1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Uniform convergence for heat kernel on unit circle

  1. Mar 10, 2013 #1
    1. The problem statement, all variables and given/known data

    I would like to use the Weierstrass M-test to show that this family of functions/kernels is uniformly convergent for a seminar I must give tomorrow.


    H_{t} (x) = \sum ^{-\infty}_{\infty} e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} .


    2. Relevant equations

    3. The attempt at a solution

    I just need to find a sequence of positive numbers that will always be greater than the heat kernel Ht(x) for all x of course. But must it be greater than or equal to Ht(x) for all t as well? That being said, it might prove difficult to find an appropriate sequence...

    Can I include t in my sequence of positive numbers? It might make it easier. At first I was just thinking of something as simple as (15/16)^n...if someone could guide me in the right direction, I would appreciate it, thanks!
  2. jcsd
  3. Mar 11, 2013 #2
    Does this work:

    Using the Weierstrass M-Test, we'll consider the sequence of terms

    [itex]| e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} | = \frac{| e^{2 pi i n x} |}{e^{-4 \pi ^{2} n^{2} t}}[/itex]

    Notice that on a ring of radius 1, for all time greater than or equal to zero, this sequence will be largest when x = 1 and t = 0. Obviously when t = 0, the sequence does not converge. But we are not concerned at time t=0 since we are given initial condition u(x, 0) = f(x).

    We wish to find a sequence of terms, [itex]M_{n}[/itex], so that [itex]| e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} | \leq M_{n}[/itex] for all n.

    So we have

    [itex]| e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} | = | \frac{ e^{2 \pi i n x} }{e^{4 \pi ^{2} n^{2} t}} | = \frac{| e^{2 \pi i n x} |}{e^{4 \pi ^{2} n^{2} t}} \leq \frac{| e^{2 \pi i n} |}{e^{4 \pi ^{2} n^{2} t}} = \frac{1}{e^{4 \pi ^{2} n^{2} t}} = M_{n}.[/itex]

    My question is, since the heat kernel on the circle is defined as a variable of x, [itex]H_{t}(x),[/itex], can my [itex]M_{n}[/itex] include the "variable" t? Will this allow me to use the M-test to show uniform convergence? I ask because uniform convergence cannot depend on the variable x. But can it depend on t?

    Is my argument above correct? If someone could just have a quick read over what I wrote and tell me where I have gone wrong, I would really appreciate it.
    Last edited: Mar 11, 2013
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted