Trigonometric series

A_{0} + \sum_{n = 1}^{\infty} A_{n} \cos (n x) + B_{n} \sin (n x),

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

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

A_{n} = \frac{1}{π} \int_{0}^{2 π} f (x) \cos (n x) d x

B_{n} = \frac{1}{π} \int_{0}^{2 π} f (x) \sin (n x) d x

Examples

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

\begin{matrix} A_{n} & = \frac{1}{π} \int_{- π}^{π} x \cos (n x) d x = 0, n \geq 0. \\ B_{n} & = \frac{1}{π} \int_{- π}^{π} x \sin (n x) d x \\ = - \frac{x}{n π} \cos (n x) + \frac{1}{n^{2} π} \sin (n x) |_{x = - π}^{π} \\ = \frac{2 (- 1)^{n + 1}}{n}, n \geq 1. \end{matrix}

Which gives an example of a trigonometric series:

2 \sum_{n = 1}^{\infty} \frac{(- 1)^{n + 1}}{n} \sin (n x) = 2 \sin (x) - \frac{2}{2} \sin (2 x) + \frac{2}{3} \sin (3 x) - \frac{2}{4} \sin (4 x) + \dots

However, the converse is false. For example,

\sum_{n = 2}^{\infty} \frac{\sin (n x)}{\log n} = \frac{\sin (2 x)}{\log 2} + \frac{\sin (3 x)}{\log 3} + \frac{\sin (4 x)}{\log 4} + \dots

is a trigonometric series which converges for all $x$ but is not a Fourier series.

One more chapter is free with a Lably account.

Examples

Uniqueness of trigonometric series

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