## Swedish Mathematician Lennart Carleson Wins Abel 144

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."*
## Except at some negible points? (Score:3, Insightful)

That's when I decided that statistics is more my kinda speed.

## Re:Except at some negible points? (Score:5, Informative)

## Re:Except at some negible points? (Score:2, Informative)

Well they mean "almost everywhere", which has a very precise meaning.The original article on Sci-Tech Today does use the phrase "almost everywhere" instead of the incorrect "negligible". I suppose it is excusable for someone who is not familiar with the area not to realise that it has a precise meaning, so maybe it would have been better to use the more common "presque partout", and let people reach for their French dictionaries.

## Re:Except at some negible points? (Score:1)

Maybe the French is more common in your part of the world, but around here (let's say USA and Ontario) people tend to use almost everywhere (or a.e.) almost everywhere. In the technical sense, I mean.

## Re:Except at some negible points? (Score:4, Informative)

## Re:Except at some negible points? (Score:1)

## Re:Except at some negible points? (Score:1)

## Re:Except at some negible points? (Score:1)

## Re:Except at some negible points? (Score:1)

The existence of a dense countable set is well known.

Given these facts, (and the vacuously true observation that the measure of a union is not greater than the sum of the two measures) do you really think its a tricky leap to find a dense, uncountable set of measure zero?

## Re:Except at some negible points? (Score:1)

I'd already been corrected on that fact.

Too repeat myself for the reading impaired : The post adds nothing non-obvious to this post, which is its direct parent [slashdot.org], except the observation that the union of dense set with an uncountable set is both dense and uncountable.

My point is that that fact is completely obvious.

## Re:Except at some negible points? (Score:2)

## no (Score:2)

by definition you

can'ttake the measure of an unmeasurable set.measure is defined on measurable sets. the word 'measurable' gives it away a bit.

nowhere in your link to the banach-tarski paradox does it say anything about taking the measure of the unmeasurable sets used in the construction.

## Re:no (Score:2)

m(E) = m(E intersect A) + m(E intersect A complement)

noting that E may or may not be measurable. This is, in fact, the way I learned measure theory.

We can equally well define a measure to be a function def

## squabbling over details in definitions (Score:2)

it's similar to a measure, i'll grant you that. but AFAIK the term 'measure' is specifically defined to be countably additive etc.

i wasn't aware that the term measure is used anywhere in the literature to mean an outer measure or

## Re:squabbling over details in definitions (Score:2)

I guess the lesson is to always be very precise because there will always be someone with a

## Re:Except at some negible points? (Score:2)

Find a Banach space X and two sets A and B such that A and B are disjoint, convex and dense and A U B = X.

(Yes, this was an assignment problem, but I've already finished that course)

## Re:Except at some negible points? (Score:1)

## Re:Except at some negible points? (Score:2)

## Re:Except at some negible points? (Score:1)

## Re:Except at some negible points? (Score:3, Interesting)

So you could have a continuous function which diverges from the sum of its Fourier series in all computable points?

## Re:Except at some negible points? (Score:1)

## Re:Except at some negible points? (Score:2)

## Re:Except at some negible points? (Score:1)

The Cantor set is the canonical uncountable set of measure zero.

## Re:Except at some negible points? (Score:2)

(And what's even more embarassing is that I originally *did* think of the Cantor set, but then dismissed that thought because "that one's countable". Ouch - I'm *really* stupid today. >_<)

## Re:Except at some negible points? (Score:1)

## Re:Except at some neglible points? (Score:2)

Andy

Hoping that he hasn't misapplied some math that he hasn't used since his degree more than 35 years ago.

## Re:Except at some negible points? (Score:2)

## Re:Except at some negible points? (Score:2)

## negligible? (Score:1, Insightful)

negligiblepoints."Negligible? In engineering, maybe. In mathematics, never.

## Re:negligible? (Score:3, Informative)

## hogwash (Score:2)

Rigor is very well for the rigorous mathematician. For the rest of us, and particularly for the purposes of talking to the layman (i.e. slashdot), it is a useless pedantry.

