Материал готовится,

пожалуйста, возвращайтесь позднее

пожалуйста, возвращайтесь позднее

All right, so today I have two things in mind today. I want to wrap

up our discussion of some of the theoretical aspects of Fourier series. We’re skimming

the surface on this a little bit, and it really, you know, kind of kills me because it’s such

wonderful material and it really is important in its own way.

But as I’ve said before and now you’ll hear me say again, the subject is so rich and so

diverse that sometimes you just have to, you can’t go into any – if you went into any one

topic, you could easily spend most of the quarter on it and it would be worthwhile, but

that would mean we wouldn’t do other things which are equally worthwhile.

And so it’s always a constant trade-off. It’s always a question of which choices to make.

So again, there are more details in the notes than I’ve been able to do in class, and will be

able to do in class, but I do want to say a few more things about it today. That’s one

thing.

And the second thing is I want to talk about an application to heat flow that’s a very

important application historically, certainly and it also points the way to other things that

we will be talking about quite a bit as the course progresses. All right, so let me wrap up

and again, some of the sort of the theoretical size of things.

And I’ll remind what the issue is that we’re studying, and so this is our Fourier series

fine, all right? Last time we talked about the problem in trying to make sense out of

infinite sums, infinite Fourier series, and the important thing to realize is that that’s not

by no means the exception, all right?

We want to make sense of infinite sums of complex exponentials sum from K equals

Minus Infinity, Infinity, cK, either the 2 pi, KT. I'm thinking of these things as Fourier

coefficients, but the problem is general. How do you make sense of such an infinite sum?

And the tricky thing about it is that if you think in terms of sines and cosines, these

functions are oscillating.

All right, everything here in sight is a complex number and complex functions, but think

in terms of the real functions, sines and cosines where they’ oscillating between positive

and negative, so for this thing to converge, there’s got to be some sort of conspiracy of

cancellations that making it work.

Of course, the size of the coefficients is going to play a role as it always does when you

study issues of convergents. But it’s more than that because the function is bopping

around from positive to negative, see, all right and that makes it trickier to do. That

makes it trickier to study.

And again, realize that this is by no means the exception, and so in particular if F of T

again is periodic, period 1, we want to write with some confidence that it’s equal to its

Fourier series.

We want to write with some confidence, at least we want to know what we’re talking

about, that F of T, say is equal to its Fourier series going from minus [inaudible] 2 p i KT,

and again, it’s really if you want to deal with any degree of generality, it’s going to be the

rule rather than the exception that you’ll have an infinite sum because any small lack of

smoothness in the function or in any of its derivatives is gonna force an infinite number

of terms.

A finite number of terms, a finite of trigometric sum will be infinitely smooth. The

function and all its derivatives will be infinitely differentiatable, so if there’s any

discontinuity in any derivative you can’t have a finite sum.

up our discussion of some of the theoretical aspects of Fourier series. We’re skimming

the surface on this a little bit, and it really, you know, kind of kills me because it’s such

wonderful material and it really is important in its own way.

But as I’ve said before and now you’ll hear me say again, the subject is so rich and so

diverse that sometimes you just have to, you can’t go into any – if you went into any one

topic, you could easily spend most of the quarter on it and it would be worthwhile, but

that would mean we wouldn’t do other things which are equally worthwhile.

And so it’s always a constant trade-off. It’s always a question of which choices to make.

So again, there are more details in the notes than I’ve been able to do in class, and will be

able to do in class, but I do want to say a few more things about it today. That’s one

thing.

And the second thing is I want to talk about an application to heat flow that’s a very

important application historically, certainly and it also points the way to other things that

we will be talking about quite a bit as the course progresses. All right, so let me wrap up

and again, some of the sort of the theoretical size of things.

And I’ll remind what the issue is that we’re studying, and so this is our Fourier series

fine, all right? Last time we talked about the problem in trying to make sense out of

infinite sums, infinite Fourier series, and the important thing to realize is that that’s not

by no means the exception, all right?

We want to make sense of infinite sums of complex exponentials sum from K equals

Minus Infinity, Infinity, cK, either the 2 pi, KT. I'm thinking of these things as Fourier

coefficients, but the problem is general. How do you make sense of such an infinite sum?

And the tricky thing about it is that if you think in terms of sines and cosines, these

functions are oscillating.

All right, everything here in sight is a complex number and complex functions, but think

in terms of the real functions, sines and cosines where they’ oscillating between positive

and negative, so for this thing to converge, there’s got to be some sort of conspiracy of

cancellations that making it work.

Of course, the size of the coefficients is going to play a role as it always does when you

study issues of convergents. But it’s more than that because the function is bopping

around from positive to negative, see, all right and that makes it trickier to do. That

makes it trickier to study.

And again, realize that this is by no means the exception, and so in particular if F of T

again is periodic, period 1, we want to write with some confidence that it’s equal to its

Fourier series.

We want to write with some confidence, at least we want to know what we’re talking

about, that F of T, say is equal to its Fourier series going from minus [inaudible] 2 p i KT,

and again, it’s really if you want to deal with any degree of generality, it’s going to be the

rule rather than the exception that you’ll have an infinite sum because any small lack of

smoothness in the function or in any of its derivatives is gonna force an infinite number

of terms.

A finite number of terms, a finite of trigometric sum will be infinitely smooth. The

function and all its derivatives will be infinitely differentiatable, so if there’s any

discontinuity in any derivative you can’t have a finite sum.

Загрузка...

Выбрать следующее задание

Ты добавил

Выбрать следующее задание

Ты добавил