Wikipedia:Reference desk/Archives/Mathematics/2011 July 1

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July 1

Fourier series

I was reading about Fourier series and have a doubt concerning it. The book I am reading from does not seem to help. As I understand, $\{e_{0}={\frac {1}{\sqrt {2}}},e_{1}=sin(x),e_{2}=cos(x),e_{3}=sin(2x),e_{4}=cos(2x)\cdots \}$ is a basis for the inner product space of piecewise continuous functions in $[-\pi ,\pi ]$ with inner product $\langle f,g\rangle ={\frac {1}{\pi }}\int _{-\pi }^{\pi }f{\bar {g}}$. Hence any function in this space may be represented by $f=\sum _{k=0}^{\infty }\langle f,e_{k}\rangle e_{k}$. My question is what happens at points of discontinuity x. As f is identical with the series $f=\sum _{k=0}^{\infty }\langle f,e_{k}\rangle e_{k}$ (which by the way is unclear to me as to why it coverges) shouldn't f(x) be identical with the series at x, i.e. $f(x)=\sum _{k=0}^{\infty }\langle f,e_{k}\rangle e_{k}(x)$. But Dirichlet's theorem (stated without proof in my book) says that at points of discontinuity, the series $\sum _{k=0}^{\infty }\langle f,e_{k}\rangle e_{k}(x)$ converges to ${\frac {f(x-)+f(x+)}{2}}$ and not to f(x). Why is this so? Thanks-Shahab (talk) 01:41, 1 July 2011 (UTC)

It comes from the proof of Foruier Series itself. Where do the coefficients come from? Why do they involve integrals? If a function satisfies Dirichlet's conditions then its Fourier series converges to

${\frac {1}{2}}\left(\left[\lim _{x\to a^{-}}f(x)\right]+\left[\lim _{x\to a^{+}}f(x)\right]\right)$

for all $a\in [-L,L].$ If the function is continuous then both summands are equal and it converges to the value. If it's not continuous at a then it converges to the average. I suppose that that's the fundamental thing to realise; that it always converges to the average of the two limits. — Fly by Night (talk) 02:34, 1 July 2011 (UTC)

I think you misunderstood my post. Forget about Dirichlet's conditions for the moment. I claim that the coefficients are $\langle f,e_{k}\rangle$ from linear algebra and that they involve integrals by definition of my inner product. Now I hope my question is clear.

Also, what kind of convergence do we generally have in case of infinite bases?-Shahab (talk) 02:59, 1 July 2011 (UTC)

I don't think I misunderstood. I just try to suggest a way to answer the question that you asked. Why should they involve integrals? Why should the series have any meaning whatsoever? Does your book have a section that shows where the Fourier series comes from, and why it is defined the way it is? 03:17, 1 July 2011 (UTC)

That inner product can't distinguish between two functions that only differ at a single point (or more generally at any set of measure zero). If f₁ and f₂ are two square-integrable functions that are equal except at the point x₀, then $\langle f_{1},g\rangle =\langle f_{2},g\rangle$ for any square-integrable function g. It should be clear that f₁ and f₂ have the same Fourier series. We can't reasonably expect that Fourier series to converge at x₀ to both f₁(x₀) and to f₂(x₀). To maybe rephrase the point made by Fly by Night, the Fourier series can't "see" what happens at a specific point, only what happens in the vicinity of a point, because that's how integrals work and the inner product is defined in terms of integration. Rckrone (talk) 03:15, 1 July 2011 (UTC)

I think that the space of piecewise continuous functions in [ − π,π] is not inner-product, since it has nonzero elements with a norm of 0. So $f=\sum _{k=0}^{\infty }\langle f,e_{k}\rangle e_{k}$ needn't apply. -- Meni Rosenfeld (talk) 05:03, 1 July 2011 (UTC)

Also piecewise continuous functions in [−π, π] are not necessarily square-integrable, so the Fourier coefficients may not even be well-defined. Rckrone (talk) 05:41, 1 July 2011 (UTC)

Hmm...I think the book is wrong for it explicitly states that the space of piecewise continuous functions is inner product.-Shahab (talk) 12:24, 1 July 2011 (UTC)

You have weak convergence. Count Iblis (talk) 15:12, 1 July 2011 (UTC)

Actually, it's strong convergence in the Hilbert space L². A relevant article is convergence of Fourier series. 15:30, 1 July 2011 (UTC)

Yes, of course. My brain malfunctioned :) .Count Iblis (talk) 16:14, 1 July 2011 (UTC)

@Rckrone: Correct me if I am wrong, but I believe all piecewise continuous functions are (Riemann) integrable and so the Fourier coefficients will all exist-Shahab (talk) 06:41, 2 July 2011 (UTC)

I made a mistake when I said the functions needed to be sqaure-integrable, rather than integrable, for the Fourier coefficients to be defined. But here's an example of a piecewise continuous function that's not integrable: let f(x) = 0 for -π ≤ x ≤ 0, and f(x) = 1/x for 0 < x ≤ π. Rckrone (talk) 15:13, 2 July 2011 (UTC)

Your example involves an infinite discontinuity. My definition of piecewise continuous (from the book of G.F. Simmons) says that only jump discontinuity is permitted. Is there an example involving jump discontinuity?-Shahab (talk) 12:39, 3 July 2011 (UTC)