Anyone you might care to name who understands mathematics well enough to be able to understand a distinction of rigor most does not need anyone to tell the difference between what is "propper" and what is easy to say.

## Young people today (Score:5, Funny)

## Re:Young people today (Score:5, Interesting)

Hardy's saying is a bit of slight against all female mathematicians too, come to think of it...

## Re:Young people today (Score:1)

You people these days just don't get it do they?

## Re:Young people today (Score:2)

I guess some folks are normal to reality.

## Re:Young people today (Score:2)

Using the word "man" to refer to people in general goes back thousands of years. But, you know, I think you're right. In THIS case, he meant it as a clever, ambiguous attempt at women bashing.

## Re:Young people today (Score:1)

## Re:Young people today (Score:2)

## Re:Young people today (Score:2)

Easier to plan careers (or consider university programmes) if you don't have a mythical six-year limit hanging over your shoulders.It's also good to recognize that you don't need to solve 150 year-old problems, or create powerful new branches of mathematics to have a career. While it would be great to do some earthshaking work, and earn a reputation as a first-class intellect, you can have a decent life doing more humble mathematics. It's a good thing, too, because if only top-notch mathematicians coul

## Re:Young people today (Score:2)

## Re:Young people today (Score:2)

I was only referring to the oft-quoted stereotype that a mathematician's most productive years are those before he turns 30.I understood that. And, actually, the stereotype I've always heard isn't about productivity, but about significant contributions. The notion being that any important mathematics will by done by age 30. Older mathematicians have always, I think, produced a sufficient volume of work... but historically the great breakthroughs have come from the young. Until recently, anyway.

That

## Wiki Article (Score:5, Informative)

Wiki Article on the Breakthrough [wikipedia.org]

## Sweden (Score:1)

## Re:Sweden (Score:1)

## Re:Sweden (Score:1)

The new article was about grits right?

## Re:Sweden (Score:1)

## Re:Sweden (Score:1)

It is Björn in swedish, Bjørn us norwegianIt is "is" in English, "us" is Klingon, or something. ;-)

## Re:Sweden (Score:2)

Norway: Blonde-bearded, blue-eyed horned assistants in viking helmets

## iPod? (Score:2)

## Re:iPod? (Score:1)

## Fourier on the iPod is at least partially legit (Score:1)

## i i i i ... am not lame (Score:1)

## nice work, but no iPod (Score:5, Informative)

## Re:nice work, but no iPod (Score:1)

First of all, for many engineering purposes, it only matters that it works, not that you can prove that it works theoretically.- Proclaimed the engineer after one successful run of his simulation/program, before he ran it again with a different set of initial conditions/parameters only to see it fail because he didn't understand the math behind what he was doing.

There is a Platonic dilemma dealing with the necessity of proof for a mathematical idea to "exist," which is all well and philosophical, but

## Re:nice work, but no iPod (Score:2)

There is a Platonic dilemma dealing with the necessity of proof for a mathematical idea to "exist," which is all well and philosophical, but that's not to say proof shouldn't matter for engineers.There is no "dilemma"; knowing a mathematical proof is neither necessary nor sufficient to determine that an engineered system works.

Proclaimed the engineer after one successful run of his simulation/program, before he ran it again with a different set of initial conditions/parameters only to see it fail because he## Re:nice work, but no iPod (Score:2)

It's not like almost every major engineering disaster in history has been caused by ignorance to the assumptions of their design.Actually, most engineering disasters in history have probably been caused by unexpected violations of known assumptions.

Sure. Let's just blindly test all of the conditions. Since we're not going to bother with proof, we might as well abandon all of the analysis.Well, with that attidue, you'll never be an engineer. But the rest of your comments have to lead us to that conclusion a

## Re:nice work, but no iPod (Score:2)

Saying that Fourier analysis becomes, when you go to the discrete domain, "simple linear algebra" shows very clearly that you do not understand the complexities of discrete fourier analysis.Wow, you know a lot of big words, but you are apparently incapable of reading even two paragraphs carefully. I didn't say that "Fourier analysis [...] becomes simple linear algebra", I said that the theorems mentioned in the article do. Also, you seem to have trouble understanding the meaning of the word "domain"; you'

## Re:nice work, but no iPod (Score:2)

The reference to iPod in the article was the usual journalistic overreach; it's sill

## Obligatory CmdrTaco quote on the subject (Score:4, Funny)

His theorems have been helpful in creating iPod.No wireless. Less decimals than pi. Lame.

