Lecture 12 | The Fourier Transforms and its Applications

this presentation is delivered by the Stanford center for professional development okay the saga continues let me remind 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 and I want to tell you what we're going to do with it so the best class of functions for free a transforms and 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 one C of X I'll call I think your last time F this time I think I'll call it fee because I'm going to be tending to use that term that letter in a lot of the discussion doesn't matter we have to be flexible in our alphabets V of X is infinitely differentiable so as smooth as you could want as as many derivatives as you can want and more different differentiable and secondly that as I said any derivative decreases faster than any power of X so let me write that down and say another word about it for any M and N greater than equal to 0 X to the N DN DX to the N the nth derivative of fee of X and absolute value also tends to 0 as X has the plus or minus infinity those two properties let's see if I got that right yes infinitely differentiable and this says that any any derivative this is the M say M am I don't know whether I use M it in that last time you display or the other way doesn't matter an M and n are independent here so this says like a third rule there's nothing mysterious here you 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 whatever that's a natural scale of measurement for how the 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 then x times that fact 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 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 a reasonable statement all right and as it turns out there are plenty of functions that satisfy this property well it wasn't obvious by any means and again nature provides you with so many different phenomena how do you pick out the one really to base your definition on you know why this for properties of Fourier transforms for that as the best class of functions for Fourier transforms are not something else well and say 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 that this is what you really need and that the story will spin out as we as we go on so why the best for free a well one reason is that if he isn't rapidly decreasing function then so is its Fourier transform is also in s alright 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 so is true for Fourier transform also if the function is infinitely differentiable so is its Fourier transform all the properties are preserved all those analytical properties are preserved by Fourier transforms that's very important again why is this so important and why this is why these particular things work so smoothly for for developing the theory you'll see and the second property is the free inversion works that is if V is in s then the Fourier transform the inverse Fourier transform the Fourier transform of T is equal to fee and the same if I go the other direction that is the Fourier transform the inverse Fourier transform fee is also known and by the I mean the usual Fourier transform here defined in terms of the integral right and there's no problem with that interval converging because the function is dying off all right so that's again that's where we were last time but if this is such a darn fine class of signals for the Fourier transform how come it doesn't include some of the signals we would really like it to include like for instance the most basic example of all the rectangle function alright the rectangle function is not in so Y but if so good if s is so good for instance the rectangle function is not in s the most basic example it's not in s because it's not continuous never mind differentiable or anything else it's not even continuous the triangle function is not ns it's continuous but it's not differentiable you know the fun examples that we started our whole study on don't even fit into the supposed class of the best functions for Fourier transforms and not to mention the other class of functions that we might want to consider like constant functions sines and cosines and so on none of those are in the class s constant functions trig functions many others are not in s alright so how do you resolve this that is how do you get to defining the Fourier transform or how do you get back to how do we sure we haven't lost anything and then what is gained by considering this very good class so how can we be sure for assured maybe I should say we haven't lost anything I mean it looks in the surface that we've lost something by restricting ourselves to consider this class when we've lost the rectangle function we've lost the triangle function who knows what else lost anything and furthermore and have gained and will gain even greater generality all right so that takes a little while to tell that story and we'll get most of the way there today but answering that question causes us to take up another strain of development that was happening around the same time all right so to answer this this we have to pick up another line of development another line of development the two will come together triumphantly but only after we follow this path for a little while and that is Delta functions and so on and so on that is the idea of idea of generalized functions so-called generalized functions they're also referred to as distributions and that's probably the term that I'll use also known as and this use of the word distribution has nothing to do with the way we used it earlier when we talk about probability it's just one of those clashes of terminology that comes up every now and then you can't do anything about it there are only so many words to go around and I don't know I actually don't know the origin of the word distribution in this context but that's what's used so generalize function distribution are synonymous terms and I'll probably find myself slipping into using distribution rather than generalized function although both terms are in currency and current use and what I mean by here is so ie or typify by by the Delta function the Dirac Delta function which really should be called the