Oh ok. I think in that case the function must be bounded so the integral will converge. Suppose f is not bounded, then there's a sequence of points x₁,x₂,... with lim_i f(x_i) = ±∞. Since this sequence is on a compact set it has a subsequence that converges to a point x and then there's an "infinite discontinuity" at x. Rckrone (talk) 17:17, 3 July 2011 (UTC)

Montenegro football team

The Montenegro national football team currently ranks 16th in the FIFA World Rankings with 915 points, as of 29 June. They were 24th on 18 May, but drew a match against Bulgaria, which helped them climb 8 places up. I was curious about what would have happened if Montenegro had won instead, and by following what is there at the FIFA World Rankings article I tried to calculate it. My result shows that with a win over Bulgaria, Montenegro would currently rank second in the world with 1685 points, only behind Spain. Could someone confirm or deny this? I wasn't familiar with the calculation method of the ranking system and only read it from the article, and I might have misunderstood or miscalculated something (because I'm not too good at mathematics). --Theurgist (talk) 17:53, 1 July 2011 (UTC)

Just by looking at the numbers it seems wrong. Drawing against Bulgaria gave them a total of 915 points, while a win would have given them a total of 1,685 − that's almost double. The Fifa rankings are calculated over a four year period of time. So your numbers suggest that they would get more extra points from turning a draw into a win than they have made in all of their previous games over the last four years. The position in the list is affected by how other teams move around; the fact is that many of the teams won't have played. They had 820 points in May, a draw gave them 915, i.e. an extra 95 points. You said they'd have 1,685 points, i.e. an extra 865 points. Are you saying a draw gets 95 points and a win gets 865? — Fly by Night (talk) 19:23, 1 July 2011 (UTC)

But notice that more recent games are evaluated higher than less recent ones. Notice also the "Win, draw or defeat" section, which explains that a win multiplies the point earnings form a game by three, a draw multiplies it by one, and a loss multiplies it by zero. This means that it makes no difference whether a team loses a game or doesn't play at all. Therefore, if a team doesn't play or constantly loses, its record will be gradually decreasing until it reaches zero on completing four years without a win or draw.

Here are the calculations I made, based entirely on the article.

Montenegro had 820 points prior to the match with Bulgaria. Had they lost, they would now have fewer than 820 points; I didn't try to calculate how many exactly.

All other teams would have still the same number of points that they actually have (except Bulgaria, but this makes no difference).

The match status was a continental qualifier, so the multiplier is 2.5.

Bulgaria was ranked 46th at the time of the match, so the opposition strength multiplier is (200 − 46) ÷ 100 = 1.54.

Both teams are European, so the regional strength multiplier is 1.

The draw gave Montenegro 100 × 2.5 × 1.54 × 1 = 385 points, to form their current amount of 915 points. If they had won, such a result would have given them three times of that. So the points that they actually failed to get is an extra 2 × 385 = 770 points.

And finally, adding up the current FIFA rankings record of the Montenegro team to the extra record they could have earned but didn't actually earn gave me 915 + 770 = 1685 points, which made the team occupy the second position, narrowly overtaking the Netherlands.

Could someone verify those calculations? --Theurgist (talk) 22:37, 1 July 2011 (UTC)

The key thing that you don't seem to include is as follows: "Teams' actual scores are a result of the average points gained over each calendar year…" So, if they scored zero, zero, zero and 1,000 points in four games in one year, they would be credited with 250 points to their score. The influences of the previous years' average tails off by the factors you mention. Look at the tables that you linked to. They had 820 points before the draw and 915 afterwards. Your calculations above say they should get and extra 385 points for the draw, when in reality they got an extra 95 points. It should be clear that you'd made a mistake by applying the rule of common sense. How could (or should) Montenegro beating Bulgaria put them second in the world, ahead of the Netherlands that were runners up in the FIFA world cup? — Fly by Night (talk) 23:30, 1 July 2011 (UTC)

Yeah, indeed, I must have overlooked that somehow. So the team's score for the past year is the average points gained per match played July 2010 through June 2011, right? That should mean that a team which plays a single match during the year against a strong opposition and wins it will have a better annual record than a team which plays and wins 10 matches, not all of which against such strong opponents. I had never before attempted to calculate the consequences of such an imaginary situation, and just tried to get familiar with and apply the calculation procedure as it was there in the article, out of curiosity. My result shocked me, but I didn't know what could have gone wrong. It seemed both commonsensically wrong and mathematically right. I wasn't sure and that's why I asked here; if I had been sure, I wouldn't have asked. Thanks for the assistance :) --Theurgist (talk) 08:04, 2 July 2011 (UTC)

And furthermore, that key thing seems to prove that it actually does make a difference whether a team loses or doesn't play at all. --Theurgist (talk) 08:15, 2 July 2011 (UTC)

That's an obvious loop hole that I imagine will have been legislated for. I think there are rules to stop that. Teams will have to play a certain number of games per year. For example the FIFA match calender mentions dates and competitions, but no teams. You'll notice that when teams play internationals, there are often several other games going on, and when qualifying games for one regional cup, e.g. the Euros, then members of other regional associations, e.g. CONCAF, will play friendlies. — Fly by Night (talk) 19:18, 2 July 2011 (UTC)