 So, in the last capsule, we introduced the class of tempered distributions. What are tempered distributions? We had the Schwarz class S of r, we put a topology on this Schwarz class and it is a topological vector space, it is a locally convex topological vector space, the dual space is pretty rich and the members of this dual space are the Schwarz distribution and we saw number of examples of Schwarz distribution, the Dirac delta, the derivative of Dirac delta, the principal value distribution, every L cube is a tempered distribution and so on and so forth. Now, in this capsule, we shall talk about operations on a tempered distribution, a distribution can be differentiated as many times as we please. So, how to differentiate a distribution, there is a first item today and let us look at the slides, one can differentiate a distribution and the following theorem clarifies this notion. Recall that if f is in the Schwarz class S of r, then f prime is also in the Schwarz class and we proved last time that f going to f prime, the derivative map is a continuous operator. Now, suppose if U is a tempered distribution, then we create a new tempered distribution by 10.6, what is the prescription of this new distribution, f maps to U applied to the f prime, the original distribution applied to the derivative f prime, but we put a minus sign here, why this minus sign comes in we will see later, but the first thing to do is that this prescription 10.6 does define a continuous linear transformation, linearity of course is absolutely clear, what is not so clear is continuity and further if U is continuously differentiable, suppose for example, my original distribution is represented by a nice function, say a continuously differentiable function and suppose if U vanishes outside of a compact set, so for example, if U is of class C1 with compact set, then of course being of class C1 of the compact set, it will represent a distribution as explained in the last capsule, but will this U prime and this derivative will they agree, so let us clarify the meaning of this last clause in the theorem, namely they do a match, so denote by U prime the classical derivative of U, remember for the last clause, I need U to have compact support and continuously differentiable, so let us take U prime to be the classical derivative which means the derivative in the sense of elementary calculus and let us denote by capital D U, this new notion of derivative given by 10.6, these two notions agree okay, so because it has compact support, we may assume that the function U vanishes outside minus rr, then what is 10.6, because U is a nice classical function, this distribution is represented by a classical function, so U of f prime is basically integral from minus r to r Ux f prime x dx, but this is nothing but capital D U of f because we are using 10.6 that the derivative in the sense of distributions, now what we will do is that we will look at this equation integral minus r to r Ux f prime x dx, f is very smooth and rapidly decreasing, U is once continuously differentiable and vanishes outside minus rr, you want to integrate by parts, when you integrate by part the derivative will shift from f to U and you will get and the minus sign will go away, integral over r U prime of x fx dx, now U prime is continuous and it vanishes outside a compact set, so U prime represents a distribution and what distribution does it represent, multiply by f and integrate and that distribution is denoted by U prime applied to f, so we see that D U applied to f is the same as U prime applied to f, so D U the new notion of derivative and U prime the classical notion of derivative are the same when regarded as distributions, that clarifies the last clause in the theorem, the rest of the theorem is pretty obvious, why is it obvious that this is continuous, let us recall fn converges to f, we have to prove that U fn prime converges to U f prime, well fn converges to f and differentiation is continuous as we have seen last time, so fn prime converges to f prime, because fn prime converges to f prime and U is a continuous function it is sequentially continuous, so U of fn prime converges to U of f prime, put a minus sign there in both the places and we see that 10.6 does define a continuous map, so it is continuous linear, so 10.6 does define a temporal distribution and the theorem also tells us that the notion of distributional derivative generalizes and encompasses the earlier classical notion of derivative in elementary calculus, so with this equipped with this thing we can differentiate non-differentiable functions surprisingly, so let us apply this to the simplest non-differentiable function that we know the heavy side function, what is the heavy side function, it takes the value 0 on the negative real line and the value 1 on the positive real line, so h of x is 1 on 0 infinity and 0 on minus infinity 0, of course the heavy side function is bounded, it is in L infinity, so defines a tempered distribution and the heavy side function is denoted by capital H, let us calculate the distributional derivative, so what is the formula h prime of f will be by definition minus h applied to f prime, what is h applied to f prime, h is the function, the classical function and it is 0 on the negative real axis, so the integral will go from 0 to infinity hx f prime x dx, of course the minus sign is already there and hx is 1 from 0 to infinity, so it is simply collapses to minus integral 0 to infinity f prime x dx and you use the fundamental theorem of calculus because f is very smooth you get f of 0, but what is f of 0 the Dirac mass at f, so the derivative or the heavy side function is the Dirac delta, we have proved a very important result that h prime is Dirac delta, so the heavy side function which was earlier not differentiable in the classical or elementary sense is not differentiable in the distributional sense, not only that we know what the derivative is, the derivative is the Dirac delta. Next example show that x times Dirac delta is 0 and x times delta prime is minus Dirac delta, let us check the second one, the first one I am leaving it as an exercise, let us call the left hand set x times delta prime to be the distribution u, so I want to calculate what is u applied to f, what is u applied to f, what is u by the way, u is basically x times delta prime, so x times delta prime applied to f means remember multiplication by polynomial, the polynomial gets clubbed with the Schwarz function, so it will be delta prime times xf and how do I differentiate, it is put a minus sign and the derivative falls on the inside function xf, use