Chapter 1 Fourier Series 1.1 Introduction

Chapter 1 Fourier Series 1.1 Introduction

Definition 1. A function f on R is called periodic if there is some positive number p such that

f(x + p) = f (x).

The number p is called a period of f (x).

Theorem 1. The trigonometric system

© 1

√2π, 1

√πsin x, 1

√πcos kx : k = 1, 2, . . .ª

is an orthonormal set in L²[−π, π]. Here, for f, g ∈ L²[−π, π], the inner product is defined by

< f, g >= Z π

−π

f(t)g(t)dt.

1.2 Computation of Fourier Series

Fourier Series (Real Version): Assume that f (x) is a periodic function of period 2π and is integrable over a period. Assume further that f (x) can be written by

f(x) = a0+

∞

X

n=1

(ancos nx + bnsin nx). (1)

Then

(A) a0 = 1 2π

Z π

−π

f(x)dx (2)

(B) an = 1 π

Z π

−π

f(x) cos nxdx, n= 1, 2, · · · (C) bn = 1

π Z π

−π

f(x) sin nxdx, n= 1, 2, · · ·

The series (1) is called the Fourier Series of f (x) and the numbers (A), (B), and (C) in (2) are called the Fourier Coefficients of f (x).

Other Intervals : Let f (x) be a periodic function of period p = 2L Then its Fourier Series is of the form:

f(x) = a0+

∞

X

n=1

³

a_ncosnπ

L x+ bnsinnπ L x´

(3)

with the Fourier Coefficients

(A) a0 = 1 2L

Z L

−L

f(x)dx (4)

(B) an = 1 L

Z L

−L

f(x) cosnπ

L xdx, n= 1, 2, · · · (C) bn = 1

L Z L

−L

f(x) sinnπ

L xdx, n= 1, 2, · · ·

• See Examples 1.5.

Notes: A function g is even if g(−x) = g(x) and odd if g(−x) = −g(x).

(1) If a function g is even, then Z L

−L

g(x)dx = 2 Z L

0

g(x)dx (2) If h(x) is an odd function, then

Z L

−L

h(x)dx = 0.

(3) The product of an even and odd function is odd.

Theorem 2. The Fourier Series of an even function of period 2L is a “Fourier cosine series”

f(x) = a0+

∞

X

n=1

a_ncosnπ L x with the Fourier coefficients

a0 = 1 L

Z L 0

f(x)dx a_n = 2

L Z L

0

f(x) cosnπ

L xdx, n= 1, 2, · · ·

The Fourier Series of an odd function of period 2L is a “Fourier sine series”

f(x) =

∞

X

n=1

b_nsinnπ L x with the Fourier coefficients

b_n = 2 L

Z L 0

f(x) sinnπ

L xdx, n = 1, 2, · · ·

2

Fourier Cosine and Sine Series on a Half-Interval: Assume that f is defined on [0, L].

To express f as a Fourier cosine series, consider the even extension of f : f_e(x) :=½ f (x), if x∈ [0, L],

f(−x), if x∈ [−L, 0) (5)

Obviously,

f(x) = a0+

∞

X

n=1

a_ncosnπ

L x, x∈ [0, L]

with the Fourier coefficients a0 = 1

L Z L

0

f(x)dx a_n = 2

L Z L

0

f(x) cosnπ

L xdx, n= 1, 2, · · ·

Next, to express f as a Fourier sine series, consider the odd extension of f : f_e(x) :=½ f (x), if x∈ [0, L],

−f(−x), if x∈ [−L, 0) Then

f(x) =

∞

X

n=1

b_nsinnπ L x with the Fourier coefficients

b_n = 2 L

Z L 0

f(x) sinnπ

L xdx, n= 1, 2, · · ·

• See Examples 1.9 – 1.13.

Fourier Series (Complex Version):

Theorem 3. The system

© 1

√2πe^inx : n ∈ Z}

is an orthonormal set in L²[−π, π].

Assume that

f(x) =X

n∈Z

c_ne^inx. (6)

Then

c_n= 1 2π

Z π

−π

f(x)e^−inxdx, n∈ Z. (7)

The series (6) is called the complex Fourier Series of f (x) and the numbers cn in (7) are called the Fourier Coefficients of f (x). Here, from 2), we see that

c0 = a0, a_n = cn+ c−n and − ibⁿ= cn− c⁻ⁿ.

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1.3 Convergence Theorems for Fourier Series

Theorem 4. (Riemann-Lebesgue Lemma) Suppose that f is a piecewise continuous funtion on the interval [a, b]. Then

k→∞lim Z b

a

f(x) cos kxdx = lim

k→∞

Z b a

f(x) sin kxdx = 0.

Let

S_N(x) = a0+

N

X

n=1

³

a_ncosnπ

L x+ bnsinnπ L x´

. (8)

We say that the Fourier series of F converges to f at x if f(x) = lim

N →∞S_N(x).

Theorem 5. Let f be a continuous and 2π-periodic function. For any x where f^′(x) is defined,

N →∞lim S_N(x) = f (x).

Theorem 6. Let f be a periodic and piecewise continuous function. Suppose that x is a point such that f^′(x⁻) and f^′(x⁺) are defined. Then

N →∞lim S_N(x) = f(x⁺) + f (x⁻)

2 .

• See Examples 1.26, 1.27, and 1.29.

Uniform Convergence: We say that the Fourier series of f converges to f uniformly if S_N(x) → f(x) uniformly with S^N in (8).

Definition 2. A function f is called piecewise smooth if it is continuous and its derivative is defined everywhere except possibly for a discrete set of points.

Theorem 7. Let f be a 2π-periodic and piecewise smooth function. Then the Fourier series of f converges to f uniformly on [−π, π].

• See Examples 1.31-1.32.

Convergence in Mean:

Let

V_N = span{1, cos kx, sin kx : k = 1, 2 · · · , N}

Suppose that f ∈ L²[−π, π] and let f_N(x) = a0+

N

X

n=1

³

a_ncosnπ

L x+ bnsinnπ L x´

. (9)

Clearly, fN is the orthogonal projection of f onto the space VN, i.e., kf − f^Nk = min_g∈V

Nkf − gk.

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Theorem 8. Suppose that f ∈ L²[−π, π] and let f^N be given as in (9). Then fN → f in L²[−π, π], i.e., kf^N − fk² → 0 as N → ∞.

Theorem 9. Suppose that f ∈ L²[−π, π] and let f_N(x) =

N

X

n=−N

c_ne^inx with an = 1 2π

Z π

−π

f(x)e^−inxdx.

Then fN → f in L²[−π, π], as N → ∞.

• See Examples 1.37.

Parseval’s Equation:

• (Real Version) Suppose that f(x) = a0 +

∞

X

k=1

(akcos kx + bksin kx) ∈ L²[−π, π]. (10) Then

1 π

Z π

−π|f(x)|²dx= 2|a⁰|²+

∞

X

k=1

(|a^k|²+ |b^k|²).

• (Complex Version) Suppose that f(x) = X

k∈Z

α_ke^ikx ∈ L²[−π, π].

Then

1 2π

Z π

−π|f(x)|²dx= 2|a⁰|²+

∞

X

k=1

|α^k|². Moreover, for any f, g ∈ L²[−π, π],

1

2π < f, g >= 1 2π

Z π

−π

f(x)g(x)dx =

∞

X

n=−∞

α_nβ_n where

g(x) = X

k∈Z

β_ke^ikx

• See Examples 1.41.

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