# §1.8 Fourier Series

## §1.8(i) Definitions and Elementary Properties

Formally,

The series (1.8.1) is called the Fourier series of , and are the Fourier coefficients of .

If , then for all .

If , then for all .

### ¶ Asymptotic Estimates of Coefficients

If is of period , and is piecewise continuous, then

### ¶ Uniqueness of Fourier Series

If and are continuous, have the same period and same Fourier coefficients, then for all .

1.8.8.

### ¶ Riemann–Lebesgue Lemma

For piecewise continuous on and real ,

(1.8.10) continues to apply if either or or both are infinite and/or has finitely many singularities in , provided that the integral converges uniformly (§1.5(iv)) at , and the singularities for all sufficiently large .

## §1.8(ii) Convergence

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

1.8.11

at every point at which has both a left-hand derivative (that is, (1.4.4) applies when ) and a right-hand derivative (that is, (1.4.4) applies when ). The convergence is non-uniform, however, at points where ; see §6.16(i).

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

## §1.8(iii) Integration and Differentiation

If and are the Fourier coefficients of a piecewise continuous function on , then

If a function is periodic, with period , then the series obtained by differentiating the Fourier series for term by term converges at every point to .

## §1.8(iv) Transformations

### ¶ Parseval’s Formula

when and are square-integrable and and are their respective Fourier coefficients.

### ¶ Poisson’s Summation Formula

Suppose that is twice continuously differentiable and and are integrable over . Then

An alternative formulation is as follows. Suppose that is continuous and of bounded variation on . Suppose also that is integrable on and as . Then

As a special case

## §1.8(v) Examples

For collections of Fourier-series expansions see Prudnikov et al. (1986a, v. 1, pp. 725–740), Gradshteyn and Ryzhik (2000, pp. 45–49), and Oberhettinger (1973).