Trigonometric series

This article is about the mathematical concept. For the book by Zygmund, see Trigonometric Series.

In mathematics, a trigonometric series is an infinite series of the form

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

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:

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

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

Examples

The Fourier series for the identity function suffers from the Gibbs phenomenon near the ends of the periodic interval.

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:

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

The trigonometric series sin 2x / log 2+ sin 3x / log 3+ sin 4x / log 4 + ... is not a Fourier series.

The converse is false however, not every trigonometric series is a Fourier series. The series

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

is a trigonometric series which converges for all ${\displaystyle x}$ but is not a Fourier series. ^[1] Here ${\displaystyle B_{n}={\frac {1}{\log(n)}}}$ for ${\displaystyle n\geq 2}$ and all other coefficients are zero.

Uniqueness of Trigonometric series

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. ^[2]

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 . ^[3]

Notes

1. Hardy, Godfrey Harold; Rogosinski, Werner Wolfgang (1956) [1st ed. 1944]. Fourier Series (3rd ed.). Cambridge University Press. pp. 4–5.
2. Kechris, Alexander S. (1997). "Set theory and uniqueness for trigonometric series" (PDF). Caltech.
3. 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. S2CID   122744778.

References

• Bari, Nina Karlovna (1964). A Treatise on Trigonometric Series . Vol. 1. Translated by Mullins, Margaret F. Pergamon.
• Zygmund, Antoni (1968). Trigonometric Series . Vol. 1 and 2 (2nd, reprinted ed.). Cambridge University Press. MR   0236587.

See also

• Denjoy–Luzin theorem