3
votes
1answer
56 views

$L^1$ norm of convolution

Let $f_{\lambda} = \frac{\lambda}{2}e^{-\lambda |x|}$. Prove that $||f_{\lambda} \ast g - g || \to 0$, when $\lambda \to \infty$, where $g \in L^1$
1
vote
1answer
53 views

Show that $\int_{\mathbb R^n}e^{|x|^{-n}}dx=$ Volume of n-sphere

I'm preparing for a calculus exam, I'd like help in solving this question. Let $x \in \mathbb R^n$, $|x|={(x_1^2+x_2^2+...+x_n^2)^{\frac{1}{n}}}$, Show that $$\int_{\mathbb R^n} e^{|x|^{-n}}dx$$ is ...
5
votes
1answer
74 views

Is there a useful relationship between pointwise and $L^2$ distance?

It would be really convenient to get a bound on the point-wise closeness of functions by knowing their $L^2$ distance. Clearly, if two functions are close in the $L^2$ sense, you cannot get a general ...
0
votes
0answers
57 views

limit of p norm as p goes to 0!

Suppose we have a measure $\mu$ and a space $X$ such that $\mu(X)=1$, and a function $f \in L^r$ for some $r > 0$, where $L^r$ is defined in the usual way even for numbers less than $1$. Show ...
0
votes
1answer
64 views

proofread $\left \| f -g\right \|_{p}^{p}=p(p-1)\int_{0}^{\infty}\int_{0}^{t}[m(f>t/g>s)+ m(g>t/f>s)]|t-s|^{p-2}dsdt$

Is this argument correct? $\left \| f -g\right \|_{p}^{p}=$ $\int_{0}^{\infty}m(|f-g|>t^{\frac{1}{p}}) dt\stackrel{c.o.v.}{=}$ $p\int_{0}^{\infty}m(|f-g|>t) t^{p-1}dt\stackrel{t=|u-s|}{=}$ ...
2
votes
1answer
45 views

Continuous functions. Second norm

Let $f:[a,b]\rightarrow \mathbb{R}^{d}$ be continuous. I need to prove that $\left \| \int_{a}^{b}f(x)dx \right \|_{2}\leq \int_{a}^{b}\left \| f(x)) \right \|_{2}dx$.
0
votes
1answer
37 views

Integral of a function is not affected by altering the function values at zero-measure set

I'm studying about Fourier analysis from a book Fourier analysis and its applications, Folland 1992 and I have one point in the source I need clarification about: On page 69 it is stated that: "The ...
0
votes
1answer
34 views

Integral like a norm instead of sum

A Rienmann integral is defined as: $\int_a^b f(x)\ dx=\displaystyle\lim_{n\rightarrow\infty}\displaystyle\frac{1}{n}\displaystyle\sum_{i=0}^{n}f\left(\displaystyle\frac{b-a}{n}\cdot i+a\right)$ I ...
1
vote
1answer
53 views

Hints on calculating an integral

Suppose $x(t)$ and $y(t')$ are curves traversing the boundary of $[0,1]^2$ in $R^2$ counterclockwise. What is the integral of the following: $$\int\limits_{t,t'}{\frac{dt\,dt'}{\|x(t)-y(t')\|}}$$ I ...
2
votes
0answers
156 views

Step functions dense in Integrable functions with respect to $L_2$

Let $I$ be a bounded interval. Prove that $\{\text{step functions }I \to C\}$ is dense in $\{\text{integrable functions }I \to C\}$ (Riemann Integrable) with respect to $\|.\|_2$ ($L_2$ norm)
3
votes
1answer
134 views

Bizarre formula for arc length

I'm reading a book on computer science/math and I found this formula for arc lengths that I've not been able to decipher: $$\left|\int_p^q\left\| {df(x)\over dx} \right\| dx\right|$$ where $\lVert ...
1
vote
1answer
236 views

Looking for an inequality related to the Cauchy-Schwarz inequality

From the Cauchy-Schwarz inequality, we can prove that $$\lVert w(x)\rVert^2_{L^2_{[0,1]}}=\int_0^1 w(x)^2\, dx \leq \sqrt{\int_0^1 w(x) \,dx}\cdot \sqrt{\int_0^1 w(x)^3\, dx}.$$ Is it possible to ...
1
vote
2answers
1k views

Inequality involving norm of matrix integral

This question seems basic but I could not find an answer. I have seen the inequality $$\left\|\int_a^b x(t) dt \right\| \leq \int_a^b \left\| x(t) \right\| dt $$ where $x(t) \in \mathbb{R}^n$ is a ...