## Re:Obligatory CmdrTaco quote on the subject (Score:2)

His theorems have been helpful in creating iPod.At 06:42 AM EST on Monday March 27, 2006, iPod became self-aware.

## iPod? (Score:4, Insightful)

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.

## Re:iPod? (Score:2)

## Re:iPod? (Score:4, Insightful)

I fail to understand how Carleson's theorems have been used in making the iPod.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 ;-)

## Re:iPod? (Score:1, Insightful)

Not only that, I think it's a goodwill gesture borne out of tremendous respect for the work these guys do. Now, I know my share of really cool people that could never have gotten higher educations because of whatever reason, and somehow, they sometimes feel l

## iPod was still not a suitable reference (Score:1)

Why not mention signal processing, that makes it possible to filter out unimportant data from sound so that iPods (and it's likes) can store more music and MP3 and Vorbis files (and their likes) doesn't can be as small as they are. See there, iPod is still mentioned, but not in a way that makes you think there was something special (mabye mechanically) about the construction of an iPod that the world never before had seen.

When seeing the iPod referenche, I at first thought there mabye was some

## Re:iPod? (Score:2)

Engineer 1: Ok, the prototype is almost finished, but we have a problem.

Engineer 2: What's that?

Engineer 1: Well, we can't prove

that every continuous function (one with a connected graph) is equal to the sum of its Fourier series except perhaps at some negligible points.Engineer 2:

Engineer 1: Come one, man! Focus! If we can't figure this

## Two years later...in an alternate math universe (Score:2)

Engineer 1: Who wants to listen to ... oh, never mind. What's the problem?

Time passes. Testing occurs.

Engineer 1: It turns out that the keyboard her band uses isn't encodable because the particular waveform it produces yields a Fourier series that doesn't converge. Net result -- the keyboard makes the codec explode. Too bad we didn't know that some fourier series don't c

## Re:Two years later...in an alternate math universe (Score:1)

## shannon's theorem = ipod connection (Score:1)

Admittedly, throwing the ipod reference in was a troll, bu

## iPod Reference Misleading (Score:4, Informative)

## Re:iPod Reference Misleading (Score:1)

## WTF does this have to do with iPods?! (Score:5, Informative)

His theorems have been helpful in creating iPod.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 itCredit where credit is due, and none is due here.

If you want credit, how about: Shannon [wikipedia.org], Fourier [wikipedia.org] and Huffman [wikipedia.org]. 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.

## Re:WTF does this have to do with iPods?! (Score:2)

You seem to ascribe to Wikipedia a degree of authority and completeness that even the Encyclopaedia Britannica doesn't claim. Just because a connection isn't documented there doesn't mean it doesn't exist. This is not to say Carlsson's Theorem has anything to do with digital signal processing (it doesn't, of course).

## Re:WTF does this have to do with iPods?! (Score:1)

You seem to ascribe to Wikipedia a degree of authority and completeness that even the Encyclopaedia Britannica doesn't claimI ascribe no such thing. It is merely a readily available and publically readable example which, incidentally, Britannica is not.

## a continuous function is NOT (Score:2, Informative)

## Indeed (Score:1)

For those not in the know, continuity at a point p for a function f means

If the function

is continous for all x in the interval [a,b] the function itself is said to be continuous about that interval. This is sometimes mis-stated as 'you can draw the graph without taking your pen off the paper'.

## Re:Indeed (Score:3, Interesting)

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)

## Connected vs. path connected (Score:2)

It's not that informative, but definitely correct in the sense that a function (real etc.) is continuous iff the graph is path-connectedThere is a minor difference between what you said and what the article text said, although no one except math PhDs would be likely to care. (For the record: your statement is correct, but the article text is not.)

