of ...

proved that if a trigonometric series is convergent to a function

f(x)

on the interval

, 2\pi /math>, 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 ''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 In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the least ...

, which appeared as the subscripts ''α'' in ''S''_''α'' .

The zeros of a trigonometric series

References

See also