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CONVERGENCE of FOURIER SERIES Contents 1. Introduction 1 2. Basic Definition of the Fourier Series 1 3. Summability and Kernels

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CONVERGENCE of FOURIER SERIES Contents 1. Introduction 1 2. Basic Definition of the Fourier Series 1 3. Summability and Kernels CONVERGENCE OF FOURIER SERIES BRYAN RUST Abstract. The purpose of this paper is to explore the basic question of the convergence of Fourier series. This paper will not delve into the deeper ques- tions of convergence that measure theory illuminates, but requires only the basic principles set out by introductory real and complex analysis. Contents 1. Introduction 1 2. Basic Definition of the Fourier Series 1 3. Summability and Kernels 2 4. The Fourier Series of a Continuous Function 7 5. The Mean-Square Convergence 8 Acknowledgments 10 References 10 1. Introduction The Fourier series of an integrable function is defined, and with the definition of Ces`aromeans and Abel means, this allows the Fourier series to be recognized as an approximation of the function. We can define a sense of convergence for continuous periodic functions and a \mean-square convergence" over the entire function for Riemann integrable functions. 2. Basic Definition of the Fourier Series We shall refer to all complex-valued 2π-periodic functions on R as functions on the circle. We will also assume functions on the the circle are Riemann integrable on some interval of [a; b] with b − a = 2π. This assumption may be repeated by saying simply a function on the circle is integrable. Definition 2.1. The Nth Fourier coefficient of f is defined to be Z π 1 −inx= f^(n) = f(x)e dx; n 2 Z: 2π −π Note that since f is 2π-periodic, we may replace the endpoints in the above integral with any real a < b such that b − a = 2π. Date: August 19, 2013. 1 2 BRYAN RUST Definition 2.2. The Fourier series associated with the function f(x) is 1 X f^(n)einx: n=−∞ We shall denote this association as 1 X f(x) ∼ f^(n)e2πinx=2π: n=−∞ The Nth partial sum of the Fourier series or the truncated Fourier series of f is defined to be is N X 2^πinx=2π SN f(x) := f(n)e : n=−N 3. Summability and Kernels It is useful to define certain notions of \means" which will aid in the question of the convergence of Fourier series. Specifically, convolutions and the notion of a family of \good kernels" are useful for questions of convergence. Definition 3.1. The Nth Ces`aromean or the Nth Ces`arosum of the sequence of P1 partial sums fσng of the series of complex numbers k=0 ck is defined to be a + a + ::: + a σ = 0 1 N−1 : N N th where the aN are the N partial sums of the given series of complex numbers P It is well known that if ck converges to a limit σ, then the Ces`aromeans converge to σ. P1 Definition 3.2. A series of complex numbers, k=0 ck, is said to be Abel sum- mable to s if for every 0 ≤ r < 1, the series 1 X k A(r) = ckr k=0 converges, and lim A(r) = s: r!1 The Abel means of the series are defined to be the quantities A(r). Definition 3.3. The convolution of two functions f and g on the circle (denoted by f ∗ g) is defined to be 1 Z π (f ∗ g)(x) = f(y)g(x − y)dy: 2π −π And as before, since the convolution is a function on the circle we may choose limits of integration [a; b] such that b − a = 2π. Proposition 3.4. Given that f, g, are h are integrable functions on the circle, we have (1) f ∗ (g + h) = (f ∗ g) + (f ∗ h); (2) (cf) ∗ g = c(f ∗ g) = f ∗ (cg) for any c 2 C; (3) f ∗ g = g ∗ f; (4) (f ∗ g) ∗ h = f ∗ (g ∗ h); CONVERGENCE OF FOURIER SERIES 3 (5) f ∗ g is continuous (if either f or g is continuous, (6) f[∗ g(n) = f^(n)^g(n): Proof. (1)-(4) follow easily from definitions (5) Suppose g is continuous and given > 0; there exists δ > 0 so that if js−tj < δ, then jg(s) − g(t)j < . We have jx1 − x2j < δ implies j(x1 − y) − (x2 − y)j < δ for any y. So Z π 1 j(f ∗ g)(x1) − (f ∗ g)(x2)j ≤ f(y)[g(x1 − y) − g(x2 − y)]dy 2π −π 1 Z π ≤ jf(y)jjg(x1 − y) − g(x2 − y)jdy 2π −π Z π ≤ jf(y)jdy ≤ B 2π −π where jf(x)j ≤ B for all x 2 [−π; π]. (6) This follows from Fubini's theorem 1 Z π f[∗ g(n) = (f ∗ g)(x)e−inxdx 2π −π 1 Z π 1 Z π = f(y)g(x − y)dy e−inxdx 2π −π 2π −π 1 Z π 1 Z π = f(y)e−iny g(x)e−inxdx dy 2π −π 2π −π =f^(n)^g(n): 1 Definition 3.5. A family of integrable functions fKngn=1 on the circle is said to be a family of good kernels if it satisfies the following three properties: (a) For all n ≥ 1, 1 Z π Kn(x)dx = 1: 2π −π (b) There exists M > 0 such that for all n ≥ 1, Z π jKn(x)jdx ≤ M: −π (c) For every δ > 0, Z jKn(x)jdx ! 