 Hello, welcome everyone to this course on Fourier analysis and applications and this is a course that is very popular among undergraduate students. No doubt you are seen vestiges of Fourier series in your undergraduate curriculum particularly while solving boundary value problems for various partial differential problems, particularly the Laplace's equation, the heat equation and the wave equation. So, you solve the Laplace's equation on a circle on a disk or the wave equation on a square etcetera and the technique is very routine. You try to expand the function as a Fourier series, a fairly general function f of x, you try to write it as a infinite superposition of sines and cosines namely f of x equal to a naught plus summation n from 1 to infinity a n cos n x plus b n sin n x the displayed equation 1.1 on the slide. Before we plunge into the theory behind these kinds of expansions, the Fourier expansions let us trace back to the early genesis of this theory of Fourier series. So, let us look at certain early trials and a little history. It always helps to start a course with a little historical background as to how it all began. A very important historical source from which I learned some of these things is this book of Ivor Gratin Guinness, Convolutions in French Mathematics 1800 to 1840 volume 2. This is in 3 volumes, the second volume is what we are talking about the turns. The book was published by Berkhauser in 1990. Let us look at a few aspects during the nascent stages of the development of Fourier analysis. The page numbers refer to this book of Gratin Guinness. Experts are expressing a fairly general function f of x as a series of the form 1.1 that we just saw predate Fourier by at least 50 years about half a century in the works of Daniel Bernoulli and Leonhard Euler. Again in connection with physical problems about vibrations and such. The perplexing issue was the following. How can a series of 2 pi periodic functions? All these functions sin x sin 2 x sin 3 x cos x cos 2 x cos 3 x they are all 2 pi periodic smooth functions on the real line. How can a superposition of such 2 pi periodic functions produce in general a priori non-periodic function? This problem seems to have languished in the state of affairs until the time of 4 year. Fourier's researches were pervasive in advocating the method of separation of variables. So commonly found today in modern undergraduate courses in PDEs. The method of separation of variables and the idea of using Fourier series of the expansions in terms of sines and cosines was not popular in as much as the solution presents itself in a form in which the boundary conditions remain masked in the coefficients of the expansion. Remember what do you do? When you write this method of separation of variables you get the solution as an infrared series and the coefficients are determined using the initial conditions of the boundary conditions etcetera. So the side conditions that you have when you solve a PDE they get incorporated and it is these coefficients of the sines and cosines that encode this information about the boundary data. So that is what I that is what it means when I write that the solutions present themselves in a form in which the boundary conditions remain masked in the coefficients of expansion 1.1 the a n's and the b n's. In contrast the functional form was preferred such as the Poisson's integral representation for solutions of the Laplace's equation of D'Alembert's solution of the wave equation in one space dimension. There you see the solutions written as integrals and the integrals involve directly the boundary condition or the initial conditions. Fourier also obtained interesting series expansions such as the one displayed in equation 1.2 in the slide that you see. Here you see the number pi by 4 has been written as a series of cosines and this is valid in the interval 0 to pi. Of course Fourier was well aware of the convergence issues there are subtle issues and he established rigorously the convergence of this equation of the series that appears in 1.2 and likewise issues pertaining to completeness also cropped up. Besides trigonometric functions Fourier also employed Bessel's functions in the summons of his series. This point was acknowledged by Lommel and Heiner. Lommel and Heiner were two significant contributors to the theory of Bessel's functions and this use of Bessel's functions by Fourier was acknowledged by Lommel and Heiner proposed the name Fourier Bessel's expansions for these. Now let us go to the researches of Lesion Dirichlet, Jordan and Paul Dubois Raymond. The first significant step towards the general convergence theorem is due to Dirichlet in 1829. So what did Dirichlet do? Dirichlet proved the first point wise convergence theorem. So for what kind of functions did Dirichlet prove the theorem of point wise convergence? He took functions which are piece wise continuous and monotone. So you take the interval minus minus pi to pi you chop it up into finitely many pieces and the function is continuous on each piece and it is monotone on each piece at the junction of between two intervals the function could have a jump discontinuity. So you could have a function which is defined piece wise and each piece is monotone the pieces may not join up continuously. At points of continuity the Fourier the series converges to the function f of x. At points of discontinuity the series converges to the arithmetic mean. We shall prove this theorem of Dirichlet rigorously at a later point in the course. We need some preparations such as the mean value theorem for integrals due to Bonnet. Dirichlet was of the view that the failure