Lecture 52

Welcome back, we are doing some application of Fourier transform on earth and today we will see some of the concepts which are very useful in statistics, physics and engineering. So, how those concepts can be related to Fourier transform and so once we have that relation then one can apply to various situation when those concepts appear in their respective subjects. So, now first one would like to talk about that now if I have a function. So, now one would like to know that where the function is mainly concentrated, where exactly is the points in which F is going to have the concentration. So, for that, that is what it is called centroid. So, that is the centroid of a function is defined as, we will denote this as the centroid, this is related with the first moment. So, now this is the ratio of minus infinity to infinity x F of x dx divided by the area minus infinity to infinity F of x dx, that is the first moment area divided by the area, that is where the F is mainly concentrated. So, now this is called the centroid. So, now we know this ratio in terms of the Fourier transform. So, now what is my integral minus infinity to infinity F of x dx, this is nothing but F hat at 0 and what we know is that if I take my g of x is equal to x F of x, then g hat at xi, this is equal to minus 1 by 2 pi i F hat xi, F hat prime at xi, this is what we have seen. So, now this the centroid can be expressed in terms of minus 1 F hat prime at 0 divided by 2 pi i F hat of 0. So, that is what the centroid is, now the moment of inertia or the second moment is defined to be, I will, is defined as integral minus infinity to infinity x square F of x, this is, this is going to be minus 1 by 4 pi square, we know that F hat double prime of 0 by F Fourier transform relationship. So, therefore, this x square F, this is going to be minus infinity to infinity x square F of F x dx divided by minus infinity to infinity F of x dx, that is minus 1 by 4 pi square and then F hat double prime of 0 by F hat of 0. So, in, in the statistics, so this x square is called the variance of frequency distribution function and in physics and engineering, it is basically roughly speaking, it is used to describe the distribution of mass around an axis and how it affects the rotational characteristics of a function, this is also called the mean square abscissa. It is, now it will be, what we will observe that it will be interesting to know that how this is going to behave in terms for a convolution function. So, now you consider, so what we want to find is that x square with respect to this convolution function, F convolution of G. So, by definition, this is going to be minus F convolution of G hat double prime at 0 divided by F convolution of G hat at 0 and with a factor of 4 pi square. Now, if we simplify this, then this is minus 1 by 4 pi square F hat of 0 G hat that is what the denominator is and then this is F hat G hat double prime at 0. Now, if we take the simple derivative with the product rule, what we are going to get, this is going to be F hat double prime at 0 G hat at 0 plus 2 times F hat prime at 0 G hat prime at 0 plus F hat of 0 G hat double prime at 0. Now, this implies what this is minus 1 by 4 pi square, then the first part is going to be F hat double prime at 0 by F hat of 0 minus 2 times 1 by 4 pi square, this is F hat 0 F prime G hat prime of 0 divided by F hat 0 G hat 0 minus 1 by 4 pi square G hat double prime at 0 by G hat of 0. Now, this is nothing but the mean square of x square of F minus and this is 2 pi i because what we know is that x of F is equal to minus 1 by 2 pi i F hat prime at 0 by F hat of 0. If you multiply do that what we get that this is F and this is x of G and this is x square of G, then the variance sigma square is the mean square of mean square deviation that is what by the. So, therefore, sigma square is going to be x minus x square that is what is. So, this just for the sake of writing it this is equal to minus infinity to infinity x minus the centroid and then F of square F of x dx divided by integral minus infinity to infinity F of x dx. So, let us now look at what is going to be the variance in terms of F convolution of G. So, therefore, so now if we if you just do a preliminary calculation what you can see is that this is x minus the centroid square F of x dx this is equal to F hat 0 minus 2 x square F hat at 0 and plus. So, therefore, sigma square is going to be equal to minus F prime 0 square divided by 4 pi square F hat of 0 square. Now, therefore, when we are writing for F convolution of G sigma square of F convolution of G with if we just the calculate this then this is going to be. So, now what it also says that the variance of F convolution G is the sum of the variance of F and the sum of the variance of G. The standard deviation actually does what? It measures the spread of the function. So, what it says is that the spread of the function increases if we are considering a convolution. So, all these concepts they can actually be written in terms of the Fourier transform and hence we can bring in all our expertise in the Fourier transform techniques into those problems which appear in physics, engineering as well as in statistics. So, now we will discuss a new topic. So, what we have done? We started with the Fourier series which means on the circle group or 2 pi periodic function or one periodic function, any periodic function one can do the Fourier analysis. Then we did for the finite abelian groups, we did Fourier transform the mainly we captured the inversion theorem as well as the Planck's rule formula and next we went to real line R and we have defined the Fourier transform on R and studied the behavior of F hat. Now, we are going to do the Fourier transform in the higher dimension. Let us say we are doing it for Rn and n is greater than 1. So, now x belongs to Rn means just I will fix for some notation that x is equal to x1, x2 to xn, the n tuple and each of this xi's they are in R and we are considering if we take the mod of x, this is square root of i is equal to 1 to n xi square whole to the power 1 over 2. Now, what is then what we would like to define what a polynomial looks like in the multivariable. So, if we take the index alpha which is equal to alpha 1 to alpha n, where alpha i belongs to n union