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CONVERGENCE of FOURIER SERIES in Lp SPACE Contents 1. Fourier Series, Partial Sums, and Dirichlet Kernel 1 2. Convolution 4 3. F

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CONVERGENCE of FOURIER SERIES in Lp SPACE Contents 1. Fourier Series, Partial Sums, and Dirichlet Kernel 1 2. Convolution 4 3. F CONVERGENCE OF FOURIER SERIES IN Lp SPACE JING MIAO Abstract. The convergence of Fourier series of trigonometric functions is easy to see, but the same question for general functions is not simple to answer. We study the convergence of Fourier series in Lp spaces. This result gives us a criterion that determines whether certain partial differential equations have solutions or not.We will follow closely the ideas from Schlag and Muscalu's Classical and Multilinear Harmonic Analysis. Contents 1. Fourier Series, Partial Sums, and Dirichlet Kernel 1 2. Convolution 4 3. Fej´erkernel and Approximate identity 6 4. Lp convergence of partial sums 9 Appendix A. Proofs of Theorems and Lemma 16 Acknowledgments 18 References 18 1. Fourier Series, Partial Sums, and Dirichlet Kernel Let T = R=Z be the one-dimensional torus (in other words, the circle). We consider various function spaces on the torus T, namely the space of continuous functions C(T), the space of H¨oldercontinuous functions Cα(T) where 0 < α ≤ 1, and the Lebesgue spaces Lp(T) where 1 ≤ p ≤ 1. Let f be an L1 function. Then the associated Fourier series of f is 1 X (1.1) f(x) ∼ f^(n)e2πinx n=−∞ and Z 1 (1.2) f^(n) = f(x)e−2πinx dx: 0 The symbol ∼ in (1.1) is formal, and simply means that the series on the right-hand side is associated with f. One interesting and important question is when f equals the right-hand side in (1.1). Note that if we start from a trigonometric polynomial 1 X 2πinx (1.3) f(x) = ane ; n=−∞ Date: DEADLINES: Draft AUGUST 18 and Final version AUGUST 29, 2013. 1 2 JING MIAO where all but finitely many an are zero, then (1.4) Z 1 1 ! 1 Z 1 ^ X 2πimx −2πinx X 2πimx −2πinx f(n) = ame e dx = ame e dx: 0 m=−∞ m=−∞ 0 We can interchange summation and integration here because only finitely many an's are nonzero. Note that the integral of the last term of (1.4) is 0 if n 6= m and ^ 1 if n = m. Thus, we get f(n) = an for all n 2 Z. The fangn2Z are also called Fourier coefficients. 1 Therefore, we see that f(n) = P f^(n)e2πinx when only finite many Fourier n=−∞ coefficents of f are nonzero. In general, we want to know when the Fourier series of a function will converge to the original function. We will specify the sense of convergence of the infinite series later in this paper. It is natural to start from the most basic notion of convergence, namely that of pointwise convergence. The partial sums of f 2 L1(T) are defined as N N Z X ^ 2πinx X −2πiny 2πinx SN f(x) = f(n)e = e f(y) dy e n=−N n=−N T (1.5) Z N Z X 2πin(x−y) = e f(y) dy = DN (x − y)f(y) dy; T n=−N T N P 2πinx where DN (x) = e is called the Dirichlet kernel. If we simplify and bound n=−N the Dirichlet kernel, we get (1.6) N X e2πi(N+1)x − e−2πiNx D (x) = e2πinx = N e2πix − 1 n=−N cos(2π(N + 1)x) − cos(2πNx) + i[sin(2π(N + 1)x) + sin(2πNx)] = cos(2πx) − 1 + i sin(2πx) (cos(2πx) − 1)(cos(2π(N + 1)x) − cos(2πNx)) + sin(2πx)(sin(2π(N + 1)x) + sin(2πNx)) = 2 − 2 cos(2πx) i[(cos(2πx) − 1)(sin(2π(N + 1)x) + sin(2πNx)) − sin(2πx)(cos(2π(N + 1)x) − cos(2πNx))] + 2 − 2 cos(2πx) −2 sin((2N + 1)πx) sin(πx)(−2) sin2(πx) + sin(2πx)2 sin((2N + 1)πx) cos(πx) = 4 sin2(πx) sin((2N + 1)πx) cos2(πx) = sin((2N + 1)πx) sin(πx) + sin(πx) sin((2N + 1)πx) = : sin(πx) Looking at Figure 1, we see that DN is very large at x = 0 and as N increases, the value at x = 0 will become larger. We have DN!1(0) = 1. Also DN is π 1 bounded by 1 away from zero. When πx ≤ , x ≤ . Our previously defined jxj 2 2 domain is T = R=Z, which is the same as periodic function with period 1. Thus, CONVERGENCE OF FOURIER