Материал готовится,

пожалуйста, возвращайтесь позднее

пожалуйста, возвращайтесь позднее

Okay. Big day today. Big day, big day. It’s finally time to reap some of the

benefits of the discussion, rather general, and I think it’s safe to fair say, abstract

discussion we’ve been having. It’ll still – it may still seem a little abstract to you today, I

understand that, so just ride along, all right? And you’ll see today some amazing formulas

that come out of this effortlessly in finding the Fourier transform of some well-known

functions – things we’re really gonna have to use. It’s really just, I don’t know, a bucket

full of miracles and the fun never stops. So, I don’t want to talk today about the Fourier

of transform of a generalized function, or a distribution, also known as distribution

Fourier transform of a distribution. So let me remind you, first of all, what the setup is,

what goes into this. To define a distribution you need a class, first of all, a class of test

functions. So the setup is, you first need – you first have to define a class of test

functions, or test signals that usually have particularly nice properties for the given

problem at hand. And it can vary from problem to problem. For us, for the Fourier

transform, it’s the class of rapidly decreasing functions. So these typically have

particularly nice properties.

Sorry for not specifying that terribly carefully. But they come, generally, out of – again,

sort of, years of bitter experience with working with problems, working with a particular

class of applications and trying to decide what the best functions are for the given class.

For Fourier transforms, the class of test functions is the rapidly decreasing function, so I

won’t write down the definition again, but I’ll remind you of the main properties in just a

minute when we need it. Rapidly decreasing functions – these are the functions which are

infinitely differentiable and any derivative decreases faster than any power of x, so they

are just as nice as you could be. Rapidly decreasing functions. All right. That’s what you

need. You need a class of test functions, and then by definition, a distribution or a

generalized function is, to use the most compact terminology, a continuous linear

functional on the set of test functions. Distribution, also called a generalized function, this

term is probably not used – generalized function is probably not used as much as it once

was when the subject was new. Nowadays, people just call it distribution since Schwartz

is really unifying work on all this. Is – so let me give you the shorthand notation for this,

a continuous linear functional on test functions. All right. So that means it satisfies two

properties. So one, so you write the pairing if is a test function, and t is a distribution, we

will write – most of the time, but not all of it, not always t, , with that sort of pairing for t

operating on .

All right. So if you want, you can think of as some distribution of physical information,

temperature, voltage, current, whatever; and t is a way of measuring it. All right. So you

measure something, you get a number. All right. So t is a measuring device and is the

thing you are measuring. So t operates on in such a way that it satisfies the principle’s

super position, or that is to say, it’s linear, and it’s also continuous. So t is linear, meaning

t of 1 plus 2 is the same thing as t operating on 1 – I’ll write up here – t operating on 1

plus t operating on 2. And the same thing for scaling.

benefits of the discussion, rather general, and I think it’s safe to fair say, abstract

discussion we’ve been having. It’ll still – it may still seem a little abstract to you today, I

understand that, so just ride along, all right? And you’ll see today some amazing formulas

that come out of this effortlessly in finding the Fourier transform of some well-known

functions – things we’re really gonna have to use. It’s really just, I don’t know, a bucket

full of miracles and the fun never stops. So, I don’t want to talk today about the Fourier

of transform of a generalized function, or a distribution, also known as distribution

Fourier transform of a distribution. So let me remind you, first of all, what the setup is,

what goes into this. To define a distribution you need a class, first of all, a class of test

functions. So the setup is, you first need – you first have to define a class of test

functions, or test signals that usually have particularly nice properties for the given

problem at hand. And it can vary from problem to problem. For us, for the Fourier

transform, it’s the class of rapidly decreasing functions. So these typically have

particularly nice properties.

Sorry for not specifying that terribly carefully. But they come, generally, out of – again,

sort of, years of bitter experience with working with problems, working with a particular

class of applications and trying to decide what the best functions are for the given class.

For Fourier transforms, the class of test functions is the rapidly decreasing function, so I

won’t write down the definition again, but I’ll remind you of the main properties in just a

minute when we need it. Rapidly decreasing functions – these are the functions which are

infinitely differentiable and any derivative decreases faster than any power of x, so they

are just as nice as you could be. Rapidly decreasing functions. All right. That’s what you

need. You need a class of test functions, and then by definition, a distribution or a

generalized function is, to use the most compact terminology, a continuous linear

functional on the set of test functions. Distribution, also called a generalized function, this

term is probably not used – generalized function is probably not used as much as it once

was when the subject was new. Nowadays, people just call it distribution since Schwartz

is really unifying work on all this. Is – so let me give you the shorthand notation for this,

a continuous linear functional on test functions. All right. So that means it satisfies two

properties. So one, so you write the pairing if is a test function, and t is a distribution, we

will write – most of the time, but not all of it, not always t, , with that sort of pairing for t

operating on .

All right. So if you want, you can think of as some distribution of physical information,

temperature, voltage, current, whatever; and t is a way of measuring it. All right. So you

measure something, you get a number. All right. So t is a measuring device and is the

thing you are measuring. So t operates on in such a way that it satisfies the principle’s

super position, or that is to say, it’s linear, and it’s also continuous. So t is linear, meaning

t of 1 plus 2 is the same thing as t operating on 1 – I’ll write up here – t operating on 1

plus t operating on 2. And the same thing for scaling.

Загрузка...

Выбрать следующее задание

Ты добавил

Выбрать следующее задание

Ты добавил