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Okay. The saga continues. Let me remind

you what we did last time. Last time, I introduced the best class of functions for Fourier

transforms, or at least I asserted that it was the best class of functions for Fourier

transforms, and I want to remind you what the properties are, then I want to tell you what

we're gonna do with it. The best class of functions for Fourier transforms. We call that S,

the class of rapidly decreasing functions, and they're characterized by two properties.

They're infinitely differentiable and any derivative decays faster than any power of X. I

will write that down. First of all, Phi of X is infinitely differentiable, so as smooth as you

could want, has as many derivatives as you could want and more, differentiable and

secondly that, as I said, any derivative decreases faster than any power of X. For any M

and N greater than or equal to zero, X to the ND N DX to the [inaudible] derivative of

Phi of X [inaudible] also tends to zero as X tends to plus or minus infinity. Those two

properties. This is the M. M and N are independent here, so this says – there's nothing

mysterious here.

You've got to measure decay or growth some way, and the simplest way of measuring

growth is in powers of X. You can say a function grows linearly or grows quadratically

or grows cubically. That's a natural scale of measurement for how a function is growing,

and so to talk about a function decreasing more rapidly than any power of X, you can say,

well, if it decreases faster than linearly, than X times that function is going to go to zero.

If it decreases faster than quadratically, you look at X squared times the function. You

want that to go to zero as X tends to plus or minus infinity.

So multiplying by a positive power of X and insisting that the product of the positive

power times any derivative here tends to zero says that it goes to zero faster than any

power of X. It's a strong statement, but it's not an unreasonable statement. As it turns out,

there are plenty of functions that satisfy this property. What wasn't obvious by any

means, and again, nature provides you with so many different phenomena. How do you

pick out the one to base your definition on? Why this for properties of Fourier transforms

as the best class of functions for Fourier transforms and not something else?

Well, genius is what genius does, and as it turns out, this was the right class to single out.

In the following sense, and even here, and it may not be completely clear that this is what

you really need, and the story will spin out as we go on. Why the best for Fourier? Well,

one reason is that if Phi is a rapidly decreasing function, then so is its Fourier transform.

That is if the function decreases faster than any power of X and any derivative, it

decreases faster than any power of X so is [inaudible] Fourier transform. Also, if the

function is infinitely differential, so is its Fourier transform. All the properties are

preserved.

All those analytical properties are preserved by Fourier transform. That's very important.

Again, why is it so important and why these particular things work so smoothly for

developing the theory, you'll see. The second property is that Fourier inversion works.

That is, if Phi is an S, than the inverse Fourier transform of the Fourier transform of Phi is

equal to Phi, and the same if I go the other direction – that is, the Fourier transform of the

inverse Fourier transform of Phi is also equal to Phi.

you what we did last time. Last time, I introduced the best class of functions for Fourier

transforms, or at least I asserted that it was the best class of functions for Fourier

transforms, and I want to remind you what the properties are, then I want to tell you what

we're gonna do with it. The best class of functions for Fourier transforms. We call that S,

the class of rapidly decreasing functions, and they're characterized by two properties.

They're infinitely differentiable and any derivative decays faster than any power of X. I

will write that down. First of all, Phi of X is infinitely differentiable, so as smooth as you

could want, has as many derivatives as you could want and more, differentiable and

secondly that, as I said, any derivative decreases faster than any power of X. For any M

and N greater than or equal to zero, X to the ND N DX to the [inaudible] derivative of

Phi of X [inaudible] also tends to zero as X tends to plus or minus infinity. Those two

properties. This is the M. M and N are independent here, so this says – there's nothing

mysterious here.

You've got to measure decay or growth some way, and the simplest way of measuring

growth is in powers of X. You can say a function grows linearly or grows quadratically

or grows cubically. That's a natural scale of measurement for how a function is growing,

and so to talk about a function decreasing more rapidly than any power of X, you can say,

well, if it decreases faster than linearly, than X times that function is going to go to zero.

If it decreases faster than quadratically, you look at X squared times the function. You

want that to go to zero as X tends to plus or minus infinity.

So multiplying by a positive power of X and insisting that the product of the positive

power times any derivative here tends to zero says that it goes to zero faster than any

power of X. It's a strong statement, but it's not an unreasonable statement. As it turns out,

there are plenty of functions that satisfy this property. What wasn't obvious by any

means, and again, nature provides you with so many different phenomena. How do you

pick out the one to base your definition on? Why this for properties of Fourier transforms

as the best class of functions for Fourier transforms and not something else?

Well, genius is what genius does, and as it turns out, this was the right class to single out.

In the following sense, and even here, and it may not be completely clear that this is what

you really need, and the story will spin out as we go on. Why the best for Fourier? Well,

one reason is that if Phi is a rapidly decreasing function, then so is its Fourier transform.

That is if the function decreases faster than any power of X and any derivative, it

decreases faster than any power of X so is [inaudible] Fourier transform. Also, if the

function is infinitely differential, so is its Fourier transform. All the properties are

preserved.

All those analytical properties are preserved by Fourier transform. That's very important.

Again, why is it so important and why these particular things work so smoothly for

developing the theory, you'll see. The second property is that Fourier inversion works.

That is, if Phi is an S, than the inverse Fourier transform of the Fourier transform of Phi is

equal to Phi, and the same if I go the other direction – that is, the Fourier transform of the

inverse Fourier transform of Phi is also equal to Phi.

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