SERIES IN Lp SPACE 3 20 15 10 5 0 −5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Figure 1. The Dirichlet kernel DN for N = 9 −1 1 we can translate the domain to [ 2 ; 2 ]. Now let's bound the Dirichlet kernel for each N. Since j sin nαj ≤ nj sin αj, we know that jDN (x)j ≤ 2N + 1 ≤ CN where C π 2jxj is some finite constant. Also note that when 0 < x ≤ , j sin xj ≥ . Therefore, 2 π j sin((2N + 1)πx)j 1 1 we have jD (x)j = ≤ = . To sum up, we have N j sin(πx)j 2πjxj/π 2jxj 1 (1.7) jD (x)j ≤ C min N; ; N jxj for all N ≥ 1 and some absolute constant C. Next, we try to find a bound of the 1 L norm of DN . From (1.7) we know that Z 1 Z 1 Z jDN (x)j dx ≤ dx + N dx 1 jxj 1 0 jxj> N jx|≤ N (1.8) 1 1 = 2 log − log + 1 2 N ≤ 2(log N + 1) ≤ C1 log N: The last inequality is true if N ≥ 2. Also, if we only integrate the part where 1 j sin(2N + 1)πxj ≥ , we get 2 Z 1 1 Z 1 1 (1.9) jDN (x)j dx ≥ ≥ C2 log N: 0 2 0 j sin πxj 1 Since j sin πxj ≤ 1, we have j sin πxj ≥ 1. We are only integrating over the range 1 of x where j sin(2N + 1)πxj ≥ 2 . The length of this range has a 1=N factor. This is how the log N comes up in the last inequality above. Thus, we proved that −1 1 (1.10) C3 log N ≤ kDN kL (T) ≤ C3 log N 4 JING MIAO for all N ≥ 2 where C3 is another absolute constant. 2. Convolution The convolution of functions arises naturally in the discussion of the Dirichlet kernel. We define and discuss further properties of this operation here. Definition 2.1. If f, g are two functions on T, and both f, g are continuous, then Z Z (2.2) (f ∗ g)(x) := f(x − y)g(y) dy = g(x − y)f(y) dy T T is the convolution of f and g. It is helpful to think of g or f as dirac delta functions which are generalized functions, or distributions, on the real number line that are zero everywhere except at zero, with an integral of one over the entire real line. Here we list and prove some useful lemmas about convolution. 1 Lemma 2.3. (1) Let f; g 2 L (T). Then kf ∗ gkr ≤ kfkpkgkq for all 1 ≤ r, 1 1 1 p q p; q ≤ 1, 1 + r = p + q , f 2 L , g 2 L . This is called the Young's inequality. (2) For f; g 2 L1(T) one has that for all n 2 Z, f[∗ g(n) = f^(n)^g(n). Proof. (1) By Fubini's theorem, since f(x − y)g(y) is jointly measurable on T × T 1 and belongs to L (T × T), we know that kf ∗ gk1 ≤ kfk1kgk1 < 1. The details of the proof of the Young's inequality are shown in the appendix. (2) is a consequence of Fubini's theorem and the homomorphism property of the exponentials e2πin(x+y) = e2πinxe2πiny: Z Z f[∗ g(n) = f(x − y)g(y) dy e−2πinx dx T T (2.4) Z Z = f(w)e−2πinw dw g(t)e−2πint dt T T = f^(n)^g(n): One of the most basic as well as oldest results on the pointwise convergence of Fourier series is the following theorem. Note that it requires a function to be H¨older continuous, instead of just continuous. α Theorem 2.5. If f 2 C (T) with 0 < α ≤ 1, then kSN f − fk1 ! 0 as N ! 1. 1 Proof. One has, with δ 2 (0; 2 ) to be determined, Z 1 SN f(x) − f(x) = (f(x − y) − f(x))DN (y) dy 0 Z (2.6) = (f(x − y) − f(x))DN (y) dy jy|≤δ Z + (f(x − y) − f(x))DN (y) dy: 1 2 >jyj>δ CONVERGENCE OF FOURIER SERIES IN Lp SPACE 5 R 1 2πinx Equation (2.6) is true because 0 DN (y) dy = 1, since the integration of e over [0; 1] is 0 except when n = 0. Also since DN (x) is periodic with period 1, integration 1 1 over [0; 1] is the same as that over [− 2 ; 2 ]. We now use the bound from (1.8), i.e., 1 (2.7) jD (y)j ≤ C min N; : N jyj Here and in what follows, C will denote a numerical constant that can change from line to line. The first integral in (2.6), which we denote by A, can be estimated as follows Z Z 1 α−1 α (2.8) A ≤ C jf(x) − f(x − y)j dy ≤ C[f]α jyj dy ≤ C[f]αδ ; jy|≤δ jyj jy|≤δ with the usual Cα semi-norm: jf(x) − f(x − y)j (2.9) [f]α = sup α : x;y jyj To bound the second integral in (2.6), which we denote by B, we need to use the oscillation of DN (y).
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