Heaviside Delta function Dirac Delta now I am assuming and I will remind you of some of the properties that you probably have seen I am assuming that you've seen the Delta function in various contexts because everybody that goes through an engineering course on signals and systems anybody goes through a quantum mechanics course and I don't know where else it comes up but it's one of those things you learn to use operationally and maybe you feel a little queasy about it you know but nevertheless it gets the job done somehow and you'd rather not worry about somehow those fine points that all the statements you're making your complete all right so what are those statements that are complete you often see it to find this way a delta of X is equal to 0 for X different from 0 but 0 its infinity be the integral from minus infinity to infinity of Delta of X DX is equal to 1 this function which is 0 everywhere except at one point and its total integral is equal to 1 and see if I integrate the Delta against any other function f of X I get the value I'll call it fee just to since I'm using the term I get the value at 0 quite remarkable indeed every one of these statements is a complete alright there's just no way to make any of this make sense precisely but there's something there alright there's something there people who used it with some skill we're able to do so without avoiding any of the pitfalls and there are possible pitfalls I mean you can do you can manipulate Delta's incorrectly and you can make mistakes alright but those masters classical masters have beside some other that follow him and deer act in particular in his applications the quantum mechanics could make sense of these things and use them effectively but no one's home and because they got the right answers and because they were so effective operationally nobody wanted to admit that these statements were all complete so what is to be done there's only so much that you can stand I mean some people have some people have higher thresholds than others but at some point it had to be cleaned up there are still you know there's still cases of this around you know their very important case of this around the one that people cite most often now our fine and integrals Fineman path integrals for quantum electrodynamics I guess is where they used I mean nobody can make sense of them mathematically but you can't deny that they work all right and it's a real it is now an acutely felt challenge to somehow give it a give a rigorous foundation for finding path integrals it doesn't exist yet as far as I know I don't think is any way of doing it it's the same sort of thing you know in in the right hands you can effectively compute with them but you know feel a little queasy about it somehow now the fact is that operation you can understand what's going on here all right Delta is supposed to represent a function which is concentrated at a point all right and this was probably even said to you when you first learned about Delta and their various ways of approaching this alright and you may have seen some of these I'll give you a sample calculation or sampling example this so various ways of doing this but it's always but it always involves a limiting process it's always via a limiting process alright so for example what I mean by limiting process is what you do is consider typically families of legitimate functions that are getting narrower and narrower and still satisfy these sort of basic properties so and it's their various ways of doing it but let me give you one very simple on EEG you consider a one-parameter family of shrinking rectangle functions or concentrating rectangle functions consider a one-parameter family family of rectangle functions all right I'll write it down so I'm going to look at one over epsilon epsilon of X all right now that's the family that I want to consider in the parameter here is epsilon and I think of epsilon as small as tending to zero ultimately tending to zero now one of those functions look like well the rectangle function we know what that looks like and it's not hard to see what happens when you scale it like that the right the ordinary rectangle function again is one between minus 1/2 and never mind what habitat the endpoints that's not important here it's 1 between minus 1/2 and 1/2 and it's zero outside that interval that's PI of X if I scale it 1 over Epsilon pi PI epsilon of X that function is one from minus epsilon over to 2 epsilon over 2 so again think of epsilon is being small here and if I multiply by 1 over epsilon I'm scale I'm making it large in the vertical direction so the height is 1 over Epsilon all right that's what the graph of that looks like and it's still the case that the area is 1 if I integrate this function I get 1 if I integrate this function it's the area the rectangle the rectangle has base Epsilon and height 1 over Epsilon the area is 1 it's as epsilon is getting smaller and smaller it is approximating what you think of as an ideally concentrated function those properties defining least the first two properties defining the Delta function it's zero again think of epsilon getting smaller so it's centering around the origin here sending around zero so it's nonzero it's it's it's zero outside a small interval around the origin it's becoming steeper and steeper right at the origin and what about the fight well let me write down its integral is equal to 1 integral from minus infinity to infinity of pi epsilon of X DX is one of our Epsilon the integral from minus epsilon over 2 to epsilon over 2 1 DX that's the only place where it's nonzero that's equal to 1 all right and what about that final property if I integrate this scaled rectangle function against a function V of X what happens right so if I look at the integral from minus infinity to infinity of PI epsilon of x times V of X DX what happens there well let's take the case where fee is smooth enough say you can actually do it more generally but just to get a simple idea imagine expanding fee in a Taylor series expansion that is write this as again this is zero except on the interval from minus epsilon over 2 to epsilon over 2 so this is equal to the integral