0 δ≤|x|≤π as n ! 1: The significance of good kernels in the context of Fourier analysis is seen in their connection with convolutions. 1 Theorem 3.6. Let fKngn=1 be a family of good kernels, and let f be an integrable function on the circle. Then whenever f is continuous at x, lim (f ∗ Kn)(x) = f(x): n!1 If f is continuous, then convergence is uniform on [−π; π]. 4 BRYAN RUST Proof. If > 0 and f is continuous at x, choose δ such that if jyj < δ, then jf(x − y) − f(x)j < . Using the first property of good kernels, we can write 1 Z π (f ∗ Kn)(x) − f(x) = Kn(y)f(x − y)dy − f(x) 2π −π 1 Z π = Kn(y)[f(x − y) − f(x)]dy: 2π −π Taking the absolute value, 1 Z π j(f ∗ Kn)(x) − f(x)j =j Kn(y)[f(x − y) − f(x)]dyj 2π −π 1 Z ≤ jKn(y)jjf(x − y) − f(x)jdy 2π jyj<δ 1 Z + jKn(y)jjf(x − y) − f(x)jdy 2π δ≤|y|≤π Z π 2B Z ≤ jKn(y)jdy + jKn(y)jdy; 2π −π 2π δ≤|y|≤π where B is a bound for jfj. The second property of good kernels shows that Z π M jKn(y)jdy ≤ ; 2π −π 2π for all n ≥ 1. The third property of good kernels shows that 2B Z jKn(y)jdy ≤ , 2π δ≤|y|≤π for all n ≥ N(δ): Therefore we have j(f ∗ Kn)(x) − f(x)j ≤ C. This proves the first part of the theorem. For the second part, we note that if f is continuous on [−π; π], then δ can be chosen independent of x. The Nth Dirichlet kernel is defined to be N X inx DN (x) = e n=−N Its definition is motivated by writing the Nth partial sum of the Fourier series of fas a convolution: N X ^ inx SN (f)(x) = f(n)e n=−N N X 1 Z π = f(y)e−inyd einx 2π n=−N −π N ! 1 Z π X = f(y) ein(x−y) dy 2π −π n=−N =(f ∗ DN )(x): CONVERGENCE OF FOURIER SERIES 5 If the Dirichlet kernels were a good family of kernels, then we could develop a sense of convergence from the partial sums of the Fourier series. However, we will see that the family of Dirichlet kernels is not a family of good kernels. In particular, the Dirichlet family of kernels does not satisfy the second property of good kernels. First, we have a closed form expression for the Nth Dirichlet kernel, sin((N + 1=2)x) D (x) = N sin(x=2) We can derive this using the geometric series identity and trigonometric identities, N X inx DN (x) = e n=−N N N−1 X X = einx + e−i(n+1)x n=0 n=0 1 − ei(N+1)x 1 − e−iNx = + 1 − eix 1 − e−ix sin((N + 1=2)x) = sin(x=2) Now Z π Z π sin((N + 1=2)x) jDN (x)jdx = −π −π sin(x=2) We can bound the integrand from below and change variables to obtain Z π Z π j sin((N + 1=2)x)j jDN (x)jdxj ≥c dx −π −π jxj Z Nπ j sin(x)j ≥c dx π jxj N−1 X Z (k+1)π 1 =c k k=1 kπ ≥c log(n) where c > 0. This implies that the Dirichlet kernels are not a family of good kernels. However, using the summing techniques previously developed, we can define good families of kernels related to the Dirichlet kernels and the Fourier series. We now present two families of kernels that are good kernels: (1) the Fej´erfamily of kernels and (2) the Poisson family of kernels (1) Our first example's definition is motivated by writing the Nth Ces`aromean of the Fourier series as a convolution. S (f)(x) + S (f)(x) + ::: + S (f)(x) σ (f)(x) = 0 1 N−1 : N N We have Sn(f) = f ∗ Dn. Hence σN (f)(x) = (f ∗ FN )(x) The Nth Fej´erkernel is defined to be D (x) + ::: + D (x) F (x) = 0 N−1 : N N 6 BRYAN RUST A closed form expression for the Fej`erkernels is given by 1 sin2(Nx=2) FN (x) = : N sin2(x=2) sin((N+1=2)x) We have shown previously that DN = sin(x=2) : Using the first expression for the Nth Fej´erkernel, N−1 X NFN (x) = Dn n=0 sin(x=2) sin((N − 1=2)x) = + ::: + sin(x=2) sin(x=2) sin2(x=2) sin(3x=2) sin(x=2) sin((N − 1=2)x) sin(x=2) = + + ::: + : sin2(x=2) sin2(x=2) sin2(x=2) We have the identity sin((k + 1=2)x) sin(x=2) = 1=2(cos(kx) − cos((k + 1)x): Thus, we have telescoping, and our previous expression becomes 1=2(1 − cos(Nx)) NFN (x) = sin2(x=2) or 1 sin2(Nx=2) FN (x) = : N sin2(x=2) (2) The definition of the second family is motivated by writing the Abel mean P1 inx of the Fourier series as a convolution f(x) ∼ n=−∞ ane , which is 1 X jnj inx Ar(f)(x) = r ane n=−∞ This is defined to be Ar(f)(x) = (f ∗ Pr)(x) The Nth Poisson kernel is defined to be 1 X jnj inx Pr(x) = r e : n=−∞ If 0 ≤ r < 1, then we have a closed form expression 1 − r2 P (x) = r 1 − 2r cos(x) + r2 To see this, split the kernel into the expression: 1 1 X n X n reiθ + re−iθ n=0 n=1 By DeMoivre's formula and by summing a geometric series, we obtain the desired closed form.
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