TheFourierTransformAndItsApplications-Lecture02 Instructor (Brad Osgood):All right. A few announcements – a few housekeeping announcements. Thomas checked the – Thomas John, our estimable, one of the Eskimo Bowl TA’s for the course, checked the website and we only had about 80 or so students who have signed up on the website, out of 130 or 140 or so that are actually signed up for the course of access. So please register for the website; that’s the way you’ll be able to get email messages and important announcements and post things on the bulletin board. So go do that. Student:[Inaudible] Instructor (Brad Osgood):Pardon me? Student:[Inaudible] room number, I don’t – Instructor (Brad Osgood):It should be now. I think there was a problem yesterday, briefly. It was not – there was a setting I had to change on the website, but if you haven’t checked – if you haven’t tried it since the first time you tried it, try it again. Okay? You should be okay. And Thomas also wanted to make an announcement about when the review sessions and office hours are gonna be. Do you need a microphone? Student:Okay. So the review sessions have been set. The first review session will be on Friday, this coming Friday from 4:15 to 5:05, in Skilling 191, it’s the room just on top. Now, we are not expecting every one of you to show up. And please, not all of you show up because we can only accommodate 30 people or so. Now, these review sessions will be available on the SCPD website. And what we will be covering – well, main topics over the week and also you giving you hints for the homeworks. Second – our second main – our third main thing would be the office hours for the TA’s have been set. Information is available on the course website under the link of course staff. You’ll see on the left-hand side there’s a link to course staff, and our individual office hours have been set, and they will start on Monday, October 1st. Instructor (Brad Osgood):Thank you. Student:Sorry. Instructor (Brad Osgood):Sorry. Student:[Inaudible] Student:Room number for the review sessions? Skilling 191.Instructor (Brad Osgood):Skilling 190 – 191, did you say for the review session? Student:No audible response. Instructor (Brad Osgood):Okay. Also, I forgot to mention that the homework – first homework set has also been posted up on the web and it is due next Wednesday. Okay. Any questions about anything? Anything on anybody’s mind? We haven’t done much yet, so there shouldn’t be that many questions. All right. So today, I wanna continue our study and begin a real – serious mathematical study of the question of periodicity. Remember that we are essentially identifying the subject of Fourier series with the study – with the mathematical study of periodicity. And last time I went on, at some length, about the virtues of periodicity, about the ubiquitous nature of periodic functions – periodic phenomena in the physical world, and also in the mathematical world. And we made a distinction, perhaps a little bit artificial but sometimes helpful, between periodicity in time and periodicity in space. Those sort of two phenomena seem to be, or often come to you in different forms, and it’s sometimes useful in your own head to sort of ask yourself which kind of periodicity are you looking at? But in all cases actually, periodicity is associated with the idea of symmetry. That’s the topic that will come up from time to time, and if I don’t mention it explicitly, as with many other things in this course, it’s one of the things that you should learn to sort of react to or think about yourself – see what aspects of symmetry are coming up in the problem, how does a particular problem fit into a more general context because, as I’ve said before and will say it again, one of the wonderful things about this subject is the way it all hangs together and how it can be applied in so many different ways. All right. If you understand the general framework and put yourself – and orient yourself in a certain way, using the ideas and the techniques of the class, you’ll really find how remarkably applicable they can be. Okay. So I said – as I said last time – as we finished up last time, when we’re sort of still just crawling our way out of junior high, a mathematical course of periodicity is possible because there are very simple mathematical functions that exhibit periodic behavior, namely the sine and the cosine. But that’s also the problem because periodic phenomena can be very general and very complicated, and the sine and the cosine are so simple. So how can you really expect to use the sine and the cosine to model very general periodic phenomena? And that’s really the question I want to address today. So how could we use such simple functions – sine of t and cosine of t – to model complex periodic phenomena? Now, first, the general remark is, how high should we aim here? I mean, how general can we expect this to be? So how general? All right. That’s really the fundamental question here. And in answering that question, led both scientists and mathematicians very far from the original area that they were investigating. Let me say – well, let me say right now, pretty general, all right. And we’ll see exactly how general – I’ll try to make that more precise as we develop a little bit more of the terminology that really – that will apply and allow us to get more careful statements. But we’re really aiming quite high here, all right? And we’re really hoping to apply these ideas in quite general circumstances. Now, not all phenomena are periodic. All right. And even in the case of periodic phenomena, it may not be a realistic assumption. I think that’s important to realize here what the limits may be, or how far the limits can be pushed. So not all phenomena, naturally, although many are, and many interesting ones are periodic. Andeven periodic phenomena, in some sense, you’re making an assumption there that is not really physically realizable. So even for periodic phenomena or at least functions that are periodic in time, even phenomena – soon I think I’ll start talking in terms of signals rather than phenomena, but phenomena sounds a little grander at this point. Even phenomena that are periodic in time – real phenomena, they just die out eventually. We only observe – or at least we only observe something over a finite period of time,