the product rule and evaluate at the origin, so minus xf prime 0 and one of the terms will involve an x and that is uninteresting and the other term which does not involve an x is minus f of 0 and that is minus of Dirac delta at f, so we have proved that u of f is minus Dirac delta of f, so as distributions u is minus delta not or x delta not prime is minus delta not, that has been checked, so in exactly the same way prove that x times pv 1 over x is 1, so this principal value distribution has the property that when I multiply it with the polynomial I get 1, log mod x, log mod x is a tempered distribution, it grows like a polynomial as x becomes larger and larger and near the origin it is integrable, so log mod x is a tempered distribution, because I can multiply log mod x by a element to the Schwarz class integral over the real line, log mod x fx dx that integral will make perfect sense, show that it defines a tempered distribution, find the derivative of log mod x, the distributional derivative of log mod x that will turn out to be exactly pv 1 upon x and compute the second derivative also that is also an interesting distribution, so derivative of Dirac delta, so we are taken pains to talk about the Dirac delta, the derivative of Dirac delta, the secondary to the Dirac delta etc, is this some kind of dumb manipulations or does it have any significance in other parts of mathematics and science, it does have deep significance, this Dirac delta and its cousins, they have a lot of important applications and it has interpretations, these have become indispensable tool, for example the derivative of Dirac delta has a nice physical interpretation, it can be interpreted as an ideal electric dipole, the second derivative of Dirac delta is the ideal electric quadrupole, what is an electric dipole, you have studied it in physics, you have taken two charges which are equal and opposite and these charges are very close to each other and then you call it a dipole and in physics books like Rissnik and Halliday you compute the dipole moment and all those kinds of things, ideally these charges must come indefinitely close to each other and that is the ideal electric dipole, that ideal electric dipole is represented by delta prime and the electric quadrupole will be delta double prime, a very nice description of this can be found on page 35 of this book by Richards and Yuan that I cited earlier, I have given you two references for distribution theory, the distribution theory and non-technical approach that contains a very nice description of this electric dipoles and electric quadruples and so on and so forth, of course Richards and Yuan have restricted themselves to one variable and the problem becomes more interesting in several variables and you can talk about multiple expansions and a nice reference for that is L0 which some a book that I referred to earlier in connection with Shebyshev polynomials, so look at that citation and there you will see multiple expansion, these are also related to Dirac delta and its derivatives, pairing, we have been writing u of x where u is a map, a linear map, linear map from where to where, from s of r to c and x basically is a element of the schwar space, we used f because it is a function which is rapidly decreasing, so this pairing between an element of the dual space and an element of the original vector space that appears very frequently in functional analysis, it is customary in functional analysis to denote this u of x by triangular bracket ux, where u comes from v star and x comes from v, this is called the pairing of v and v star and duality arguments, this notation is very useful in that context and this notation is very useful in the context of distributions, since temper distribution is an element of the dual space of s of r and we shall use this notation in what follows, we are going to abandon this old notation u of x and we are going to use a triangular bracket, why is this triangular bracket preferable? Now suppose this distribution is represented by an LP function or if this distribution is represented by a polynomial qx, go back to the early examples of temper distributions and what are these temper distributions? Suppose if u is an LP function then it is simply take ux, take fx, multiply and integrate 10.7, so you see that this distribution it seems to me that it is a generalization of integration, so a distribution is basically something like a density ux times dx and that gives you a way to integrate a function f of x. In classical physics textbooks they would use this integral notation even when u is not an integral in the classical sense, it is a generalization of the notion of integrals, in particular a regular Borel measure is a distribution, so instead of ux dx it will be d mu. So, since distributions are generalizations of regular Borel measures it is meaningful to use this triangular bracket notation 10.7 and you can think of this as a generalization of integration, many of the intuitive ideas for integration will carry over in the context of distribution, for example let me cite one such example, when you integrate in classical analysis or in undergraduate calculus you differentiate under the integral sign, similarly will that notion carry over to distributions there is a way to differentiate under the distribution, this notion is very useful, so there are these analogies with integration theory and which are useful to keep in mind and we shall use this triangular bracket notation in what follows. Now we come to another notion called the support of a distribution, suppose I take a continuous function, let us begin with a nice example of a continuous function from Rn to C, the target space is the complex numbers nothing fancy then what is the support of a function, support of a function is obtained as follows, first look at the places where u is non-zero, now because I am assuming that the function is continuous this is an open set and you take a closure of the set and support of the function is a closure of the set of points x in Rn where ux is different from 0 and the support of a continuous function is denoted by sub u, loosely speaking the support of u is the smallest closed set outside which the function is 0 or the complement of the support is the largest open set such that the function restricted to that open set is 0. So let us convert this into an exercise and let us see how to recast this notion of support so that it will generalize conveniently in the context of distributions, so suppose for example p lies outside the support of u, now this happens if and only if there is a neighborhood np of