Lecture 56

Welcome back, in this lecture we are going to discuss Fourier series in the higher dimension. Although most of the result what we are going to state this is they go their proof goes parallely analogously in the dimension as in the dimension 1. Nevertheless, first let us define the symbols and what we are trying to do in the higher dimension. Remember that in dimension 1 we are dealing with 2 pi periodic function which means we are dealing with the 0 to 2 pi interval or 0 1 periodic function which is 0 to 1 interval and then when we are joining the 0 and 1 together then we used to get the circle. That means the sum modulo 1 if we are trying to give the operation in the interval 0 1 then this we have identified with the circle. So, in the same fashion let us define what is called n torus. So, we will denote it as T n and this we will identify with 0 1 cross 0 1 which is we denote the cube of length 1, this is n times. So, the points here are essentially x 1 to x n and now we say that the 2 points we are going to identify, we will identify x 1 0 x n and then x 1 x 2 in place of 0 if I am putting 1 here then whenever 0 and 1 appear in the same coordinate. So, more precisely we say that x is identified with y if x minus of y varies over Z n then we write in that case x congruent to y mod 1. So, here the 1 is the vector 1 which is consist of n tuple of constant 1. One can easily see that this is an equivalence relation that partitions the whole R n into equivalence class. This is easy to see the equivalence of the relation. So, this equivalence relation partition R n into equivalence classes. So, now we denote this as T n which is identified with R n co centred with Z n. In the case of n equal to 1 what we have seen is the circle and if I take n equal to 2 just imagine that this is 0 1 0 1 cube. So, first what you are doing the left and the right sides they are together you are going to get a kind of a cylinder. Now the top and bottom edge they are joined. So, what you are going to get a ring ball or a vada. So, this is this is what is the torus what we will get. So, in while doing Fourier analysis what we have seen is that we need a structure of a group. Now this T n we can consider it is an additive group with the some modulo 1 that is what exactly as in the dimension 1. And it also can be identified e to the power 2 pi i x 1 e to the power 2 pi i x n which is an element in C n such that x 1 to x n this belongs to the cube this with this notation fixed. Now we need to define what is do we mean by the periodic function. So, f is let us take a function from R n to C such that f of x plus m this is equal to f of x for all x in R n and m in Z n. So, we call this as one periodic function and therefore, easily one can see that the integral over T n f of x dx as usual when we are talking about the integral over T n what we mean is that this is the Riemann integration what we are defining over the cube and identifying it the T n with the cube and so that is by definition what this is what I mean by that. This is going to be equal to you take any cube with the length 1 wherever it is positioned in R n that is also going to give us because of the periodicity of f is going to give us the same result that means this is a 1 a 1 plus 1 cross a n a n plus 1 and then this is f of x dx for all a i belongs to R. So, exactly this is what is the periodicity it does not matter as long as you are in the cube of length 1. So, it does not matter where your cube is in R n the integral is going to be same if one is dealing with a periodic function. Now, analogously we will define the Fourier coefficient suppose f is a continuous function. So, Fourier coefficient this is defined here there you know it was defined on Z and here we are going to define on Z n here the building block is going to be e to the power 2 pi i m dot x. So, for m belongs to Z n we define f hat of m this is equal to integral over T n f of x e to the power minus 2 pi i x dot m dx. This is also e to the power 2 pi i x dot m that one can easily check that this is a one periodic function in on R n. So, this is the natural definition of the Fourier transform. So, easy to see or naturally it will just you follow the proof we will just note it down that f plus g hat at m this is equal to f hat at m plus g hat at m and alpha times f hat at m this is equal to alpha f hat of m this is true for all m. Now, if I take f bar instead of f and I take the m as what we have seen is that if we take the bar above outside the integral then you are going to get that this is f hat of minus of m and then bar. Now, the because it has a group structure. So, it is going to have a translation. So, now tau y of f if you are defining this is equal to f of x minus of y then what we get is that tau y f hat at m this is equal to e to the power minus 2 pi i m dot y and f hat of m and similarly if I multiply e to the power 2 pi i k dot x and with f and then if I am taking the Fourier transform then this is going to be translated in the side of the Fourier transform. So, this exactly the proof is the same as this. Now, we define the important concept of f convolution of g it has a group structure. So, we can define it x minus of y g of y dy and just changing the order of integration we can arrive the product formula for the convolution. So, already we have noticed what do we mean by the derivative in the higher dimension. So, we say that with just to recall d alpha where alpha is a multi index where each of this alpha i belongs to N union of 0 alpha is a non negative and mod of alpha d alpha of f. Now, we have introduced this notion while doing. So, now if I take the Fourier transform this is going to be m, m here is in Z n and then the multi index alpha to the power f hat of m for m varies over Z n. We know that what is m to the power alpha that means m 1 to the power alpha 1 m 2 to the power alpha 