Wikipedia:Reference desk/Archives/Mathematics/2007 April 19

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

April 19

More Hölder inequality fun

I have another question that relates to Hölder's inequality. This is from a qualifying exam a year ago (that I failed, unfortunately).

Suppose $f\in L^{p}(0,\infty )$ for some $p\in (1,\infty )$. We have two statements to prove:

1. Show that $\left|\int _{0}^{x}f(t)\ dt\right|\leq ||f||_{p}\ x^{1-1/p}$ for all x > 0.
2. Show that $\lim _{x\to \infty }{\frac {1}{x^{1-1/p}}}\int _{0}^{x}f(t)\ dt=0$.

Now, I can do number 1. Let 1/q = 1 - 1/p, and note that for any x > 0, the function $g(x)=1$ is in $L^{q}$. Use Hölder's inequality with the original f and this g, and you have the result. (Since g(x) is a constant function, the Lebesgue and Riemann integrals agree, so it's not hard to get the right hand side.)

For number 2, I'm just at a loss. I feel like I want to use number 1, and have some squeeze theorem result or some such, but I can't get anywhere. Any hints would be greatly appreciated. (Again, this is not homework, this is just a problem from a qual that I could never do.) Thanks in advance! –King Bee ^{(τ • γ)} 14:21, 19 April 2007 (UTC)

Ooh, I just got an idea. How about this: Number 1 gives you that $f\in L^{1}$. Note that for all x, $\int _{0}^{x}f\leq \int _{0}^{x}|f|$. Now let $a_{n}$ be any sequence such that $a_{n}\to \infty$ with n. Set $f_{n}=|f|\chi _{(0,a_{n})}$. Then $\lim _{x\to \infty }\int _{0}^{x}|f|=\lim _{n\to \infty }\int _{\mathbb {R} }|f|\chi _{(0,a_{n})}$. Since $f_{n}\leq |f|$ and $f_{n}\to |f|$, we can apply the monotone convergence theorem to get $\lim _{x\to \infty }\int _{0}^{x}|f|=\lim _{n\to \infty }\int _{\mathbb {R} }|f|\chi _{(0,a_{n})}=\int _{0}^{\infty }|f|=||f||_{1}$. So this limit exists and is finite, and since $\lim _{x\to \infty }{\frac {1}{x^{1-1/p}}}=0$, the limit in question is 0.

Commentary. I see no flaw here. Can someone step in and let me know if this looks good to them as well? Thanks! –King Bee ^{(τ • γ)} 14:39, 19 April 2007 (UTC)

(I suppose I should add that $\lim _{x\to \infty }{\frac {1}{x^{1-1/p}}}\int _{0}^{x}f(t)\ dt\leq \lim _{x\to \infty }{\frac {1}{x^{1-1/p}}}\int _{0}^{x}|f(t)|\ dt$, since I didn't state that explicitly.) –King Bee ^{(τ • γ)} 14:42, 19 April 2007 (UTC)

Unfortunately, part 1 does not show that $f\in L^{1}$. (It does show that it is in $L_{\mathrm {loc} }^{l}$ but that is not good enough for the rest of the argument.) Stefán 18:20, 19 April 2007 (UTC)

You are correct. It shows $f\in L^{1}(0,x)$ for all x. Any hints on getting around this snafu? –King Bee ^{(τ • γ)} 19:55, 19 April 2007 (UTC)

Take the quantity ${\frac {1}{x^{1-1/p}}}\int _{0}^{x}f(t)\ dt$ to the p-th power and look at the integral $\int _{0}^{\infty }\left({\frac {1}{x^{1-1/p}}}\int _{0}^{x}f(t)\ dt\right)^{p}{\frac {1}{x}}dx$. Then read the article on Hardy's inequality. Stefán 19:10, 20 April 2007 (UTC)

Ahh, thanks for the help. I was really hoping for something a little more obvious; I have never even heard of Hardy's inequality. Thanks! –King Bee ^{(τ • γ)} 22:05, 22 April 2007 (UTC)

Significant_figures

I am calculating a percentage of work items cleared on any particular day. For example if I had cleared 319 out of 405 work items then the ratio cleared would be 0.787654321. Now I know from the Significant_figures article and the Spurious Accuracy reference that I should round my result to 3 significant digits since .788, or 78.8%. My question is this: I know that there is a reporting error of around 2-3% for the number of work items and number of cleared work items. Would this reduce my allowed significant digits to 2 or would I still be allowed to use 3 significant digits? How exactly does the margin of error in my initial mesurements effect the significant digits? Thanks, -Czmtzc 14:34, 19 April 2007 (UTC)

You should give the number of significant digits that is supported by the accuracy of your input values - 3 significant digits is only a rough guide. In you case, you say you have uncertainties of up to 3% in each of your two input figures, so your total uncertainty could be up to 6%. In other words, 319 out of 405 could in fact be anywhere from 314 out of 411 (76.4%) to 324 out of 399 (81.2%). To correctly reflect this level of accuracy, I suggest you round your results to the nearest 5% - so 78.8% would be rounded up to 80%, whereas 76.2% would be rounded down to 75%. Gandalf61 14:54, 19 April 2007 (UTC)

I agree with what Gandalf said (although the two errors aren't strictly additive, but it's close enough for two low error rates around 2-3%). However, there was an error in what you said, Czmtzc. You said that, even if there was no error, you should round to three significant digits. While that would be true if dealing with real numbers given to 3 significant figures, integers are different. In the case where an exact count is known, this is infinite accuracy. So, you can report as many significant digits as you like. If 2 of 3 cars are running, I don't have to round that to 70%, I can say 67%, or 66.7%, or 66.66666666666667%, if I prefer. StuRat 19:40, 19 April 2007 (UTC)

I see what you are saying StuRat about exact counts. That is why I was careful to mention that I didn't have perfect confidence in the source data, so I also appreciate the answer you gave Gandalf61. Thanks to All. --Czmtzc 20:38, 19 April 2007 (UTC)

You're welcome. StuRat 04:36, 20 April 2007 (UTC)

Your question highlights the problem of using the number of figures to inform the reader about accuracy. As you might have realised, it works flawlessly only when the number that is the center of the interval has a finite decimal expansion (in the base you are using) and changing the last of those figures by one takes you the edges of the interval. In all other cases, you'll have to throw away some accuracy or pretend to have more of it than you actually do. :-( And, of course, it might break down even more if an interval isn't a good description of the uncertainty in the first place. —Bromskloss 12:13, 20 April 2007 (UTC)