of point wise convergence would stem from some lack of integrability or the function f whose expansion is being sought. Jordan generalized the results of Dirichlet to include functions of bounded variations. We shall discuss later in detail what functions of bounded variations mean and we shall give that precise definitions later. In the year 1873 Paul Dubois Raymond produced an example of a continuous function whose Fourier series failed to converge. So even continuity or the function does not guarantee the convergence of the Fourier series. It is a very important point to be kept in mind. Continuity alone does not suffice and I have given a reference to a book by Kehane and others and I have listed all the three authors. The complete reference will be available in a few slides to follow. Let us look at some 20th century milestone. In 1926, A. N. Kolmogorov found a function in L1, L1 is the Lebesgue space, L1, L2, Lp these are the space of functions which are Lebesgue integrable on the closed interval minus pi pi. So Kolmogorov found a function which is integrable which is Lebesgue integrable on minus pi pi whose Fourier series diverges everywhere and N N Luzin conjectured that the Fourier series of a function in L2 must converge point wise almost everywhere. In particular, if you take a continuous function from minus pi to pi, a continuous function is in L2, right. A continuous function from minus pi to pi is obviously in L2. So although Dubois-Reymond established that the Fourier series of a continuous function may diverge, it must converge almost everywhere. That is the set of places where the Fourier series diverges has set of measure 0. The conjecture of Luzin was finally settled decades later by Leonard Carlson, it is a monumental achievement, later it was generalized by Hunt for Lp functions where p is strictly bigger than 1. These are some of the 20th century milestones. Let us move a little further and let us see after Dirichlet, in fact there is a bit of anachronism, we are going back to the 19th century here. We are going back to Bernard Riemann. After Dirichlet, the study of Fourier series was significantly furthered by Bernard Riemann in his habilitation shift which he wrote in 1854. In 1859, there appeared his epoch making memoir on the distribution of primes and the Riemann zeta function, where he gives a proof of the famous functional equation that bears his name and he gives two proofs in fact for the functional equation and one of them uses Fourier analysis. Basically what he does is that he derives the theta function identity of Jacobi and he uses the he via the melaintransform that theta function identity goes over to the functional equation for the zeta function and the theta function identity itself can be established via Fourier methods, namely you solve the heat equation in two ways using Fourier series and Fourier transforms and you compare the results invoking some uniqueness theorem for the initial value problem for the heat equation with periodic initial conditions. The theta function identity is as I said is related to the functional equation of the zeta function. Of course Riemann's memoir opened up as we know a new vista in the theory of analytic number theory. The pervasive nature of Fourier analysis, the theory of Fourier series and Fourier transforms has been profoundly generalized and its use extends far beyond the purpose for which it was originally invented, namely for solving partial differential equations. Today Fourier analysis is an indispensable tool in several diverse areas of mathematics such as probability theory, number theory, geometry to name a few. Besides the book of Gratton Guinness, the authoritative account of Kahane and Lemaire ought to be consulted for any serious understanding of the historical developments of the subject. The three of this book of Kahane and Lemaire is devoted to the work of Riemann. The discussion of Riemann's work can be found in this particular chapter 3 of this book. Let us now look at the impact of Fourier analysis in the late 19th and early 20th century mathematics. Fourier analysis provided much impetus for the development of point set topology as well as the development of measure theory. In fact, the significant researchers of Georg Cantor in set topology in the late 19th century and early 20th century was due to his interest in problems originating in Fourier analysis. There are two references for this that are very important. One is by A. Keshiris set theory in the uniqueness of trigonometric series and this a reading of this would be an excellent project for a student in the master's level. The second reference that I would like to point out is Roger Cook's beautiful article a somewhat lengthy article, but a very very comprehensive article on the uniqueness of Fourier series and descriptive set theory and he spans more than a century 1870 to 1985. It appeared in 1993 in the archive of the history of exact sciences and let us begin with the first chapter the basic notions. Let us go through the plan of the chapter. In this chapter we focus on the following. We look at a class of 2-py periodic functions of the real line. There are Lipschitz continuous and Holder continuous of exponent alpha. Now by the way these words which appear in blue are new concepts and concepts and like Lipschitz continuity, Holder continuity, Riemann, LaBaglama they will be highlighted in blue and names of references will be highlighted in purple. So, Lipschitz continuous functions and Holder continuous functions they will be defined later. We prove a fundamental theorem called the Riemann LaBaglama for functions in L1. The Riemann