§1.8 Fourier Series

Formally, if $f(x)$ is a real- or complex-valued $2\pi$-periodic function,

The series (1.8.1) is called the Fourier series of $f(x)$, and $a_n,b_n$ are the Fourier coefficients of $f(x)$.

If $f(-x)=f(x)$, then $b_n=0$ for all $n$.

If $f(-x)=-f(x)$, then $a_n=0$ for all $n$.

Let $f(x)$ be an absolutely integrable function of period $2\pi$, and continuous except at a finite number of points in any bounded interval. Then the series (1.8.1) converges to the sum

at every point at which $f(x)$ has both a left-hand derivative (that is, (1.4.4) applies when $h\to 0-$) and a right-hand derivative (that is, (1.4.4) applies when $h\to 0+$). The convergence is non-uniform, however, at points where $f(x-)\ne f(x+)$; see §6.16(i).

For other tests for convergence see Titchmarsh (1962b, pp. 405–410).

If $a_n$ and $b_n$ are the Fourier coefficients of a piecewise continuous function $f(x)$ on $[0,2\pi ]$, then

If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$.

Suppose that $f(x)$ is twice continuously differentiable and $f(x)$ and $\left|f^{\prime \prime }(x)\right|$ are integrable over $(-\mathrm{\infty },\mathrm{\infty })$. Then

It follows from definition (1.14.1) that the integral in (1.8.14) is equal to $\sqrt{2\pi }ℱ\left(f\right)\left(-2\pi n\right)$.

An alternative formulation is as follows. Suppose that $f(x)$ is continuous and of bounded variation on $[0,\mathrm{\infty })$. Suppose also that $f(x)$ is integrable on $[0,\mathrm{\infty })$ and $f(x)\to 0$ as $x\to \mathrm{\infty }$. Then

As a special case

For collections of Fourier-series expansions see Prudnikov et al. (1986a, v. 1, pp. 725–740), Gradshteyn and Ryzhik (2015, §§1.44–1.45), and Oberhettinger (1973).