Fejer's Theorem S. Ziskind 1 Discussion We provide a very brief overview of Fourier Series, and prove Fejer's The- orem, which illustrates the use of Cesaro Means in establishing a pointwise convergence property of such series. Several consequences of the theorem will also be noted. The overview is drawn from [1], while the proof of Fejer's Theorem is based on [2]. It must be noted that we ignore technical issues involving the definition of integrals using Lebesgue measure instead of Riemann sums. This is discussed in [1] and [3] among many other books, but is out of scope for this memo. 2 Fourier Series The general goal of harmonic analysis is to represent functions as the sum of simpler functions. Here we consider expressing periodic functions on a fixed interval as the sum of sines and cosines, which leads to the study of Fourier Series. To make things specific let us take the interval to be [−π; π], with the un- derstanding that any function of interest satisfies f(x + 2π) = f(x). The obvious periodic functionsp on the interval are sin nx and cos nx, where n ≥ 0 is an integer, and i = −1. Because of Euler's identity eiθ = cos θ + isin θ, we have eiθ + e−iθ eiθ − e−iθ cos θ = and sin θ = 2 2i which says that we can equivalently break periodic functions into a sum of the exponentials einx, for integral values of n, both positive and negative. Being easier to work with, we switch to this approach, and define the functions 1 inx en(x) = e , for integral n. Note that en = e−n, where the overbar indicates complex conjugation. A direct calculation shows that 1 Z π ek(x)ej(x)dx = δkj 2π −π where δkj, the Dirac delta "function", is either 1 or 0, depending as to whether j = k or not. Recalling basic calculus, the integral should make us think of a Riemann sum of products, and linear algebra tells us that the sum of products is a dot product (a.k.a inner product, a.k.a scalar product), where the conjugation reminds us to think of complex Euclidean space of many dimensions, Cn. When working in n complex dimensions we define the inner product of 2 vectors as < X; Y >= Pn x y , and then use this to define the norm of a p 1 k k vector as kXk = < X; X >, and distance via kX − Y k. These definitions align with the ordinary geometric notions in 2 and 3 dimensions, and extend angle measure and the like into n dimensions. It is natural to extend the notions of inner product, norm and distance into infinitely many dimensions, i.e. the space of infinite sequences of complex numbers, C1, by letting 1 1 X 2 X 2 < X; Y >= xkyk and kXk2 = jxkj k=1 k=1 but we face a new problem: infinite series don't always converge. The solution is obvious: restrict attention to (2 sided) infinite sequences for which 1 X 2 jxkj < 1 k=−∞ In addition to making norms finite, the restriction also makes all dot products finite. The space of such sequences, along with the associated dot product, is known as l2, the space of square summable sequences (extending in both directions). 2 The natural continuous analogy replaces a sum with an integral, and as before we need to restrict attention to those functions on [−π; π] for which 1 Z π jf(t)j2dt < 1 2π −π and we define the inner product by 1 Z π < f; g >= f(t)g(t) dt 2π −π These functions comprise the space L2, the space of square integrable func- tions. (The integrals are taken in the sense of Lebesgue, not Riemann.) Working again by analogy to the finite dimensional spaces, we see that the functions ek are orthogonal and of unit length. Thus we can treat them as basis vectors in L2 and form the projection of a vector onto any of these vectors as Z π Z π Z π ^ 1 1 1 −int f(n) = f(t)en(t) dt = f(t)e−n(t) dt = f(t)e dt 2π −π 2π −π 2π −π The numbers f^(n) are the Fourier Coefficients of f, and the sequence of all ^ 1 of them, ff(n)g−∞ is the Fourier Series of f. The following results, given without proof (see [1]), show that L2 and l2 behave strikingly like Cn. 2 1 Completeness: L is complete, with feng−∞ acting as an orthonormal basis. ^ Plancherel's Theorem: kfk2 = kfk2 Parseval's Theorem: R f g = P f^g^ Theorem: If we define k=+n X ^ sn(x) = f(k) ek(x) k=−n then kf − snk2 ! 