Fourier Analysis

Fourier Coefficient

Suppose that

f \in L^{1} ([- π, π])

and

n \in ℤ

. The $n$ -th Fourier coefficient of

f

\hat{f} (n) : = \frac{1}{2 π} \int_{- π}^{π} f (x) e^{- i n x} d x .

Fourier Series

Suppose that

f \in L^{1} ([- π, π])

. The Fourier series of

f

is the formal series

\sum_{n \in ℤ} \hat{f} (n) e^{i n x} .

Dirichlet Kernel

For

n \in ℤ^{+}

, the Dirichlet kernel is

D_{n} (x) : = \sum_{k = - n}^{n} e^{i k x} .

Fejer Kernel

For

n \in ℤ^{+}

, the Fejer kernel is

F_{n} (x) : = \frac{1}{n} \sum_{k = 0}^{n - 1} D_{k} (x) .

Riemann-Lebesgue Lemma

Suppose that

f \in L^{1} ([- π, π])

. Then

\lim_{| n | \to \infty} \hat{f} (n) = 0.

Fejer's Theorem

Suppose that

f

is continuous and

2 π

-periodic. Then the Cesaro means of the Fourier series of

f

converge uniformly to

f

Fourier Series Converges in

L^{2}

Suppose that

f \in L^{2} ([- π, π])

. Then the partial sums of the Fourier series of

f

converge to

f

L^{2}

Plancherel Identity for Fourier Series

Suppose that

f \in L^{2} ([- π, π])

. Then

\frac{1}{2 π} \int_{- π}^{π} {| f (x) |}^{2} d x = \sum_{n \in ℤ} {| \hat{f} (n) |}^{2} .