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Swedish Mathematician Lennart Carleson Wins Abel
William Robinson writes "Sci Tech is reporting that Swedish mathematician Lennart Carleson has won the Abel Prize on Thursday for proving a 19th century theorem on harmonic analysis. His theorems have been helpful in creating iPod. Prof Carleson's major contributions have come in two fields - the first has subsequently been used in the components of sound systems and the second helps to predict how markets and weather systems respond to change. One of Carleson's many triumphs was settling a conjecture that had remained unsolved for over 150 years. He showed that every continuous function (one with a connected graph) is equal to the sum of its Fourier series except perhaps at some negligible points."
My math prof used to shoot chalk pieces onto me for saying something like that!
That's when I decided that statistics is more my kinda speed.
We used to have a Bill of Rights. Now, with the rights gone, all we have left is the bill.
Young people today. You tell them about a deep result in real analysis, and the only thing they're interested in is how it relates to their iPod. And get off my lawn.
Athletic Scholarships to universities make as much sense as academic scholarships to sports teams.
Wiki Article on the Breakthrough
"Sure there's porn and piracy on the Web but there's probably a downside too."
It would have been better if they had said "almost everywhere". That is, the set of points at which a Fourier series diverges has Lebesgue measure zero. There is a quasi-converse, due to the Israeli analyst Katznelson, which, given any set of Lebesgue measure zero -- let's call it Bill -- constructs a continuous function whose Fourier series diverges everywhere in Bill. For more info, see Tom Korner's excellent "Fourier Analysis".
The result he proved is nice mathematics, but you don't need it for iPods or audio coding. First of all, for many engineering purposes, it only matters that it works, not that you can prove that it works theoretically. Secondly, audio coding is done over discretely sampled signals, and most of those theorems become simple linear algebra in that case.
No wireless. Less decimals than pi. Lame.
Yeah, it might be that the MPEG-4/AAC/H.264 algorithms are based in part on Fourier analysis, but I fail to understand how Carleson's theorems have been used in making the iPod. Cupertino is hardly knowledgable in the more esoteric realms of theoretical mathematics, and there is simply no need to incorporate such stuff in an mp4 player.
This is bad journalism, written by bad reporters who lack the most basic understanding of mathematics and engineering. He just thought it might be cool to clam in an iPod in the mess.
The iPod reference is completely misleading, as simple harmonic analysis is way bigger than just an iPod. It's merely talking about this guy proving that Fourier was basically right, validating harmonic analysis and expanding the horizons for signal processing. That's the biggie: signal processing, not the bloody iPod. The stupid article probably includes iPod just for the sake of hits.
Oh really? Search Wikipedia entries, the articles, all links - no mention of iPod except in those annoying side adverts. Why? Because it has nothing to do with it
Credit where credit is due, and none is due here.
If you want credit, how about: Shannon, Fourier and Huffman. Then there's all the folks involved in working out noise masking and all the oddities of human hearing that I don't have the names of.
I seriously need a "No iPod mentions whatsoever" checkbox for my slashdot profile to pull some more signal out of the slashdot article noise.
one with a connected graph. For instance the function equal to sin(1/x) for x != 0 and 0 for x = 0 does have a connected graph but is NOT continuous.
Wondering why i am doing so strange posts? I am trying to get a "+5,Flamebait" or "-1,Insightful" rating.
...was published in the Swedish maths journal Acta Mathematica and is calles On the convergence and growth of partial sums of Fourier series.
Swedish plasma phys. PhD student; MSc EE; knows maths, programming, electronics; finance interest; seeks opportunities
This is sometimes mis-stated as 'you can draw the graph without taking your pen off the paper'.
That's not a mis-statement in the case of a real function of a real variable. It's not that informative, but definitely correct in the sense that a function (real etc.) is continuous iff the graph is path-connected (i.e. every two points on the graph can be connected by a continuous path (and by saying 'continuous path' I have of course made the definition self-referential and thus silly, but it is still true)).
Yes, you do recall correctly! ;-) Your version is a more general one. In a metric space, where your parent's version is applicable, the two are equivalent. Sorry, I probably shouldn't tell you this, you certainly knew it already. However, someone else might be interested.
Swedish plasma phys. PhD student; MSc EE; knows maths, programming, electronics; finance interest; seeks opportunities
The iPod reference got this story greenlighted on slashdot. Otherwise it might not have made it. If you want to guarantee acceptance, mention something bad about M$, something good about Linux, or anything about Apple (preferably good, but there is the odd flame article).
This advice was brought to you by someone with a 100% submission record. (ok, one of one ;-)
Your version is a more general one. In a metric space, where your parent's version is applicable, the two are equivalent.
Math geek warning!
Actually the equivalence goes much further than for metric spaces. In all topological spaces you have a sense of convergence of a sequence, and so it makes sense to ask the question "Does x_n->x imply f(x_n)->f(x)?". If f is continuous, the answer is always yes, but the converse need not be true in general - it is however true if the topological space is a so-called 1st-countable space, e.g. if the space is metric (as you said).
For even more general topological spaces you need the concept of a net, but in that setting the equivalence is actually total; a function f is continuous if and only if x_i->x implies f(x_i)->f(x) for every net (x_i).
This is interesting, please give us more!
In all topological spaces you have a sense of convergence of a sequenceI must admit I didn't know of any way to speak of convergence without the notion of a metric. How is that possible?
For even more general topological spaces you need the concept of a netMore general than what? And do you mean we need the "net" to replace the sequence? If you say so I'll believe you. However, one must still be able to define a sequence (a function from "the set of all natural numbers" to "the topological space in question"), since it doesn't really require much of the space, right?
(We need more maths on Slashdot!)
Swedish plasma phys. PhD student; MSc EE; knows maths, programming, electronics; finance interest; seeks opportunities