of 0 non-negative integers for i is equal to 1 to n, then for x to the power alpha we will denote the notation this is x1 to the power alpha 1, x2 to the power alpha 2 and xn to the power alpha n and here the mod alpha we will define to be alpha 1 plus alpha 2 plus alpha n. So, we will say a polynomial of degree k if mod alpha is k. So, we can write this as the polynomial of degree k, this is mod of alpha lesser equal to k and then C alpha x to the power alpha and if the highest coefficient when mod of alpha is equal to k if this C alpha is non-zero at least for one multi index alpha then we call that that is a polynomial of order k. So, this is polynomial in Rn and then we would like to define denote the derivative. So, the multi index D alpha this is the operator working. So, this is del alpha 1 by del x1 alpha 1 to del alpha n divided by del xn alpha n this sometimes we write also this is how we write this is for a multi index alpha denote this is the notation for the derivative what we have. So, in order to do Fourier analysis what we of course need to define the integral and we all of us we know what is an integral means just to recall. So, now how to follow the prescription of Riemann exactly in the same way what does the Riemann does he takes an interval and take a function a bounded function from the interval to R real valued bounded function. So, now what is going to be replaced by an interval in higher dimension it is going to be replaced by a rectangle what is a rectangle R is a rectangle R is equal to product of j from 1 to n some aj bj like for example if I have a1 b1 this is an interval in R and this is an interval which is a2 b2 in the y axis then the rectangle is this is the rectangle which is going to denote by this. So, this is our R this is a1 b1 cross a2 b2 and now what one would like to get a partition to define Riemann integration now for the partition is that if pj are partition of aj bj for each j then we notationally you do not p is equal to p1 to pn is called a partition of R. So, what does so now in the partition we have sub intervals because this is suppose aj bj then we have this is the partition what we are considering is the pj now this sub intervals is going to be for each j so this if sj are sub intervals of pj then you define s this is equal to s1 cross s2 cross sn this is going to be sub rectangle of the rectangular of the partition p. So, now we have fixed all the notations to define our upper Riemann sum and the lower Riemann sum. So, for a partition p the upper Riemann sum is defined to be up of for a function now I am taking a function f from R to R bounded. So, this is equal to summation over supremum of f of x where x belongs to the sub rectangle s and then what is now this is there we have a length bj minus aj or the length of the partition xj plus 1 minus xj what is the most natural thing is going to be replaced the length here we have a rectangle in n dimension if we are in 2 dimension the length is going to replace by area and in the higher dimension this is going to be replaced by the volume where the volume of s is equal to s1 cross multiplied by sn where this is the volume I am using the same notation and this is now all this s denotes the length of the interval sj. So, and then we have Lp of f this is equal to summation over infimum of x belongs to s and the sum is varying over all the s such that s sub rectangle of p partition p infimum of f of x with the volume of s. So, these are all the intervals and then these are the length and I can take the product now we say that f is Riemann integrable if for any epsilon there exist a partition p such that Upf minus Lpf is less than epsilon. So, exactly just a kind of replacing the symbol all this prescription of Riemann is going to work here too. So, one important property of this integral what we will be using quite often is that let us just write down the result let f be continuous on R we have any closed rectangle what we are taking. Now, suppose R is equal to R1 cross R2 where R1 is something in Rn1 and R2 is contained in Rn2 and n1 plus n2 is our n the two pieces. So, now we write x belongs to Rn as to be x is equal to x1 x2 where xi belongs to Ri or x belongs to R rectangle Ri i 1 and 2 then define f of x1 which means define f from a map from Rn R1 to R as f of x1 is equal to integral over R2 f of x1 x2 dx2. Now, I am restricting the f to Rn2 and I am taking the integral in Rn2 in the plane Rn2 then integral over R of f of x dx this is equal to integral over R1 then first I perform the integral in the R2 and then I perform the integral dx1. Similarly, if we define we can do the perform the integral first of R1 and then we can interchange the order of the integration and so on. So, this is and of course, what we know is that when we are doing a change of variable then if we have a matrix which is or for a map which goes from some a to b then with the change of variable what we will get is a Jacobian which is the determinant of the map derivative of the map. So, change of variable formula is also holds here and with all the this notational fixing the notation what we will do is that in order to do the Fourier transform we need to define our Schwarz space. Now, we have all the ingredient to define our Schwarz space. So, now we say that f is infinitely differentiable on Rn if d alpha of f exist for all multi index alpha and so the Schwarz space Rn this is equal to f belongs to C0 infinity of Rn that means it or C infinity of Rn such that the supremum over x and we have got here the mod of x to the power alpha d beta f of x then x belongs to Rn this is finite for all multi index alpha and beta that is our Schwarz space as the way we have computed the existence of Cc infinity function in R by just taking the product here we can define the existence of the Cc infinity function in Rn 1 2 Rn and then obviously what is going to be replaced the Gaussian which is the major row if I define f of x is equal to e to the power minus pi and now this is the minus pi there it was x square here the natural choice would be mod x square then this is going to be in the Schwarz space that will be easy and standard straightforward to check. So, we will define Fourier and transform in Rn and we will do analysis there and see some more applications. Thank you.