5
votes
1answer
205 views

Fourier expansion of Takagi-function (everywhere non differentiable function).

Let us consider Takagi-function defined by $T(x) \colon\!= \sum_{n=0}^{\infty}s(2^nx)/2^n$, where $s(x) \colon\!\!= \underset{n \in {\Bbb Z}}{\mathrm{min}} \,|x-n|$. $T(x)$ has its period $1$, so ...
1
vote
0answers
78 views

Decay rate of Discrete Prolate Spheroidal Sequences in frequency

What is the decay rate of DPSS sequences in frequency? Consider an interval $T\subset\mathbb{Z}$ of length N in time. Consider another interval $[-W,W]$ in frequency with $W<1/2$. Let $\phi_0$ ...
2
votes
0answers
346 views

L^1-convergence of convolution exponential

Consider a differential equation \begin{eqnarray*} \frac{d}{d\tau}q^{\tau}\left(x\right)=Aq^{\tau}\left(x\right)+h\star q^{\tau}=Aq^{\tau}\left(x\right)+\int ...
21
votes
9answers
12k views

Is square of Delta function defined somewhere?

Hello, every one. I am wondering whether any one knows that whether the square of Dirac Delta function is defined some where? In the beginning, this question might look strange. But by restricting ...
-2
votes
1answer
559 views

General solution to ODE [closed]

Considering the following ODE : find $f(x)$ such that $$\frac{\sigma^{2}}{2}\frac{d^2}{dx^2}f(x)+a(b-x)\frac{d}{dx}f(x)-(\rho+\lambda)f(x)=-\lambda g(x) $$ Where, ...
2
votes
3answers
795 views

Posing cauchy data for the heat equation: $t=0$ a characteristic surface?

When solving the heat equation on say $\mathbb{R}$ (or $[0,2\pi]$, whichever is easier to talk about) we are posing Cauchy data on the surface $t=0$. My understanding is that $t=$constant are ...
3
votes
1answer
1k views

When can I use Fourier Series to solve the heat/wave equation on $[0,L]$?

Hi everone, Ok so to begin with I know that for Dirichlet, Neuman and Robyn boundary conditions you can use the method of fourier series to solve the heat and wave equations given Cauchy data on ...
1
vote
2answers
275 views

Using Wavelet Transforms to Approximate Matrices

It's a long time since I worked on this kind of problem, so please bear with me. I have an approximate inverse matrix that I'm using as a preconditioner to solve the conjugate gradient method. ...
7
votes
1answer
481 views

Motivation for BMO

At the moment, I don't have access to the early 1960's paper of John and Nirenberg that (from what I understand) introduced the space BMO (bounded mean oscillation). Why were John and Nirenberg ...
5
votes
2answers
286 views

Are the asymptotics of Fourier coefficients to periodic solutions of ODE known?

The Van der Pol equation, given by $$x'' + x = g x' (1 - x^2),$$ has periodic solutions $x(t)$, with the period $T(g)$ depending on the parameter. Thus, one can expand $x(t)$ as a Fourier series ...
3
votes
2answers
510 views

Ansätze for solving PDEs with wavelets

It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features. My question: Which Ans├Ątze do you know ...