 Hello. So, in the last capsule recall that we proved that the Fourier transform as an operator on the Schwarz space S of r is continuous. That is the respect that it is new topology that we introduced on S of r. Why is it that we are interested in this? Because in chapter 4 we have already finished proving that f maps to f hat the Fourier transform is continuous with respect to the L 2 topology. You see the Schwarz space S of r is sitting inside L 2. So, L 2 is a metric space L 2 is a norm linear space and it receives a topology from L 2 and with respect to that topology the Fourier transform was continuous. And we also saw that the Fourier transform maps L 1 to L infinity and that is also continuous and it is continuous as a mapping from L 2 to L 2. Isn't that enough? Why do we want this new notion of continuity with respect to this topology that we have just introduced? There are many reasons for that. Earlier in the earlier chapters chapter 4 we were able to define Fourier transform on L 1 and L 2. Once you have it on L 1 and L 2 there is a way to define it on L p for p between 1 and 2 and that is all we can get by classical methods. What we need is to go beyond L 2 and we want to define it for any L p not just for p between 1 and 2 and even beyond these L p spaces we want to define it to a large class of entities. So, how far can we go? You can even define the Fourier transform on objects like e to the power i t a where a is a real number which doesn't have any decay. In fact we are going to define Fourier transforms of polynomials and functions that grow like polynomials. A polynomial is not going to be in L infinity unless it is a constant of course. Certain ideas from functional analysis are required and these ideas are called duality arguments. The class of objects to which we extend it is the class of tempered distributions. So, the class of entities on which we are going to define the Fourier transform are tempered distributions there is an adjective tempered. Distributions are generalized functions and notion of distributions is quite broad and we are going to look at a certain subclass of distributions. We are not going to study all the distributions we are going to only study the tempered distributions in this particular course. The general theory of distributions of course you can read Stichards or Hormander or any of the other references. So, what are tempered distributions? In chapter 7 we talked about continuous maps between normed linear spaces. In particular R itself is a normed linear space the complex plane is a normed linear space. So, we can take linear transformations from the normed linear space to R or C and talk about its continuity. We shall now generalize this from normed linear spaces to topological vector spaces such as the space S of R the space of Schwarz functions. A tempered distribution definition 108 a tempered distribution is a continuous linear map from S R to C that is it is a continuous linear map defined on the Schwarz space. Continuity here refers to the continuity that we are defined just now. When does F N converge to F in the Schwarz space it is a very strong notion of convergence in this strong sense we are talking about continuity. The set of all such continuous linear maps is denoted by S prime of R as always in functional analysis when V is a topological vector space V prime will be the dual space often denoted by V star. So, tempered distributions are the dual of S of R the dual of S of R is denoted by S prime of R and S prime of R being a dual space is a vector space. So, if V is a topological vector space the set of all continuous linear transformations from V to C is denoted by V star is called the dual of V. However, it may happen that V star may consist only of the zero element. How do I know that there is a any linear transformation at all? Of course, a zero map is certainly a linear transformation it is of course continuous being a constant map, but it may happen that that may be the only continuous linear map from V to C. Such examples exist and one such example is available on page 213 of the book of Goffman and Pedrick that I already mentioned earlier in this course. This is published by Prentice Hall in 1965 60 years have passed and yet the book still remains a gem. You will get rich dividends when you read this book of Goffman and Pedrick there are highly non-trivial theorems being proved in Goffman and Pedrick. He gives you the space and the space is basically the space of all measurable functions on the closed interval 0. So, take the closed interval 0 1 and take the Lebesgue measure nothing fancy and F is measurable G is measurable F minus G is measurable mod F minus G is measurable. Now, take mod F minus G divided by 1 plus mod F minus G that is a bounded measurable function because it is a bounded measurable function integral mod F minus G upon 1 plus mod F minus G is finite and this integral you denoted by D F G the distance between F of F N G is this integral that is a metric and the vector space operations are continuous with respect to this metric and this metric is even complete Cauchy sequences converge. So, it is a nice complete metric space but unfortunately it is dual space is just the zero element. So, there may not be anything interesting in V star. So, how do I know that there is anything at all which is interesting in S prime. This pathology that is exhibited in this example on page 213 of Goffman-Pedrick does not happen with S of R. S of R is a very nice topological vector space it is not a pathological one. There are plenty of elements in S prime of R. So, tempered distributions exist in great profusion. Why is it that it is so rich? Of course, we are going to produce examples of tempered distributions but the fact that it is a rich class S prime of R is a rich class is important. What is the secret behind this richness? The secret lies in the Hahn-Bannach theorem. Hahn-Bannach theorem remember in functional analysis is a convexity theorem and it applies when the spaces have some kind of convexity properties and the relevant property that we are looking at is local convexity. So, what are the locally convex topological vector space? It is a topological vector space first and foremost it has an origin and the origin has a local base. You must have a local base of convex neighbourhoods. So, when a topological vector space V has the properties the origin has a local base of convex neighbourhoods then you call it