Trigonometric series

Examples

Every Fourier series gives an example of a trigonometric series. Let the function f(x) = x on [-\pi,\pi] be extended periodically (see sawtooth wave). Then its Fourier coefficients are: :\begin{align} A_n &= \frac1\pi\int_{-\pi}^{\pi} x \cos{(nx)} ,dx = 0, \quad n \ge 0. \[4pt] B_n &= \frac1\pi\int_{-\pi}^{\pi} x \sin{(nx)} , dx \[4pt] &= -\frac{x}{n\pi} \cos{(nx)} + \frac1{n^2\pi}\sin{(nx)} \Bigg\vert_{x=-\pi}^\pi \[5mu] &= \frac{2,(-1)^{n+1}}{n}, \quad n \ge 1.\end{align} Which gives an example of a trigonometric series: :2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin{(nx)} = 2\sin{(x)} - \frac22\sin{(2x)} + \frac23\sin{(3x)} - \frac24\sin{(4x) } + \cdots

However, the converse is false. For example, :\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 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 f on the interval [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.

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

Notes

References

References

1. Kechris, Alexander S.. (1997). "Set theory and uniqueness for trigonometric series". Caltech.
2. (1993). "Uniqueness of trigonometric series and descriptive set theory, 1870–1985". Archive for History of Exact Sciences.