Analysis Problem

Question

Given that: For any $[a, b]\subset (-\infty,+\infty)$, $f$ is integrable in $[a,b]$, $p>0$, and ${\mid f\mid}^{p}$ is integrable in $(-\infty,+\infty)$.

Prove that

$$\lim_{h\to0}\int_{-\infty}^{+\infty }\mid {f(x+h)-f(x)}\mid ^{p} dx=0 $$

What have you attempted yourself? – Joel Jun 28 '16 at 14:26 — Joel, Jun 28 '16 at 14:26
Did you intend to include $p \in (0,1)$? – copper.hat Jun 28 '16 at 14:37 — copper.hat, Jun 28 '16 at 14:37

score 1 · Answer 1 · answered Jun 28 '16 at 18:15

1

As stated in the appendix of these lecture notes, the set of compactly supported functions is dense in $L^p$ also for $0<p<1$. Hence the usual proof applies: the property is obvious if the support of $f$ is bounded, then we approximate $f$ with compactly supported continuous functions and ewe conclude.

answered Jun 28 '16 at 18:15

Siminore

35,136

I haven't learnt functional analysis yet. This exercise is in improper integral. So I think there is a method to solve the problem without using functional analysis. – mse Jun 29 '16 at 03:47

score 0 · Answer 2 · edited Jun 28 '16 at 16:32

0

Hint: Put $h = \frac 1 n$ and then use Lebegue's dominated converge theorem.

edited Jun 28 '16 at 16:32

Alex M.

35,207

answered Jun 28 '16 at 15:31

Giovanni Siclari

183

1

This does not provide an answer to the question. Once you have sufficient reputation you will be able to comment on any post; instead, provide answers that don't require clarification from the asker. - From Review – Qwerty Jun 28 '16 at 15:50
1

@Qwerty: Hints are allowed as answers. In fact, the policy on MSE is to use comments only for clarifications and it is accepted to provide hints instead of full answers. (It is true that fewer and fewer people obey these rules - including me -, but it is good for you to know them.) In particular, this means that the above post does qualify as an answer. – Alex M. Jun 28 '16 at 16:34

Analysis Problem

2 Answers2