from minus epsilon over 2 to epsilon over 2 of V of X DX ok now imagine writing fee at or so because pi of x is equal to 1 there and it's equal to 0 outside that interval so that's equal to 1 over salon the interval from minus epsilon over two to epsilon over two say write fee of X as V of zero plus V prime of zero times X plus and so on and so on higher order terms I'm thinking about the Taylor series expansion that's assuming the function is smooth you can do it you can write a similar argument if the functions you only continuous but never mind that let me just I just want to see what the what the point what the point is here why it's concentrating all right so if I integrate that and then I get higher jump put one more term in here fee double prime is zero over two times x squared plus and so on it's on higher order terms integrate with respect to X alright what happens if I carry out the integration well the first term that's fee of 0 V of 0 times the integral of 1 from minus epsilon over to 2 epsilon over 2 1 over epsilon that just integrates to 1 and then the second term what happens here well this is a constant this is a constant so if I integrate X I get x squared all right so I'm going to get epsilon Squared's here times 1 over epsilon that's going to give me a term of order epsilon if I integrate xq x squared I'm going to get an X cubed and if I waited between minus epsilon over 2 and epsilon a 2 I'm going to terms of order epsilon cubed times 1 over epsilon that's going to be in terms of order epsilon squared all right so the result is that beyond the first term beyond the constant term here I'm going to get terms of order epsilon or higher order terms all right epsilon epsilon squared epsilon cubed and so on and so on so what happens is epsilon tends to 0 this term goes away okay so as epsilon goes to 0 you can say that the limit as epsilon epsilon goes to 0 this tends to fiy of 0 that is to say the limit as epsilon tends to 0 of this integral 1 over epsilon interval 4 minus epsilon over 2 epsilon over 2 pi so no doing do it like this we write the whole thing down here the integral from minus infinity to infinity of high epsilon of X V of X DX that's the interval just computed is equal to C of zero all right that's what's meant by concentration via a limiting process now again I'm assuming actually you can tell me if I'm wrong but you probably saw this calculation at some point I mean when somebody was trying to justify the Delta function and somebody talked about it is somehow ideal concentration they probably looked at it pretty much in this way all right so again just to make sure you you see what that you understand what the issue is here okay to say this to consider this the limit is epsilon tends to 0 1 over Epsilon Pi epsilon of X if you consider this limit of this function this sequence of functions depending on the parameter epsilon it makes no sense all right to consider this limit makes no sense all right but to consider operationally what it means when I integrate this scale function against an ordinary function and take the limit of the integral that does make sense and it produces the value constant to the value at 0 this limit epsilon tends to 0 integral from minus infinity to infinity or I'll put that I'll do it I'll do it like I did before PI epsilon of X V of X DX to say that that's equal to fee of 0 this does make sense ok all right that's ok and the fact is by experience the ways that Delta appeared in applications weren't so much this way 'lord naked you know just it just in sort of a lake a limit of a sequence of functions really it occurred operationally when it was paired with another function but it was paired with a legitimate function and somehow the idea was by a limiting process you were concentrating things and you were just pulling out the value at the origin saying all right that's really operationally how it appeared and that was a extremely important thing to realize all right so these again these statements somehow again individually these statements just don't make sense and can't be can't be made to make sense all right but in practice the way it was used you replace Delta you replace what you what you think of as idealized Delta by some sequence of functions which are concentrating and you consider them as paired with a function via integration and then you can do everything you want to do in a context that you have a certain amount of confidence in yeah all right cede my there pardon me outside the integral is there a 1 over Sigma we're here yeah no because I define PI oh oh I guess I definitely I wanna thank you sorry whatever I'm sorry sorry sorry thank you yeah thank you all right now we're almost there all right and once again this is one of these tipping points where you know you look at the accumulated body of evidence you say to yourself what's really going on here and you know again the mathematical modus operandi somehow is to turn the solution of a problem into the definition to turn around yeah sorry one but yeah shouldn't ya interest is you scale this thing right this doesn't make sense I mean this should be there this one of reps alone should be there for me you missed it over there but I mess it up somewhere else on the bear right here okay now is everything okay over there over where up there no trouble is I was thinking I guess I was thinking of the scale when I put the scale in here I was thinking about as already there's also scaling the outside and I was wrong there I'm sorry but everything's okay is everything okay now I'm just going to walk back and forth year one to two more times and then like myself I don't expect the camera to follow me maybe once more okay all right all right now this was a big conceptual step and again it follows a mathematical modus operandi of turning the solution of a problem into a definition so we're going to concentrate instead of concentrating on somehow the limiting behavior and so on and so on the idea is to concentrate on the operational effect or the outcome of concentration in one case and then more generally more general operations all right so I want to change the point of view it really requires a