p such that integral ux fx is 0 for all f in cc infinity of np, in other words you have taken a smooth function with support in np and you are integrating this f against u and I am getting 0, so this is an equivalent way to reformulate, so take all those points p with this property the point p has the property that p has a neighborhood np such that this happens these points evidently form an open set and the complement of this is the support. So this previous equation can be recast is the pairing of u and f is 0 for all f compactly supported in n smooth functions, the advantage of this reformulation is that it will carry over very easily for distributions definition 110, let u be a tempered distribution then the support of u denoted by sub u is a complement to the set of points p having a neighborhood np such that the distributional pairing u tested with f is 0 for all f is smooth compact support and support is contained in np, remember that a smooth function with compact support is sitting inside the schwarz space, schwarz space s of r contains all smooth function with compact support, so this f certainly lies in the schwarz space and this triangular bracket u f certainly makes sense. So this definition 110 is basically the definition of the support of a tempered distribution we took the classical definition of support tweaked it and we got a formulation that will readily generalize. Obviously by the very definition these points having this property is an open set and the complement is the closed set, so support is a closed set, the heavy side function, the heavy side function is 0 on the negative real axis and it is 1 on the positive real axis, the support of the heavy side function is the interval 0 infinity closed at 0 and open at infinity. The Dirac distribution delta naught has support origin, it is clear that when I differentiate a function the support will go down, what happens when I differentiate the heavy side function I get the Dirac delta, what is the support of the heavy side function interval 0 infinity closed at 0, what is the support of the Dirac delta singleton 0, so when I differentiate the support can go down and the inclusion can become strict. The 0 function has support as the empty sets, in fact the support of the function is empty if and only if it is a 0 distribution or 0 function, so the Dirac delta has a smallest possible support namely the origin. The derivative of the Dirac delta also has support at the origin, these are exercises show that the support is contained in the singleton 0 and show that the singleton does indeed align the support. The distribution u has compact support, then we say that u is a compactly supported distribution, the Dirac delta and its derivatives are compactly supported distribution, you take any distribution u and you multiply it by cc infinity function you will get a compactly supported distribution. Distribution with compact support play an important role in the theory of distribution there are distinguished class of distributions. Now we state a very important theorem, theorem 111, what are the distributions whose support is the singleton 0, Dirac delta has support as origin, the derivatives of the Dirac delta also have point support and therefore finite linear combinations of these Dirac deltas. Summation j from 0 to n, cj delta 0 to the power j that distribution u also has point support namely the origin. The converse is also true that is remarkable if a distribution u has point support namely the origin, then that distribution is a finite linear combinations of Dirac delta and its derivatives, it is a finite linear combination. The fact that this is a finite linear combination is significant and what is delta not j, it is the jth derivative of Dirac delta, what is the jth derivative of Dirac delta, it is the distribution f maps to minus 1 to the power j, jth derivative of f evaluated at 0 that is a temporal distribution and that is what you see here. We will not prove this theorem, the proof is there in any book on distribution theory, you will find the proof, you could look up Armander or Stichardt's book. For completeness we also mentioned that there is nothing very sacred about the origin, we can talk about Dirac delta at any point p, what is the Dirac delta at the point p, what is the distribution, it is the distribution given by f maps to f of p, take the Schwarz function f evaluated at p and that is the Dirac delta at p and the support of this particular thing is a single element p. So, formulate an analogous theorem for distributions with point support p, we can state theorem 111 prime, suppose you use a distribution with point support p and there are constant c0, c1, cn such that u is summation j from 0 to n, cj delta p to the power j where delta subscript p means the Dirac delta at. Now, we come to Fourier analysis and we want to talk about Fourier transforms of a tempered distribution. So, before we take up the Fourier transform of a tempered distribution, we must go back and prove a little lemma concerning Fourier transforms of Schwarz functions. We are going to take f and g both lying in the Schwarz space S of r, so that both f hat and g hat have their Fourier transform in the very classical sense, f hat is also rapidly decreasing. So, f is rapidly decreasing, f hat is also rapidly decreasing, what is this triangular bracket f hat g, it is a pairing of f hat and g, this is not the L 2 in a product because when you take the L 2 in a product we put a bar on the g, this is not the L 2 pairing, but the pairing of a distribution and a Schwarz function. So, what is this left hand side, it is integral f hat of x g x dx, what is the right hand side, it is integral of f of x g hat x dx. So, these are equal, the proof is very simple and let us dispose it of LHS equal to integral over R in f hat x g x dx and put the definition of f hat x because f is in the Schwarz space, the classical definition will work, integral over R f of y e to the power minus i x y dy. Now, the next thing is obvious, we are going to switch the order of integrations and using Fubini theorem and we are going to get integral f y dy integral g x e to the power minus i x y dx and that is nothing, but integral over R f of y g hat y dy which is the right hand side. So, just a direct application of Fubini's theorem will confirm 10.8. So, what is 10.8 again, you take two elements f and g in the Schwarz class and the pairing f hat g is the same as the pairing f g hat. Now, equation 10.8 would give us a clue how to define the Fourier transform of a tempered distribution. We will take it up in the next capsule. Thank you very much.