0 as n ! 1. 3 3 Fejer's Theorem The last theorem of the preceding section may be re-stated as: \The sym- metric partial sums of the Fourier series of an L2 function converge to the function in the L2 norm." In many ways this is the most natural sense of convergence for an L2 func- tion's Fourier series, but there is a more basic form of convergence that this doesn't address. We ask: Do the partial sums converge pointwise to f: ? sn(x) ! f(x). More generally, we can ask when this happens, for what sort of functions, and how often. This turns out to be a surprisingly difficult and deep issue. It has long been known that pointwise convergence can fail at some points, even for continuous functions. Even worse, there is an example of a function R satisfying jfj < 1 for which sn(x) diverges at every single value of x! This made people suspect that the general problem was hopeless. Never- theless, Lennart Carleson proved, in 1966, that the partial sums do in fact converge for almost every value of x when f 2 L2, so in particular for con- tinuous f. This theorem is deep, complex, and far beyond the scope of this memo. Prior to Carleson, Fejer (1904) found a relatively simple way of capturing the value of a continuous function in the pointwise sense, but he needed to smooth the values of the partial sums. Given the sums sn(x), define the Cesaro Means of this sequence as n−1 1 X σ (x) = s (x) n n k k=0 Fejer's Theorem: If f is a bounded periodic function on [−π; π] that is continuous at x, then the Cesaro means of the symmetric partial sums of its Fourier series converges pointwise to f at x. i.e. σn(x) ! f(x). If f is continuous on the whole interval then the convergence is uniform. Proof - Step 1: We start the proof of Fejer's Theorem by evaluating n n X X 1 Z π s (x) = f^(k)e (x) = e (x) f(t)e (t)dt n k k 2π k −n −n −π 4 n 1 Z π X 1 Z π = ( e (x − t))f(t)dt = f(t)D (x − t)dt 2π k 2π n −π −n −π where Dn(y), the Dirichlet Kernel, is defined and evaluated (noting it to be a geometric series) as n X sin(n + 1 )y D (y) = eiky = ··· = 2 n sin(y=2) −n We will not use it in this form, but simply note its graph for n=10. 25 20 15 10 5 0 -5 -4 -2 0 2 4 Figure 1: Dirichlet's Kernel, n=10 5 It should be noted that both sn and σn are both expressed as integrals of the form Z f(t)g(t − x)dt where g is either the Dirichlet kernel or the Fejer kernel. In general, an integral of this form is called the convolution of f and g. In essence it takes the function f and overlays it with the function g, but with g shifted so that it is centered at x. Because of periodicity, any part of either D or K that is shifted over the edge of [−π; π] simply wraps around and reappears on the other side of the interval. Much more can be said about convolution, and it is important for many applications (such as signal processing). An equivalent form of Dn(y), more useful for the next step of the proof, is that n n n X X X 1 − ei(n+1)y 1 − e−i(n+1)y D (y) = eiky = eiky + e−iky − 1 = + − 1 n 1 − eiy 1 − e−iy −n 0 0 (1 − ei(n+1)y)(1 − e−iy) + (1 − e−i(n+1)y)(1 − eiy) = − 1 (1 − eiy)(1 − e−iy) cos ny − cos(n + 1)y = ··· = 1 − cos y Proof - Step 2: Next we observe that n−1 n−1 1 X 1 X 1 Z π σ (x) = s (x) = f(t)D (x − t) dt n n k n 2π k 0 0 −π n−1 1 Z π n 1 X o 1 Z π = f(t) D (x − t) dt = f(t)K (x − t) dt 2π n k 2π n −π 0 −π where Kn(y), the Fejer Kernel, is defined as n−1 1 X K (y) = D (y) n n k 0 Using the second expression for the Dirichlet kernel, we note that cos ny − cos(n + 1)y (n + 1)K (y) − nK (y) = D (y) = n+1 n n 1 − cos y 6 Given this expression for Kn, we note first that K1(y) ≡ 1, and then that 1n cos y − cos 2y o 1n1 − cos 2y o K (y) = 1K (y) + = 2 2 1 1 − cos y 2 1 − cos y 1n cos 2y − cos 3y o 1n1 − cos 3y o K (y) = 2K (y) + = 3 3 2 1 − cos y 3 1 − cos y and by induction (and the double angle formula) we find 1 h1 − cos ny i 1 hsin(ny=2)i2 K (y) = = n n 1 − cos y n sin(y=2) Here is its graph for n = 10.