A path connected graph is not the same thing as a connected graph. There exist examples of graphs which are connected but not path connected. The article text

## Re:Indeed (Score:2, Interesting)

I recall correctly: a continuous function from one space to another is a function such that the inverse images of open sets are open.>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.

## Re:Indeed (Score:2, Informative)

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-

## What is convergence in a non metric space? (Score:2, Interesting)

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 net [wikipedia.org]More 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 natur

## Re:What is convergence in a non metric space? (Score:2, Informative)

I must admit I didn't know of any way to speak of convergence without the notion of a metric. How is that possible?In a topological space you have a notion of a neighbourhood of a point, i.e. a set containing the point in it's interior. You then say that the sequence (x_n) converges to the point x if for every neighbourhood U of x, there is a number N, such that x_n is in U whenever n>N. Basically this is a translation of the epsilon-N-formalism for convergence in metric spaces (since in a metric spa

## Re:What is convergence in a non metric space? (Score:1)

In a topological space you have a notion of a neighbourhood of a pointEh.. I actually had to look up the definition of "neighbourhood", since I still thought it required a metric. I had forgotten about the more general definition (with open sets).

More general than 1st-countable spaces.Ok, I thought we were already talking about a general topological space. I'm not very familiar with first-countable spaces (or, again, have forgotten).

Furthermore, that net thing seems neat. Gotta look into it. Thank

## Re:a continuous function is NOT (Score:2)

## Re:a continuous function is NOT (Score:1)

## His Fourier paper... (Score:2, Informative)

Acta Mathematicaand is calles On the convergence and growth of partial sums of Fourier series [actamathematica.org].## Thank heavens (Score:1)

## Re:Thank heavens (Score:1)

I thought the standard size was the foot.

The size of Kobe Bryan's foot.

Ben

## Bah (Score:1)

## Sweet (Score:1)

His theorems have been helpful in creating iPod.Wow

## Bad journalism (Score:1)

It's like saying Einstein's special theory of relativity helped to invent the automobile. After all it deals with motion.

For practical purposes lot's of convergence theorems for Fourier series had been known before this one and those would be more than enough to show that in practice things would work.

Take for instance

http://en.wikipedia.org/wiki/Riesz-Fischer_theorem [wikipedia.org]

from 1907.

And before that even others, though this one is quite nice.

S

## Re:Bad journalism (Score:2)

It's like saying Einstein's special theory of relativity helped to invent the automobile.No, his special theory of relativity also helped create the iPod...

## Hold the phone! (Score:2)

Swedishor is he aNorwegianliving in Sweden? I believe the submission is in error on a non-negligible point!## Swedish mathematician... at UCLA (Score:2)

## Re:Isn't this known already or assumed at least (Score:2)

## Re:What a coincidence (Score:2)

This guy Carleson helped create the iPod and his brotherHelped create the ipod?

Carleson's work on fourier series paved the way for many advances in modern technology (including many compression techniques).

Calling him 'creator of the ipod' is like calling John Bardeen [wikipedia.org] creator of the radio alarm clock.

## Re:What a coincidence (Score:1)

Finally, a name that I can curse at in about... 2 hours.

## Re:What a coincidence (Score:1)

Maybe it's because people pronounce it CIN-CI-NATI (or, for those that have never seen it written, SIN-SI-NATTY)

Don't blame us because your home city is hard to spell. Heck, try living in Gloucester and having to hear Americans try and pronounce it..

## Re:This sentence no article (Score:1)

His theorems have been helpful in creating Windows.

iPod is a proper noun, just like Windows, it works, and it's how Apple refers to their product. I'm not saying I agree with it, but don't yell about a grammar mistake that isn't there.

## Re:This sentence no article (Score:2)

iPod is a proper noun, just like Windows, it works, and it's how Apple refers to their product.No no, grandparent was entirely correct.

Windowsis a piece of software, like iTunes or Photoshop. No definite article required.iPodis a class of discrete objects, Like computer, Walkman or automobile. Try saying "wheels have been useful in creating automobile"... does that sound OK to you?You're right that it isn't a mistake (it's intentional of course), but it is still a grammatical error.