fundamental change of point of view here so to capture this idea and to include much more and how is it going to include much morale again I'll have to explain to you in just a second all right and to include more we need to change a point of view we need a fun a really fundamental change of point of view all right it becomes operational it becomes an emphasis on what on the outcome rather than you know sort of a process alright the focus is on the outcome rather than on the process what I mean by this is in the case of Delta the outcome was at the end of the day it was a constant it was it concentrated and pairing this approximating function thus approximating sequence of functions gave you the value of the function you're interested in at zero and that that had to be done the process was taking a limit there was a limiting process involved in that we want to concentrate on the outcome actually getting the value of the origin rather than the process so here is how you set that up there are several aspects to it so what I'm going to do really is write down the definition or axioms for a class of generalized functions a class of distributions which are going to include the Delta function is going to capture the essential nature the Delta function and it's actually going to as it turns out include much more all right so there are several aspects of the definition one this is the definition generalized functions or distributions distributions distributions all right so first you start out with a class of test functions so you start with now what are called test functions now when the Fourier transform comes back into the picture this is going to be the class of rapidly decreasing functions but for other problems you might consider different class of functions but generally speaking these are the sort of best functions for the properties you're worried about all right so you think of these as the best functions I'll call them feet all right best functions for the problem at hand for the given let me just say area of application all right so again for Fourier transforms is going to be the Schwarz functions the rapidly decreasing functions for other functions it may be those functions which I mentioned last time of compact support the functions which are actually not just tengah 0 outside some finite interval which are identically 0 outside some finite interval all right then two associated with these test functions is a class or called generalized functions or distributions all right a distribution I'll call it t is a linear operator on the tetete on the test functions that produces a number so one says in the biz it is a linear functional the distribution T is a linear functional I'll say what this means just a second functional on the test functions alright what that means is I give you or you give me for a given for a test function fee T of V T operates on fee produces a number and it's linear is a number typically you allow complex number zero and T is linear that is to say T of the sum of two functions is T of T 1 plus T of e to and T of elba is a principle of superposition T of alpha times fee is alpha times T of T I'll get out of the way in just a second here all right so a distribution is a linear operator on the class of test functions you start by defining the class of test function that somehow is going to have all the nice properties you could possibly imagine for your problem a distribution or a jet also known as a generalized function is an operator on those functions it produces a number and the final property is you don't want to give up taking limits completely because limits do come in to the subject and so you assume that these linear operators are continuous in the following sense three the final property is continuity property that is if VN is a sequence of functions which converges to a function fee then that implies that if I operate on it with one of these generalized functions or distributions the phat converges to C of fee I'll say more about this in just a second actually this is the most problematic part of the definition but I won't I'll say more about it later ok the continuity property means that if you have a sequence of test functions that are converging to another test function if he n converges the fee then that implies that supposed to be a implies and if I operate on this sequence with it with a with a distribution with a generalized function that produces a sequence of numbers does so on the right on the left hand side is a convergence of a sequence of functions that's hard actually and again I'll come back and talk a little more about that later on the right hand side is just a sequence of numbers that's easy it's easy to talk about a sequence of convergence of sequence of numbers so these numbers converge to that number all right this is because you don't want to abandon taking limits completely because it does come up in them it does come up in any applications all right now I want to introduce a little again a little terminology here and notation that's that's used in this subject and that is you often write you often say added distribution is paired with a test function instead of saying its operating on test function that's what's going on you often say it's paired with a test function again you'll see a reason for this in just a second as a reason where we're at specific pairings that come up and you often write a notation for the pairing is often written like this with angle angle brackets T is paired with fee this notation is supposed to just indicates an alternate notation for writing T operating on feet both notations are in use this notation is probably a little bit more common all right this is not an inner product alright it's supposed to indicate that T is somehow operating on fee to produce a number and the operation is linear if I take two functions to your fee 1 plus V 2 is T 1 plus T 2 and T of alpha fee is alpha times T of fee all right I know this sounds there's no deep waters here but when you see how this works and you see how effectively can compute with this it's really quite stunning was not so easy to do I mean this was no you know adopting this point of view