Recent content by laonious

Thank you for the insight, it was very helpful! To apply dominated convergence I think we have to be careful because f isn't integrable, but I think we can say something like \int_a^{\infty} f \leq \sum_i \int_{a_i}^{a_{i+1}}\frac{1}{x^2}+\frac{\epsilon}{2^i}\,dx<\infty.

Hi all, Let f:[0,\infty)\rightarrow\mathbb{R} be a bounded measurable function such that \lim_{x\rightarrow\infty}x^2f(x)=1. Find an integral expression for \lim_{\lambda\rightarrow 0^+}\frac{\int_0^{\infty}(1-\cos(x))f(\frac{x}{\lambda})dx}{\lambda^2}. This one is really bizarre to me...

namphcar, Thanks for the great response, it was quite helpful. Would you mind being a little more explicit in your last step? It's not clear to me that \left(1-f(0)+\frac{||f''||_{\infty}}{2}\right)^2\leq 4, since a function of the form e^{kx} would have an arbitrarily large second...

Hi all, I'm really banging my head on this problem: Let f be a real-valued measurable function on the measure space (X,\mathcal{M},\mu). Define \lambda_f(t)=\mu\{x:|f(x)|>t\}, t>0. Show that if \phi is a nonnegative Borel function defined on [0,infinity), then...

I think the fundamental issue is that the notion of area doesn't really make sense here. If you're dealing with a single complex variable, then the space is 1-dimensional over C. So the only way to get an idea of area is to treat it as R^2 and look at z=x+iy, but then you've changed the...

Sorry, here's a fixed-up rewrite.

Hello, I would appreciate any assistance with the following question: Suppose f \in C^2[-1,1] is twice continuously differentiable. Prove that |f'(0)|^2 \leq 4 ||f||_\infty (||f||_\infty + ||f''||_\infty), where ||f||_infty is the standard sup norm. At first I thought Taylor expansion...

Wow, that's clever breaking it up into a finite and an improper integral. I'd tried using the mean value theorem but wasn't sure how to show integrability. Thanks for your help!

Hi everyone, I am studying past analysis prelim exams to take in the fall and have run into one which really has me stumped: Let f be a real-valued Lebesgue integral function on [0,\infty). Define F(x)=\int_{0}^{\infty}f(t)\cos(xt)\,dt. Show that F is defined on R and is continuous on R...