首页 Fourier Analysis

Refers to Fourier Analysis by Stein

# Applications to some partial differential equations

# The time-dependent hear equation on the real line

Definition 5.2.1 The following patrial differential equation is called the heat equation

∂u∂t=∂2u∂x2\dfrac{\partial u}{\partial t}=\dfrac{\partial ^2u}{\partial x^2}

The initial condition we impose is u(x,0)=f(x)u(x,0)=f(x).

# The Poisson summation formula

Definition Given a function f∈S∈(R)f\in \mathcal S\in (\mathbb R) on the real line, we can construct a new function on the circle by follow. The function F1F_1 is called the periodization of ff

F1(x)=∑n∈Zf(x+n)F_1(x)=\sum_{n\in\mathbb Z}f(x+n)

There is another way to arrive at a "periodic version" of ff by Fourier analysis

F2(x)=∑n∈Zf^(n)e2πinxF_2(x)=\sum_{n\in\mathbb Z}\hat f(n)e^{2\pi inx}

Remark

F1,F2F_1,F_2 is periodic and continuous, for both of the sum converges absolutely and uniformly.

Theorem (Poisson summation formula) If f∈S(R)f\in\mathcal S(\mathbb R), then

∑n∈Zf(x+n)=∑n∈Zf^(n)e2πinx\sum_{n\in\mathbb Z}f(x+n)=\sum_{n\in\mathbb Z}\hat f(n)e^{2\pi inx}

In particular

∑n∈Zf(n)=∑n∈Zf^(n)\sum_{n\in\mathbb Z}f(n)=\sum_{n\in\mathbb Z}\hat f(n)

Proof

Tips: Fourier coefficient

For both of them are continuous.

# The Heisenberg uncertainty principle