to give a rigorous foundation for Delta and then actually to also develop the Fourier transform I mean was no less may be comparable no less revolutionary may be comparable to the whole shift from like classical mechanics to quantum mechanics it required a different point of view you had to look at things differently you have to do it well do you need quantum mechanics or not I mean you know it's it's it's a it's a more accurate description of the world at certain scales well theory of distributions is a more accurate way in a more effective way of dealing with the problems you really want to deal with it's just the way it is all right now again let's go back and recover Delta so let's recover Delta in this in the in the context of this definition of this definition alright so what is Delta doing operationally Delta operationally is that what is the outcome of applying Delta it is to pull out the value of the function at zero at the end of the day that's what Delta is supposed to do you wrote down this nutty integral you we everybody wrote down this nutty integral I'm not exactly trying to talk you out of it I'm just trying to say that there's another way of looking at it that makes more sense right so operationally operationally the effect of Delta is to pull out the value of the origin is to evaluate the function a function at the origin all right so you save it yourself a couple times operationally operationally the effect of Delta the effect of Delta MN is is to evaluate the function at the origin all right the mathematical modus operandi is turn the solution of a problem into the definition I send myself aha that's how I should define Delta define Delta according to this definition which I'm raising now on test functions it's supposed to be a linear functional on test functions how is it defined you give me it you give me a test function I have to tell you how the Val Delta operates on it yeah else um I thought the function of paired with not the same function it operates on you kind of suggests that no I mean I don't know what you thought but this is what I'm saying okay I mean this is this is this is a definite this is a notion of a common notion of pairing okay that is it pair the the fee is a given test function T operates on on the function fee so instead of writing T operating on fee which is sort of a functional notation you often write you just often write this notation as an alternative and it's very common actually also see this in physics physics when they talk about bra vectors and ket vectors you know a bra vector operates on a ket vector I hate saying that I feel ridiculous saying that especially when I have no idea what I'm talking about but you know what I'm talking about if you had a physics course same thing one sort of one sort of vector pairs with another kind of vector in physics and here and operator pairs with a function there's a class of functions that it operates on this is the operator all right and it's an and what I'm saying is is is that that the use of the word pairing there is appropriate I mean that's what that's what this was said all right now like I say let me go back let me go back to Delta here so you say to yourself one the operational effect of Delta is to evaluate a function of the origin turn that into a definition define Delta by Delta paired with fee is what bless your souls fee of zero he is a class of test functions that is given to you you give me a test function I have to tell you what Delta does to that test function and then I have to verify that satisfies the properties of a distribution I say Delta operating on fee is nothing but fee of zero all right now is it linear well what is Delta paired with v1 plus v2 that is V 1 plus V 2 at 0 by definition that's what Delta does it evaluates v1 plus v2 at zero but that is of course V 1 of 0 plus v2 of 0 which is Delta of Delta paired with V 1 plus Delta paired with V 2 that is Delta paired with V 1 plus Delta paired with V 2 and similarly for the scalar multiplication property all right how about continuity how about continuity again without saying precisely what I mean by a sequence of functions converging what about this statement the Delta that affi n converges to fie in a sense that you just imagined so a sequence of functions converging to another function does that imply that Delta V n converges to delta of feet or Dexy values the I'll use the pairing notation here does that imply that Delta paired with the sequence TN converges to Delta paired with feet well write out both sides what is the left-hand side what is Delta paired with VN Delta paired with fie n is just fie an of 0 by definition well if the sequence of functions fie n is converging to fie then surely fie n of 0 is converging to the value at 0 all right if BN converges to fee then surely be n of 0 converges to fee of 0 and so surely it must be the case that Delta paired with fie n Delta paired with fie n operating on fee n converges to Delta operating on a fee so it's continuous it's a continuous linear functional ok now I want you to think about this a little bit and I hope appreciate it really because this mysterious Delta function that was defined by these ridiculous properties it's zero everywhere except at one point where it's infinite its total integral is equal to 1 and it pulls out the value of the function when you integrate it against it I mean those ridiculous properties have been in effect captured in the simplest possible distribution all right this complicated limiting operation that we talked about in terms of concentration operationally is defined completely airtight by evaluation at zero all right the simplest sort of operation evaluated zero and that captures this that captured this mysterious notion of Delta is very impressive now if that was all you could do you say that's a lot of work and I want to change my completely change my worldview you know so you can define this one little distribution that I was perfectly happy taking limits of any way and I know those statements were but I'm happy enough with I can I can tolerate ambiguity you know why do you really put me through this and the fact is that it goes far beyond this