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Last time, we had gotten right to the verge of discovering the Fourier transform as a

limited case of Fourier series and I just wanted to pick up where we left off and finish

that off and then launch into a more formal treatment of Fourier transforms and talk about

how we’re going to proceed.

So we are about to get the Fourier transform as a limiting case of – we’ll actually there’s

two aspects to it. There’s a limiting case of the Fourier coefficient and the Fourier series;

the analysis part and the synthesis part of the Fourier coefficient and Fourier series.

So when I talk about the Fourier transform, I’m sort of thinking of the two things

together, but there’s really two parts to it and they both come out of more or less the same

analysis.

If I solely remind you what the set up was and who and why and what I mean by a

limiting case, by I mean a non-periodic phenomena I want to model as a periodic

phenomena as the period tens to infinity. All right?

And I took a special case of this, or just a case for illustration, that is, I suppose I have

some function which dies off eventually so it’s zero outside some interval, now that’s –

ultimately we’re gonna drop that restriction, I’m just taking this as a case to sort of model

what the situation should look like. So I’m going to take the case and, again, this was a

set up from last time. So take a case that looks like this. Have some function as F of T,

which is zero outside some interval. I’m drawing it like it’s positive but it can be very

general, but the fact is that it dies off at some point, and then I periodize it so – you can

image it’s a very big interval to begin with – but finite. And the I periodize it – I take an

even bigger number – Capital T, and I look at it from say minus T over T to 2 over T,

that’s supposed to represent one complete period, and then periodize this function. So the

pattern repeats. All right?

So periodize to make periodic of period T. So think of T as big, and eventually we think

of T as going off to infinity. So I won’t draw the picture, but again, the idea is I just take

the same pattern and repeat it over and over again.

All right then, you can write down the formula for the Fourier coefficient and you can

write down the formula for the Fourier series. So what the series looks like – the Fourier

coefficient looks like this: CK is 1 over T times the interval from minus T over 2 –

instead in integrating from 0 over T, I integrate over any period, so I take the period from

minus T over 2 to T over 2 and then the formulate is either the minus 2 PI I K over T, T,

that’s how I wrote it last time, FOTDT and the Fourier series, that’s the analysis part, has

decomposing F into its components and the complex – the corresponding complex

exponentials are these.

And then the Fourier series is to recover the function from its components as a sum from

minus infinity to infinity, C sub K of each of the 2 PI I, K over T, times T. All right?

And what you would like it you would want to take a simple-minded approach of saying

a Fourier transform inverse Fourier transform and so on, is a limited case of Fourier

series as you let the period ten to infinity you just let T ten to infinity but that doesn’t

quite work.

All right, so you would like to just let T ten to infinity and low and behold, the formula

has emerged, but it doesn’t work. Doesn’t quite work. All right?

limited case of Fourier series and I just wanted to pick up where we left off and finish

that off and then launch into a more formal treatment of Fourier transforms and talk about

how we’re going to proceed.

So we are about to get the Fourier transform as a limiting case of – we’ll actually there’s

two aspects to it. There’s a limiting case of the Fourier coefficient and the Fourier series;

the analysis part and the synthesis part of the Fourier coefficient and Fourier series.

So when I talk about the Fourier transform, I’m sort of thinking of the two things

together, but there’s really two parts to it and they both come out of more or less the same

analysis.

If I solely remind you what the set up was and who and why and what I mean by a

limiting case, by I mean a non-periodic phenomena I want to model as a periodic

phenomena as the period tens to infinity. All right?

And I took a special case of this, or just a case for illustration, that is, I suppose I have

some function which dies off eventually so it’s zero outside some interval, now that’s –

ultimately we’re gonna drop that restriction, I’m just taking this as a case to sort of model

what the situation should look like. So I’m going to take the case and, again, this was a

set up from last time. So take a case that looks like this. Have some function as F of T,

which is zero outside some interval. I’m drawing it like it’s positive but it can be very

general, but the fact is that it dies off at some point, and then I periodize it so – you can

image it’s a very big interval to begin with – but finite. And the I periodize it – I take an

even bigger number – Capital T, and I look at it from say minus T over T to 2 over T,

that’s supposed to represent one complete period, and then periodize this function. So the

pattern repeats. All right?

So periodize to make periodic of period T. So think of T as big, and eventually we think

of T as going off to infinity. So I won’t draw the picture, but again, the idea is I just take

the same pattern and repeat it over and over again.

All right then, you can write down the formula for the Fourier coefficient and you can

write down the formula for the Fourier series. So what the series looks like – the Fourier

coefficient looks like this: CK is 1 over T times the interval from minus T over 2 –

instead in integrating from 0 over T, I integrate over any period, so I take the period from

minus T over 2 to T over 2 and then the formulate is either the minus 2 PI I K over T, T,

that’s how I wrote it last time, FOTDT and the Fourier series, that’s the analysis part, has

decomposing F into its components and the complex – the corresponding complex

exponentials are these.

And then the Fourier series is to recover the function from its components as a sum from

minus infinity to infinity, C sub K of each of the 2 PI I, K over T, times T. All right?

And what you would like it you would want to take a simple-minded approach of saying

a Fourier transform inverse Fourier transform and so on, is a limited case of Fourier

series as you let the period ten to infinity you just let T ten to infinity but that doesn’t

quite work.

All right, so you would like to just let T ten to infinity and low and behold, the formula

has emerged, but it doesn’t work. Doesn’t quite work. All right?

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