Trigonometric series

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In mathematics, trigonometric series are a special class of orthogonal series of the form^[1][2]

${\displaystyle A_{0}+\sum _{n=1}^{\infty }A_{n}\cos {(nx)}+B_{n}\sin {(nx)},}$

where ${\displaystyle x}$ is the variable and ${\displaystyle \{A_{n}\}}$ and ${\displaystyle \{B_{n}\}}$ are coefficients. It is an infinite version of a trigonometric polynomial.

A trigonometric series is called the Fourier series of the integrable function ${\textstyle f}$ if the coefficients have the form:

${\displaystyle A_{n}={\frac {1}{\pi }}\int _{0}^{2\pi }\!f(x)\cos {(nx)}\,dx}$

${\displaystyle B_{n}={\frac {1}{\pi }}\displaystyle \int _{0}^{2\pi }\!f(x)\sin {(nx)}\,dx}$

Every Fourier series gives an example of a trigonometric series. Let the function ${\displaystyle f(x)=x}$ on ${\displaystyle [-\pi ,\pi ]}$ be extended periodically (see sawtooth wave). Then its Fourier coefficients are:

${\displaystyle {\begin{aligned}A_{n}&={\frac {1}{\pi }}\int _{-\pi }^{\pi }x\cos {(nx)}\,dx=0,\quad n\geq 0.\\[4pt]B_{n}&={\frac {1}{\pi }}\int _{-\pi }^{\pi }x\sin {(nx)}\,dx\\[4pt]&=-{\frac {x}{n\pi }}\cos {(nx)}+{\frac {1}{n^{2}\pi }}\sin {(nx)}{\Bigg \vert }_{x=-\pi }^{\pi }\\[5mu]&={\frac {2\,(-1)^{n+1}}{n}},\quad n\geq 1.\end{aligned}}}$

Which gives an example of a trigonometric series:

${\displaystyle 2\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}\sin {(nx)}=2\sin {(x)}-{\frac {2}{2}}\sin {(2x)}+{\frac {2}{3}}\sin {(3x)}-{\frac {2}{4}}\sin {(4x)}+\cdots }$

However, the converse is false. For example,

${\displaystyle \sum _{n=2}^{\infty }{\frac {\sin {(nx)}}{\log {n}}}={\frac {\sin {(2x)}}{\log {2}}}+{\frac {\sin {(3x)}}{\log {3}}}+{\frac {\sin {(4x)}}{\log {4}}}+\cdots }$

is a trigonometric series which converges for all ${\displaystyle x}$ but is not a Fourier series, i.e., while ${\displaystyle B_{n}={\frac {1}{\log(n)}}}$ for ${\displaystyle n\geq 2}$, it is undefined for ${\displaystyle n=1}$.^[3]

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 ${\displaystyle f}$ on the interval ${\displaystyle [0,2\pi ]}$, which is identically zero, or more generally, is nonzero on at most finitely many points, then the coefficients of the series are all zero.^[4]

Later Cantor proved that even if the set S on which ${\displaystyle f}$ 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 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 .^[5]

• Bari, Nina Karlovna (1964). A Treatise on Trigonometric Series. Vol. 1. Translated by Mullins, Margaret F. Pergamon.
• Hardy, G. H.; Rogosinski, Werner (1999). Fourier series. Mineola, N.Y: Dover Publications. ISBN 978-0-486-40681-7.
• Zygmund, Antoni (1968). Trigonometric Series. Vol. 1 and 2 (2nd, reprinted ed.). Cambridge University Press. MR 0236587.

• Denjoy–Luzin theorem