Recent content by tom_rylex

This class is making my head hurt. I could use some help. Homework Statement Show that if f(x) is a function of slow growth on the real line, \lim}_{\substack \varepsilon \rightarrow0^+} \langle f(x)e^{- \varepsilon |x|}, \phi (x) \rangle = \langle f, \phi \rangle where \phi (x) is a test...

Homework Statement I am looking at the derivation of the D'alembert equation, and I'm having trouble with understanding where the limits of integration come in. Homework Equations Given the 1-d wave equation: u_{tt} = c^2u_{xx} , with the general solution u(x,t)= \theta(x-ct) +...

My set of test functions meet the following criteria: * function has a finite region of support, inside of which \theta(x) \neq 0 , outside of which \theta(x)=0 * \theta(x) has derivatives of all orders.

Homework Statement Show that if {x_k} is any sequence of points in space R^n with |{x_k}| \rightarrow \infty , then \delta(x-x_k) \rightarrow 0 weakly Homework Equations The Attempt at a Solution I'm still trying to grasp the concept of weak convergence for distributions. It...

As in: <fn-f,g> --> 0 (absolutely convergent series are convergent) <fn,g> - <f,g> --> 0 (linearity property wrt the first variable for inner products)

Homework Statement (I'm posting this because my proofs seem to be lousy. I want to see if I'm missing anything.) Show that if f_n \in L^2(a,b) and f_n \rightarrow f in norm, then <f_n,g> \rightarrow <f,g> for all g \in L^2(a,b) Homework Equations L^2(a,b) is the space of...

Homework Statement Prove that if f(x) is Holder continuous, i.e, \sup_{a<x , y<b} \frac{\abs{f(x) - f(y)}}{\abs{x-y}^\alpha} = K^f_\alpha<\inf with \alpha > 1 , then f(x) is a constant function Homework Equations The Attempt at a Solution I've been staring at this for a...

[Solved] Linear Differential Operator order Thanks. I just needed a nudge to get in the right direction. Some parts of DiffEq have been a while for me. I set up the linear differential operators L = \frac {d}{dx} M = \frac {d}{dx} + x to show that L(M(u)) \neq M(L(u)) for all u. I...

[Solved] Linear Differential Operator order Homework Statement I'm misunderstanding something basic about how this works: Give examples of linear differential operators L and M for which it is not true that L(M(u)) = M(L(u)) for all u. Homework Equations Since it's arbitrary, I...