a locally convex topological vector space and this Schwarz space S of R is an example of locally convex topological vector space. Why is local convexity important? Because take a convex neighbourhood of the origin and you can have the Minkowski gauge functional and that becomes a semi norm with that semi norm apply the Hahn-Bannach theorem and you will get a good supply of elements in the dual space. I have given you a brief account as to why the space of tempered distributions is pretty rich. That is a functional analysis way of approaching the problem but for us we are going to take a more concrete route and we are going to produce concrete examples of tempered distributions. The most basic example is the Dirac delta and its close cousins. For the cousins of the Dirac delta the derivative of the Dirac delta and there is a distribution called pv1 over x and there is a heavy side function the signum function these are close cousins of the Dirac delta. So, first let us define the Dirac delta. The Dirac delta is a map from S of R to C given by f maps to f of 0. So, what is the linear transformation? The input is the Schwarz function f of x rapidly decreasing function f of x the output is simply the value of the function at the origin. This is evidently continuous with respect to the topology of S of R. What is the topology of S of R? Remember fn converges to f means what? fn converges to f means it in particular fn converges to f uniformly and much more of course. So, uniform convergence the consequence if fn converges to f uniformly fn of 0 will converge to f of 0 straight away. So, the Dirac delta is a continuous linear form on S of R that is a basic example of a tempered distribution. Now let us see the next example take a polynomial q of x, q of x is a fixed polynomial. Take an element f in S of R take a rapidly decreasing function and take a rapidly decreasing function f and multiply it with a polynomial. What happens q x times f of x is again rapidly decreasing. So, multiplication by q of x is certainly gives you again an element of S of R. But once I take this product q x into f of x because it is in the Schwarz space it is integrable. So, integrate q x times f of x over the real numbers. So, f maps to the integral q x f x that is a linear transformation from S of R to the real numbers or the complex numbers that is continuous. To check continuity how do you check that fn converges to 0 will mean the integral q x f n x must converge to 0 that is display 10.4 that you see because of linearity a linear map is continuous if and only if it is continuous at the origin. So, we need to check continuity at the origin. So, we have to show that limit as n tends to infinity integral over R q x f n x dx goes to 0. But how do I do that multiply and divide the usual trick multiply and divide by 1 plus x squared to the power n multiply and divide by 1 plus x squared to the power n. And what do you get after multiplication by 1 plus x squared to the power n and multiplication by q x that is called the product of these two polynomials R. We know that supremum of fn x R n x goes to 0, but definition of convergence and now estimate. So, what do you get we get that integral q x f n x dx mod less than or equal to I multiply and divide by 1 plus x squared to the power n and there is R x times fn x and take the supremum of this and take it outside the integral and the integral of dx upon 1 plus x squared to the power n that is an innocent constant and the other factors supremum of mod fn x R x over R goes to 0. So, that proves that you can multiply by a polynomial and integrate that gives you a continuous linear transformation or it gives an element of the dual space or you get a tempered distribution in other words. Now, let us take the next example the Schwarz space S of R sits inside LP of R for every p right every p bigger than or equal to 1. Now, suppose if I take capital F which is an LQ suppose I take a capital F in LQ there I can multiply capital F with the little f which I pick from the Schwarz space Schwarz space is sitting inside LP. So, little f is in LP capital F is in LQ what is Q? Q is the dual exponent 1 upon p plus 1 upon q equal to 1 Q is the exponent dual to p and then the integral fx times capital fx dx that is a real number. So, f maps to this integral display of 10.5 that gives you a tempered distribution. So, in short every LQ function gives you a tempered distribution namely take the LQ function multiply by little f and integrate over the real numbers. In the previous example we multiplied by polynomial and we integrated in this example we multiply by capital Fx and we integrate. In these examples we said that the distributions 10.4 and 10.5 are represented by Qx and f of x. In short the distribution 10.5 is represented by f of x and the distribution 10.4 is represented by the polynomial Q of x. In other words we shall make no distinction between the distribution given by 10.3 and Qx and we shall make no distinction between the distribution 10.5 and capital F of x. But we have to check one small detail and what is that small detail? How do I know that if I take two elements of LQ suppose I take two distinct elements G and H in LQ they may represent the same distribution or the resulting distribution if instead of G and I take little H do I get two distinct distributions or do I get the same distribution? Is this identification and injective mapping? So, if I take two elements from LQ and look at the corresponding distributions and the corresponding distributions are equal then the original two functions were also equal. In other words integral over R fx Gx dx equal to integral over R fx Hx dx for all f should imply G equal to H almost everywhere or you put capital F of x equal to Gx minus Hx and the hypothesis is integral over R 10.6 capital Fx little fx dx is 0 for all f in the Schwartz class and the conclusion should be that capital F must be 0 almost everywhere. So, we have to conclude that capital F is 0 almost everywhere we need to use standard techniques from measure theory. How do you show that capital F is 0 almost everywhere? Look at the points where capital Fx is bigger than 0 look at the points where capital Fx is less than 0. We shall show that the measure of this set A where f of x is positive is 0. Suppose not suppose this measure is positive suppose this measure is positive then how do we arrive at a contradiction? Pick a compact subset B of A of positive every set of