particular example the fact that it captures this particular example so easily and so effectively is already a good thing but if it were the only thing it would have withered on the vine all right but it doesn't it allows us to define very robustly Fourier transforms and everything else and we'll get to that not right now but we will let me give you one other one other slight version of this you have also probably seen a shifted Delta function in your work with Delta functions in other classes you've also probably seen I imagine some statement it looks like this the interval from minus infinity to infinity of Delta X minus y f of Y dy is equal to what f of X all right you've probably seen statements that look like that right Delta sifts through the values and so on okay now that statement is to use afraid never mind all right that doesn't make any sense what do you want to define here operationally you wanted to find a shifted Delta function all right if this is the Delta function based as the origin based at 0 Delta applied to fiy pulls out the value at 0 what do you want to define to capture this statement precisely as a distribution I give you you give me a test function I tell you what the pairing should be to pull out the value at number X so what do you define what do you want to define here tell me what the distribute tell me give me a dick give me a name for distribution and tell me how it operates I'll be afraid I think bad will happen not the boards are not going to fall off a wall or anything like that if you get it wrong you want to capture this property you want to capture this property of a shift to Delta function right or you don't think of this I mean I know you're trying to figure this in terms of convolution probably don't think about in terms of convolution just think about operationally trying to pull out the value of the function at some other point other than the origin what is the distribution that will do that define a distribution that will do that I can wait right alright so I define it's a different distribution alright so define let's say Delta sub a as a distribution by the formula Delta sub a paired with fee is V of a okay so the case we had before was when a is equal to zero now is that linear is that continuous you can check but it's the same idea all right so Delta a does is this defines health as a distribution I give you give me a test function I have to tell you how the distribution operates on that test function to produce a number what do I tell you you give me fee I say Delta a operating on fee produces the value of the value at a period airtight airtight alright no ridiculous statement like this no limiting processes involved nothing all right it is a straight definition based on what you want the outcome to be okay now all right so take a deep breath hold it exhale good now I claim that we have gained something we've certainly gained something in clarity here or rigor if you want I mean this mysterious Delta function has now emerged operationally as the simplest possible distribution but there's a but oh the struggle I mean I had to introduce these complicated things and so on and so on either you want me to change my whole point of view and that's asking a lot of me and so on and why why why why Lord why one question you can at you might well ask at this point is have you lost anything all right I mean yes you've defined Delta so we've gained Delta but have we lost anything all right what I mean by this is you know ultimately the test functions are very restricted alright the test functions might be a very restricted class these rapidly decreasing functions they don't include you know constant functions they don't include the rectangle folks they don't include the triangle function and so on I mean how are those functions going to get back into the scene you know how are AG rectangle function the triangle function trig functions etc going to come back in how they going to come back into the picture I mean you know Delta is this bizarre thing you know and true was defined in a pretty simple way but you know I really want to get to the point where I can consider the functions that I really want to consider triangle functions rectangle functions signs and co-signs and so on can I consider those in the context of generalized functions and more to the point when we get to it can I actually find can I consider those in the in the context of actually taking Fourier transforms because that's what I want to get to I want to get to defining more general Fourier transforms so the Fourier transform of a Delta is going to make sense so the Fourier transform of constant functions is going to make sense so the Fourier transform of sines and cosines is going to make sense I mean how can I do that if I can't do it classically by an integral how can I do that in this context or can I do in this context all right so the first so I want to make I want to explain now how generalized functions include in a natural way the sort of ordinary functions that is to say I haven't lost anything I haven't lost those functions that I really want to consider they're there they are in there but they're in there in a slightly different way so you can consider so this is the suspension of how we consider how to consider ordinary functions in this context all right so again I want to consider for instance the constant the constant function one so eg how to consider the constant function take the very take a very simple function the constant function one as a generalized function as a distribution okay all right so now you buy the premise you buy the gag all right you want to consider this as a distribution what does that mean always means the same thing if you want to consider something as a distribution that's that says you give me a test function or I'll give you a test function you have to tell me how your how your new thing here is operating on that test function alright so how do we pair so given a test function fee in whatever class you're considering how do we define a pairing assign a pairing of one and fee all right well it's actually very simple but it's but it's again maybe not one of those things