Trigonometric series

Remarks

In real form the series is $\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx)$, with $a_n = c_n + c_{-n}$ and $b_n = i(c_n - c_{-n})$, and its partial sums are the same. Example: $\sum_{n=1}^{\infty} \frac{\cos nx}{n^2}$ is a trigonometric series with $c_0 = 0$ and $c_n = c_{-n} = \frac{1}{2n^2}$ for $n \ge 1$; it converges at every $x$ by comparison with $\sum 1/n^2$. Non-example: the power series $\sum_{n \ge 0} x^n$ is a series of functions, but its terms are powers of $x$ and not multiples of $e^{inx}$, so it is not a trigonometric series. A trigonometric series is the Fourier series of an integrable function only for special coefficients. For instance, $\sum_{n \ge 2} \frac{\sin nx}{\log n}$ converges at every $x$ but is not the Fourier series of a Lebesgue integrable function.