LaBaglama says that if you take a function which is in L1 then the Fourier coefficients will decay to 0 as n goes to infinity. This is a very important step in the proof of the convergence theorem. Then we will derive an expression for the so called Dirichlet kernel and the behavior of the Dirichlet kernel is extremely important with regard to the convergence of the Fourier series. We must examine the nature of the singularity of the Dirichlet kernel. The Dirichlet kernel is an oscillatory kernel. It is a highly oscillatory function near the origin and it is these oscillations and the mutual cancellations that are to be studied very carefully. And we will also try to understand the obstacle to proving point wise convergence. When the function is merely assumed to be continuous and nothing more the argument breaks down. Whereas, if you have a little extra regularity such as Holder continuity or Lipschitz continuity then we can prove the convergence theorem. The singularity in the Dirichlet kernel is to some extent destroyed by this Holder continuity or Lipschitz continuity or whatever. Let us look at a simple example of Lipschitz continuous function. What are the Lipschitz continuous function? Let us recall a function is said to be Lipschitz continuous if mod f x minus f y is less than or equal to L times mod x minus y. Formal definition will come later. I will just speak the definition out in a later slide we will put it down it will appear explicitly. So, you have probably encountered Lipschitz continuity in your courses particularly in your course on ordinary differential equation where we prove existence uniqueness under a Lipschitz condition under a Lipschitz hypothesis. So, one of the popular conditions to impose in the function is Lipschitz continuity. So, let us look at a simple example of a Lipschitz continuous function take cos A x on the interval minus pi pi is a smooth function on the on the interval minus pi pi and it is an even function being an even function it takes the same value at minus pi and pi. So, what I do is I simply take the 2 pi periodic extension. The 2 pi periodic extension is just a bunch of arches one following the other, but at the at the at the junction from minus pi to pi and from pi to 2 pi at the point pi this extension will not be differentiable, but it will be holder continuous it will be in fact, it will be even be Lipschitz continuous. So, the 2 pi periodic extension of cos A x is an example of a holder continuous function. So, the basic convergence theorem that we prove will hold for this cos A x. So, if you take this function cos A x and write the Fourier series for that we are going to get this expression we are going to derive all these things later just I am just giving you a sneak preview of what is going to come. Cos A x is when you write it as a Fourier series the constant term is sin pi A upon pi A. Since cos A x an even function in the Fourier series will only contain even functions. So, the sin n x term will not be there at all. What are the coefficient of cos n x you see displayed out there minus 1 to the power n upon A squared minus n squared and there is a common 2 A sin A pi upon pi for all the terms that has been pulled out of the summation sin. So, here is a Fourier series for cos A x and this is going to be true for all values of x because cos A x is the 2 pi periodic extension of cos A x is Lipschitz continuous on the entire real line. So, I am allowed to put whatever value of x I want and this equation will be valid. So, let us put x equal to 0 put x equal to the left hand set becomes 1 and the right hand set instead of cos n x I simply get minus 1 to the power n upon A squared minus n squared and a little rearrangement will give you this partial fraction expansion or the Metag Leffler expansion for cosecant pi A. The partial fraction decomposition for the cosecant can be derived using Fourier series and you see displayed cosecant pi A equal to 1 upon pi A plus 2 A by pi summation n from 1 to infinity minus 1 to the power n upon A squared minus n squared. Similar expansions for cotangent and other things can also be derived using just the basic convergence theorem we prove the basic convergence theorem and these are some of the corollaries that we can derive. So, it gives back some very beautiful formulas in classical analysis using these expansions using these Metag Leffler expansions we can find for instance special values of the zeta function. We shall discuss the special case of the Poisson summation formula for the using the Gaussian. We take the Gaussian and we use the Gaussian to cook up a 2 pi periodic function determine the Fourier series and will derive a special case of the Poisson summation formula and will also derive the Jacobi theta function identity. As mentioned earlier this is equivalent to Riemann's functional equation for the zeta function. We will also discuss the Bernoulli numbers and will obtain a generating function for the Bernoulli numbers and we shall get special values of zeta functions. We close the chapter with a discussion of the generating function for the Bessel's function of the first kind. We will recall what the Bessel's functions are using ordinary differential equations. So, there will be a slight bit of digression to get the requisite Bessel's function identities recursion formulas for the Bessel's function and these Bessel's functions will be embedded into a power series and a generating function will be determined in closed form using Fourier analysis. And later in a later chapter we will use these in a in an application to a problem in celestial mechanics. Now this concludes the historical