you would and it takes its cue from really how these things were how these things grew out of the classical applications and the classical view of things I'm going to define I'm going to define a by integration so two pair 1 and fie I have to get a number a distribution operates I say this over and over again this what you have to say to yourself the distribution operates on a function to produce a number it has to be linear has to be continuous so one has to operate on fee somehow to produce a number and the pairing is my integration that is to say I'm going to you you give me fee I have to tell you how one pairs with fee and here's my definition one pairs with fee as the integral from minus infinity to infinity of one times V of X DX a lot of big buildup for a very simple definition but-but-but sorry that's what it is okay that certainly produces a number and if fee is a good enough function this integral is going to converge different values of fee will give me different numbers alright so in some sense I know all about one if I know all about these integrals for different values of fee there's nothing about one you can ask me somehow that I can't answer if I integrate it against different sorts of functions of course there's not much you can ask me about one that I can't answer anyway but you know operationally speaking you know there's nothing you can ask me about one but I can't tell you if you allow me to integrate it against any old test function fee all right more generally how does more generally aren't really more more generally but the same way by the same the same token I want to include a rectangle function the triangle function everything else and and consider those as distributions in the same way so likewise for G the rectangle function the rectangle function pairs with the test function by integration integral from minus infinity to infinity PI of X V of X DX C of X may be very smooth PI of X is not smooth but the product makes sense and the interval makes sense all right that's my definition of how pi pairs with feet that identifies pi the rectangle function as a distribution this definition identifies or defines I should really say defines one the constant function one as a distribution now you can check integration is a linear operation so linearity works so does continuity that's a little bit more complicated actually that requires certain limiting theorems for integrals but nevermind I'm setting some of those details aside right now and in general same thing for trig functions say so how about sine of two pi X alright that function doesn't you can't integrate that function from minus infinity to infinity sup but you can integrate it if i pair it with a function say that is decreasing by definition sine of two pi X can be considered as a generalized function if I tell you how it operates on a test function how does it operate on a test function it does so by integration the interval from minus infinity to infinity sine of two pi X times C of X DX alright the sine of two pi X doesn't make sense that integral doesn't converge but if I multiply by function which is which is dying off the integral will converge alright so for the purposes that I'm going to want to consider sine of two pi X can be considered as a generalized function so it's a real honest-to-god function but for these purposes it can also be considered as a generalized function because I can tell you how it operates on test functions it operates by integration right again this seems like a lot of God I mean there's a lot to absorb here and are you really going to you know really been able to do anything with this and just you have to go along for the ride a little while longer and then you'll see how this works because it's really impressive R so in general most many very wild functions can be and some not so wild functions can be considered generalized functions by this pairing so if f of X is any function I won't say I'll put closer on any because not all functions it's not the case that for all functions the integral will make sense but for many of them you can consider you can consider f of X as a generalized function or distribution by defining its pairing F paired with fee is the interval from minus infinity to infinity f of X V of X DX F operates on fee by integration alright and that's a linear operation now again you know for not not all functions will this integral converge alright but that is for naught M you can stick in a really wild function here and maybe it's not going to work but for most garden-variety functions and more than most garden-variety functions for things that can be pretty wild that sort of integral will make sense because the test functions are so nice the nice you make the test functions the more the more wild functions you can stick in here all right and for Fourier transforms we're going to find that the Schwarz functions as test functions are just the right class are just the right class they're the ones that are going to allow us to include sines and cosines and deltas and all sorts of things like that all right so one says in this context that a function determines a distribution how does a function determine a distribution you give me the function I have to tell you how it operates on a test function it operates by integration and then you really have to say all right that operation is linear that operation is continuous that all is OK that all that that all works out fine and so that we haven't lost anything in the sense of the class of generalized functions includes all the functions that society needs to function no play on words no pun intended all right all right so next time we'll wrap it up now next time you're going to see how this comes together to define the generalized Fourier transform and more alright see you then

Channel: Stanford

Views: 36,184

Rating: 4.9351354 out of 5

Keywords: Electrical, engineering, computers, math, physics, geometry, algebra, calculus, technology, functions, linear, operations, sin, cosin, Fourier, transformations, series, rapidly, decreasing, decays, power, of, derivative

Id: MQ_qTG7dcJQ

Length: 52min 55sec (3175 seconds)

Published: Thu Jul 03 2008