positive measure will have a compact subset of positive measure. Also I am going to assume that P is not infinity and P is not 1 these extreme cases P equal to infinity and P equal to 1 are left as an exercise for the audience. We know that capital F is in some Lebesgue class say Lp I am assuming that P is not infinity and P is not 1. So, because P is not infinity and P is not equal to 1 capital F to the power P minus 1 that is going to be Lq where q is a conjugate exponent and q is not going to be infinity because P is not 1 q is not going to be infinity and P is not infinity. So, F to the power P minus 1 makes sense. So, I need to exclude P equal to infinity because I am going to cook up this F to the power P minus 1 and I need to exclude P equal to 1 because I want the conjugate exponent should be in Lq and q should not be infinity. Remember I repeat the Schwarz space S of r is dense in Lq provided q is not infinity. The Schwarz space is not I repeat dense in L infinity L infinity is very big space L infinity L infinity of the real line is not a separable space and the Schwarz space is not going to be dense it is going to be dense in Lp for P not equal to infinity. So, take a sequence Fn in S of r and that sequence converging to F to the P minus 1 times the characteristic function of B and this particular thing certainly belongs to Lq. So, what do we have over here let us estimate. So, mod integral capital Fx Fnx minus F to the P minus 1x characteristic function of the interval B that is less than or equal to apply the Holder's inequality the Lp norm of F times the Lq norm of the other piece. But this Lq norm collapses to 0 remember 1.6 will now give you that integral of capital F of P times the characteristic function of B is 0. What is it that we have we assume that F is positive remember on this set B F is positive on A and B is a subset of A and that is a contradiction because B has positive measure and the function is positive on the set. And there remains a case P equal to 1 and P equal to infinity which I am leaving it as an exercise to the audience. The argument would have been much simpler in case capital F were to be continuous, but we do not have that over here. So, that completes this argument if integral of capital Fx into little fx is 0 for all F then capital F must be 0. So, that gives another example of a temporal distribution. Now the derivative of the Dirac delta delta naught prime how do I define delta naught prime I define it to be F maps to minus F prime 0. That is also going to be continuous because if Fn converges to F in the Schwarz class the derivatives will converge uniformly. So, Fn prime of 0 will converge to F prime of 0. So, this also defines a tempered distribution it is not derivative or the Dirac distribution. Now another example is if u is a tempered distribution suppose if I already give you a tempered distribution and how do I multiply a tempered distribution u by a polynomial q very simple you simply throw the polynomial along with F. So, I am given a tempered distribution u and I am going to create a new distribution qx times u. So, I have to tell you what this new distribution does to a function F in the Schwarz class. What does it do to this function F multiply the function by the polynomial and then apply my u this new distribution is called qx times u that this tells you how to multiply a tempered distribution by a polynomial and 10.7 is also going to be a continuous map from the Schwarz space to the real line. So, you need to prove the continuity of this map 10.7 that furnishes another example of a tempered distribution. Show that cosine of e to the power x and sine of e to the power x are tempered distributions of course, they are L infinity functions and every L infinity function is a tempered distribution because every LP function is a tempered distribution by the previous slides. Is there a tempered distribution u which is represented by e to the power x in other words is there a tempered distribution u such that u applied to F where F is the Schwarz class is the same as integral F of x dx e to the power x. There is no such tempered distribution which has this property. It is a little bit of an effort for you but it is not difficult if you really think about it. You need to find a function in the Schwarz class which decays to 0 but when I multiply by e to the power x the decay will be compensated and the right hand integral will not make sense. So, this exponential function is growing very fast it will not define a tempered distribution. What about e to the power x cos e to the power x and e to the power x sine e to the power x? You will argue that they are also growing exponentially fast. The amplitude cosine and sine are oscillating but the amplitudes are growing. But we will see very soon that e to the power x cos e to the power x and e to the power x sine e to the power x are both tempered distributions. You try to prove it at this stage never mind if you get stuck because it will follow from general principles very soon. The next example is a very important example. If f is in the Schwarz class then what we do is we multiply f of x by 1 by x and want to integrate it. There is a problem because dx upon x will give you trouble near the origin. So, I do not allow you to go close to the origin from the real line I scoop out a closed interval minus epsilon epsilon. I scoop out a closed interval minus epsilon epsilon and I compute the integral. Once I scoop out the interval minus epsilon epsilon the integral makes sense. Of course, it will depend on epsilon and now I take the limit as epsilon goes to 0 and this limit will exist that is what you are supposed to show. This limit will exist and so you get a real number. So, f maps to this real number that is obviously linear. Does it define a tempered distribution? If fn converges to f in a Schwarz class will integral fnx dx upon x with limit as epsilon goes to 0 will that converge to this object? If so you got a yet another distribution that is called pv1 over x. It is called the principal value distribution. You may have encountered this principal value business in elementary complex analysis courses that is related to this. When you compute certain integrals you have to deal with principal values Cauchy principal value as you call it. So, this is a principal value distribution. We will continue this in the next capsule with these examples and developing the theory further. Thank you very much.