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In mathematics, trigonometric series are a special class of orthogonal 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

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

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

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.

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 ${\displaystyle }$, 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 ${\displaystyle f}$ 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 .

See also

• Denjoy–Luzin theorem

Notes

1. Hardy & Rogosinski 1999, pp. 2, 4.
2. Zygmund 1968, pp. 6–7.
3. Edwards 1979, pp. 158–159.
4. Edwards 1982, pp. 95–96.
5. Kechris, Alexander S. (1997). "Set theory and uniqueness for trigonometric series" (PDF). Caltech.
6. 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.{{cite journal}}: CS1 maint: postscript (link)

References

• Bari, Nina Karlovna (1964). A Treatise on Trigonometric Series. Vol. 1. Translated by Mullins, Margaret F. Pergamon.
• Edwards, R. E. (1979). Fourier Series. Vol. 64. New York, NY: Springer New York. doi:10.1007/978-1-4612-6208-4. ISBN 978-1-4612-6210-7.
• Edwards, R. E. (1982). Fourier Series. Vol. 85. New York, NY: Springer New York. doi:10.1007/978-1-4613-8156-3. ISBN 978-1-4613-8158-7.
• 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.

v t e Sequences and series
Integer sequences	Basic Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10 Advanced (list) Complete sequence Fibonacci sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array
Properties of sequences	Cauchy sequence Monotonic function Periodic sequence
Properties of series	Series Alternating Convergent Divergent Telescoping Convergence Absolute Conditional Uniform
Explicit series	Convergent 1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2 + 1/3 + ... (Riemann zeta function) Divergent 1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
Kinds of series	Taylor series Power series Formal power series Laurent series Puiseux series Dirichlet series Trigonometric series Fourier series Generating series
Hypergeometric series	Generalized hypergeometric series Hypergeometric function of a matrix argument Lauricella hypergeometric series Modular hypergeometric series Riemann's differential equation Theta hypergeometric series
Category

• Fourier series