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
ancosnπ
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
ancosnπ L x with the Fourier coefficients
a0 = 1 L
Z L 0
f(x)dx an = 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
bnsinnπ L x with the Fourier coefficients
bn = 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 : fe(x) :=½ f (x), if x∈ [0, L],
f(−x), if x∈ [−L, 0) (5)
Obviously,
f(x) = a0+
∞
X
n=1
ancosnπ
L x, x∈ [0, L]
with the Fourier coefficients a0 = 1
L Z L
0
f(x)dx an = 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 : fe(x) :=½ f (x), if x∈ [0, L],
−f(−x), if x∈ [−L, 0) Then
f(x) =
∞
X
n=1
bnsinnπ L x with the Fourier coefficients
bn = 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
√2πeinx : n ∈ Z}
is an orthonormal set in L2[−π, π].
Assume that
f(x) =X
n∈Z
cneinx. (6)
Then
cn= 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, an = 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
SN(x) = a0+
N
X
n=1
³
ancosnπ
L x+ bnsinnπ L x´
. (8)
We say that the Fourier series of F converges to f at x if f(x) = lim
N →∞SN(x).
Theorem 5. Let f be a continuous and 2π-periodic function. For any x where f′(x) is defined,
N →∞lim SN(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 SN(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 SN(x) → f(x) uniformly with SN 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
VN = span{1, cos kx, sin kx : k = 1, 2 · · · , N}
Suppose that f ∈ L2[−π, π] and let fN(x) = a0+
N
X
n=1
³
ancosnπ
L x+ bnsinnπ L x´
. (9)
Clearly, fN is the orthogonal projection of f onto the space VN, i.e., kf − fNk = ming∈V
Nkf − gk.
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Theorem 8. Suppose that f ∈ L2[−π, π] and let fN be given as in (9). Then fN → f in L2[−π, π], i.e., kfN − fk2 → 0 as N → ∞.
Theorem 9. Suppose that f ∈ L2[−π, π] and let fN(x) =
N
X
n=−N
cneinx 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
(|ak|2+ |bk|2).
• (Complex Version) Suppose that f(x) = X
k∈Z
αkeikx ∈ L2[−π, π].
Then
1 2π
Z π
−π|f(x)|2dx= 2|a0|2+
∞
X
k=1
|αk|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=−∞
αnβn where
g(x) = X
k∈Z
βkeikx
• See Examples 1.41.
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