2 up to m n to the power alpha m. So, then one of the most important thing is the trigonometric polynomial. So, what one would define this is summation over m varies over some subset of f and alpha m e to the power 2 pi i m dot x where f is a subset which is finite f is any finite subset of Z n. Then this is what it is going to be the trigonometric polynomial a degree of p, p is the largest number largest of m 1 plus m n such that alpha m or rather we have denoted this as mod m alpha m is not equal to 0. So, and one of the most fundamental property what we have seen is one can prove this, this is T n e to the power 2 pi i m dot x d x this is equal to 0 if m is not equal to 0 and is equal to 1 if m is equal to 0. So, therefore, if I compute the Fourier transform p hat of m this is equal to 0 if the degree of the polynomial is n and mod n is greater than n p has degree ok. So, these are the standard notation and the easy properties of the Fourier coefficients in the higher dimension which can be derived with the same idea that of in for dimension 1 ok. So, now we need to define the Dirichlet kernel. Now Dirichlet kernel is going to be the first if you recall the in dimension 1 it is minus n to n the sum varies over minus of n to n. So, way of defining this is d n of n this we can define to be summation over e to the power 2 pi i m dot x this is the dimension of n and at the point x that is what we are defining. So, now the sum there it is from minus n to n here now what we can take the value of minus m to m it now our m is in n tuple. So, suppose let us take n equal to 2. So, m 1 and m 2. So, we have various choice. So, we can take both m 1 and m 2 we can take mod of m i is lesser equal to n for each i or we can take mod of m is lesser equal to n mod of m means m 1 square plus m 2 square whole to the power 1 half this is lesser equal to n. So, now in dimension 1 both these sums they agree, but in higher dimension definitely they are going to be different. So, what we will call this as the classical Dirichlet kernel if m varies over Z n and mod of m i is lesser equal to n that is what is our classical Dirichlet and then we will call it at the spherical will denote it as d n tilde dimension is n at the point x mod of m is lesser equal to n and e to the power 2 pi i m dot x. So, for dimension 1 as we were discussing d n is equal to d n tilde, but in higher dimension they are going to be 2 different object. So, or this classical or you can take the square you can call it as the square Dirichlet kernel. So, now we will concentrate in this lecture on the square Dirichlet kernel. So, the square Dirichlet kernel as you can see d n and x this is going to be d n the single one of x 1 multiplied by d n of x n. So, this is nothing but the product of i is equal to 1 to n sin 2 n plus 1 pi x i divided by sin pi x i that is what we have seen because this is nothing but the product it is lying on the rectangle. So, it is on the product of this. So, then the fair kernel corresponding to this, this is the average. So, now what one can get this 1 by n to the power n then there will be m sum k 1 is equal to 0 to n k 2 is equal to 0 to n k n equal to 0 to n and then because if you take the sum and then take the product there will be many cross term this is x 1 to d of k n of x n this is the average of the product of the Dirichlet kernel. So, that is our fair kernel. So, one can actually one can easily write it down that this F n for the dimension at the point x this is equal to summation over m and mod of m i is lesser equal n we are taking the square Dirichlet kernel and then this is 1 minus mod m 1 by n plus 1 and then 1 minus mod m 2 by n plus 1 and it goes on 1 minus mod m n by n plus 1 and e to the power 2 pi i m dot x. This again with the similar kind of argument as in dimension 1 one can show that this is to 1 to n sin pi n plus 1 x j sin square divided by sin square pi x j F n a good kernel and for continuous F convolution of k n this converges to F uniformly as n goes to infinity and as you can see that F convolution with the fair kernel is nothing but a polynomial and so F convolution of F n which is n dependency of x this is going to be summation m varies over z n and then this is mod of m i lesser equal to n and then 1 minus mod m 1 by n plus 1 1 minus mod m n by n plus 1 F hat of m and e to the power 2 pi i m dot x. So, this is again a trigonometric polynomial and this trigonometric polynomial is going to be dense in this and to conclude what the result this is also what one can see if summation over m varies over z n mod of F hat of m this is finite then F of x because that is what we have seen F convolution of k n this is going to converse to F uniformly therefore, so this as application of the fair theorem this is what we will get and this is what is the inversion formula for us if summation over mod of F hat of n is finite it if it is summable then this happens and also for continuous function one can get that dx this is also this is going to be F hat of m square this is nothing but the Planck's rule. So, we will have the parallel one and also the partial identity if F and G are continuous then this we can write this as. So, all these are going to be true and there is no new idea is needed and almost whatever the idea is needed we have discussed this over here and so this is the Fourier series for n torus and now natural question is that we have defined the Fourier transform in the higher dimension in R n and we have the Fourier series in T n. So, like the dimension one case here we would like to ask our self in fact this is going to be true what is the Poisson summation formula and deriving is will be just as same as in dimension one then by using this Poisson summation formula what are the result we can get in higher dimension that is what we are going to focus in the next lecture as a part of our application. Thank you.