sketch of the early genesis of Fourier analysis and also the plan of the first chapter that we shall be studying. So, now let us formally begin the chapter 1 the basic convergence theorem. So, to set the stage let us begin with the 2 pi periodic function f from r to r which at present is quite arbitrary except that over the interval minus pi to pi the integral exists. Now it is extremely important to use the Lebesgue integrals in our discussion it is not convenient to use Riemann integration. The space we want spaces that are complete Cauchy sequences must converge. So, l 2 for example, so we must work with l 2 of minus pi pi in a later chapter. So, Lebesgue integrals are absolute must. So, Lebesgue theory is a must. So, for technical reasons we must use Lebesgue integrals. So, we assume that the function f is integrable in the sense of Lebesgue on the interval minus pi pi. As explained earlier the objective is to express f as an infinite series f of x equal to a naught plus summation n from 1 to infinity a n cos n x plus b n sin n x a series of the form 1.1 is called a trigonometric series. The fundamental question of course is regarding the meaning of the equality in 1.1. For instance in what sense does the series in 1.1 converge? The series can converge in a variety of ways already you know in your undergraduate courses you can talk about point wise convergence, you can talk about uniform convergence, you know that uniform convergence is very strong point wise convergence is very weak. To list some of the alternatives let us look at what are the other modes in which 1.1 can converge. So, first let us add the first 2 n plus 1 terms of the series I said 2 n plus 1 because there will be cause and there is a sign and there is a constant term. So, I will take n cosines n sines and the constant. So, a naught plus summation a j cos j x plus b j sin j x j from 1 to n. So, I take the first n terms involving the cosine the first n terms involving the sin and the constant term. So, s n of f of x is the nth partial sum of the series, but this nth partial sum really is a sum of 2 n plus 1 terms. I am going to call it the nth partial sum just for convenience. So, we say that the series converges to f of x point wise almost everywhere if this partial sum s n f x converges to f x as n tends to infinity for all x except on a set of measure 0. We say this the series f 1.1 converges to f x in mean if the difference s n f x minus f x the norm the l 2 norm converges to 0. That is if the sequence s n converges to f in l 2 the third form of convergence is Cesaro convergence that you take s 1 x s 2 x s n x you take and then you take the arithmetic means of these partial sums. The arithmetic means of these partial sums must converge to f x. These are the 3 important modes of convergence that one can think about. One can also talk about distribution convergence in the sense of distributions. One can also try to include that, but for that we need to develop the theory of distributions. I do not think you have studied distribution theory in your earlier courses. You are for sure studied point wise convergence. You also studied mean convergence or if you are an engineer you would have encountered this so called RMS root mean squared. They integrate from minus pi to pi mod g x squared dx and you normally put a 1 upon 2 pi to make it an average value and then you take the square root. So, that dimensionally everything is correct. So, it is a RMS in electrical engineering you will call it the convergence in RMS converges a root mean squared. And the third is Cesaro convergence. These 3 modes of convergence are going to be extremely important. We are going to devote a whole chapter to mean convergence and we are going to devote an entire chapter to Cesaro convergence. And Cesaro convergence will give us a very important theorem on Wiles equidistribution theory. So, we shall need of course, we are not going to get these for free. The F must satisfy additional hypothesis before we can get any of these things. For the mean convergence we will need that F must be in L 2. For point wise convergence we are going to assume that the function is holder continuous to exponent alpha or lipchitz continuous or something or we could assume that the function is piece wise continuous and monotone that is Dirichlet's theorem. So, Cesaro convergence for example, if F is continuous just continuity then we know from Paul Dubu Arayman's example that point wise convergence fails, but Cesaro convergence will happen. But not only that Cesaro convergence will happen very strongly if FFX is continuous then these Cesaro means actually converge uniformly. It is far better we get something much much better. So, there are there is a tradeoff sometimes we need additional hypothesis if you want point wise convergence we need better than continuity like lipchitz continuity holder continuity monotonosity piece wise monotonosity. Mean convergence is easy we just need F should be in L 2 and the Cesaro convergence it is not point wise convergence it is convergence of arithmetic means F is just assumed to be continuous, but the Cesaro means give you uniform convergence. Why does this happen? We will understand this better when we average the Dirichlet kernel it improves we get what is called as a Feier kernel. The last thing that I talked about is known as a Feier's theorem we will have a whole chapter on this Cesaro convergence. I think this is a very good place to stop this particular module and I think I will stop here and we will continue this in the next module we will derive things and we will develop the theory further. So, thank you very much and goodbye.