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§1.8 Fourier Series

Contents

§1.8(i) Definitions and Elementary Properties

Formally,

1.8.1 f(x)=12a0+n=1(ancos(nx)+bnsin(nx)),
1.8.2 an =1π-ππf(x)cos(nx)dx,
n=0,1,2,,
bn =1π-ππf(x)sin(nx)dx,
n=1,2,.

The series (1.8.1) is called the Fourier series of f(x), and an,bn are the Fourier coefficients of f(x).

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

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

Alternative Form

1.8.3 f(x)=n=-cneinx,
1.8.4 cn=12π-ππf(x)e-inxdx.

Bessel’s Inequality

1.8.5 12a02+n=1(an2+bn2)1π-ππ(f(x))2dx.
1.8.6 n=-|cn|212π-ππ|f(x)|2dx.

Asymptotic Estimates of Coefficients

If f(x) is of period 2π, and f(m)(x) is piecewise continuous, then

1.8.7 an,bn,cn=o(n-m),
n.

Uniqueness of Fourier Series

If f(x) and g(x) are continuous, have the same period and same Fourier coefficients, then f(x)=g(x) for all x.

Lebesgue Constants

1.8.8 Ln=1π0π|sin(n+12)t|sin(12t)dt,
n=0,1,.

As n

1.8.9 Ln(4/π2)lnn;

see Frenzen and Wong (1986).

Riemann–Lebesgue Lemma

For f(x) piecewise continuous on [a,b] and real λ,

1.8.10 abf(x)eiλxdx0,
as λ.

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

§1.8(ii) Convergence

Let f(x) be an absolutely integrable function of period 2π, 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 12f(x-)+12f(x+)

at every point at which f(x) has both a left-hand derivative (that is, (1.4.4) applies when h0-) and a right-hand derivative (that is, (1.4.4) applies when h0+). The convergence is non-uniform, however, at points where f(x-)f(x+); see §6.16(i).

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

§1.8(iii) Integration and Differentiation

If an and bn are the Fourier coefficients of a piecewise continuous function f(x) on [0,2π], then

1.8.12 0x(f(t)-12a0)dt=n=1ansin(nx)+bn(1-cos(nx))n,
0x2π.

If a function f(x)C2[0,2π] is periodic, with period 2π, then the series obtained by differentiating the Fourier series for f(x) term by term converges at every point to f(x).

§1.8(iv) Transformations

Parseval’s Formula

1.8.13 1π-ππf(x)g(x)dx=12a0a0+n=1(anan+bnbn),

when f(x) and g(x) are square-integrable and an,bn and an,bn are their respective Fourier coefficients.

Poisson’s Summation Formula

Suppose that f(x) is twice continuously differentiable and f(x) and |f′′(x)| are integrable over (-,). Then

1.8.14 n=-f(x+n)=n=-e2πinx-f(t)e-2πintdt.

An alternative formulation is as follows. Suppose that f(x) is continuous and of bounded variation on [0,). Suppose also that f(x) is integrable on [0,) and f(x)0 as x. Then

1.8.15 12f(0)+n=1f(n)=0f(x)dx+2n=10f(x)cos(2πnx)dx.

As a special case

1.8.16 n=-e-(n+x)2ω=πω(1+2n=1e-n2π2/ωcos(2nπx)),
ω>0.

§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).