Outline of Fourier Series: Math 201B February 24, 2011 1 Functions and convolutions 1.1 Periodic functions • Periodic functions. Let T = R=(2πZ) denote the circle, or one- dimensional torus. A function f : T ! C is equivalent to a 2π-periodic function f~ : R ! C. (The real line R is the universal cover of the circle T and f~ is the lift of f from T to R.) We identify f with f~. • Continuous functions. The space of continuous functions f : T ! C is denoted by C(T). It is a Banach space when equipped with the maximum (sup) norm. • Smooth functions. If k 2 N, the space k-times continuously differ- k entiable functions is denoted by C (T). This is a Banach space with the Ck-norm (the sum of the maximum values of a function and its derivatives of order less than or equal to k.) The space of smooth func- tions (functions with continuous derivatives of all orders) is denoted 1 by C (T). This is a Fr´echet space with the metric 1 X 1 kφ − kCk d(φ, ) = k : 2 1 + kφ − k k k=0 C p p • L -spaces. For 1 ≤ p < 1, the Banach space L (T) consists of all Lebesgue measurable functions f : T ! C such that Z 1=p p kfkp = jf(x)j dx < 1: T 1 The space L (T) consists of essentially bounded functions. We iden- tify functions that are equal almost everywhere. 1 2 2 • L -Hilbert space. The space L (T) is a Hilbert space with inner product Z hf; gi = f(x)g(x) dx T 1 p • Density. The space C (T) is dense in L (T) for 1 ≤ p < 1 and 1 in C(T), but it is not dense in L (T). More specifically, according to the Weierstrass approximation theorem, the same density results are true for the space P(T) of trigonometric polynomials of the form P inx jn|≤N cne . 1.2 Convolutions and approximate identities 1 1 • Convolution. If f; g 2 L (T), the convolution f ∗g 2 L (T) is defined by Z (f ∗ g)(x) = f(x − y)g(y) dy: T • Young's inequality. If 1 ≤ p; q; r ≤ 1 satisfy 1 1 1 + = 1 + p q r p q r and f 2 L (T), g 2 L (T), then f ∗ g 2 L (T). Moreover, kf ∗ gkr ≤ kfkp kgkq : This follows from Fubini's theorem. In particular, convolution with an L1-function is a bounded map on Lp, kf ∗ gkp ≤ kfk1 kgkp ; and the convolution of L2-functions is bounded (and therefore contin- uous by a density argument) kf ∗ gk1 ≤ kfk2 kgk2 : • Approximate identity. A sequence of functions 1 fφn 2 L (T): n 2 Ng 2 is an approximate identity if there exists a constant M such that Z φn dx = 1 for every n; T Z jφnj dx ≤ M for all n; T Z lim jφnj dx = 0 for every δ > 0. n!1 δ<jxj<π (Analogous definitions apply to a family of functions that depend on a continuous parameter.) • Mollification. If fφng is an approximate identity and f 2 C(T) then p φn ∗ f 2 C(T) and φn ∗ f ! f uniformly as n ! 1. If f 2 L (T), p 1 and 1 ≤ p < 1, then φn ∗ f ! f in L as n ! 1. If φn 2 C (T) 1 1 and f 2 L (T), then φn ∗ f 2 C (T). (The Lebesgue dominated convergence theorem justifies “differentiating under the integral sign.") 2 Fourier Series 2.1 L1-theory 1 • Definition of Fourier coefficients. For f 2 L (T) define the Fourier coefficients f^ : Z ! C by 1 Z f^(n) = f(x)e−inx dx: 2π T 1 • Fourier coefficients determine a function. If f; g 2 L (T) and f^(n) =g ^(n) for all n 2 Z then f = g (up to pointwise a.e.-equivalence). Thus follows from ap- proximation of a function by convolution with an approximate identity that consists of trigonometric polynomials (e.g. the F´ejerkernel.) 1 • Riemann-Lebesgue lemma. If f 2 L (T), then f^(n) ! 0 as jnj ! 1: This result follows from the estimate 1 ^ f ≤ kfkL1 `1 2π 3 and the density of trigonometric polynomials (or smooth functions) in 1 L (T). 1 • Convolution theorem. If f; g 2 L (T), then (\f ∗ g)(n) = 2πf^(n)^g(n): That is, the Fourier transform maps the convolution product of func- tions to the pointwise product of their Fourier coefficients. 2.2 L2-theory • Fourier basis. The functions 1 inx p e : n 2 Z 2π 2 form an orthonormal basis of L (T). The orthonormality is easy to verify; the completeness follow by the use of convolution with an ap- proximate identity that consists of trigonometric polynomials to ap- 2 proximate a general f 2 L (T). 2 1 • Fourier series of an L -function. A function f 2 L (T) belongs to 2 L (T) if and only if 2 X ^ f(n) < 1 n2Z and then (with the above normalization of the Fourier coefficients) X f(x) = f^(n)einx n2Z where the series converges unconditionally with respect to the L2- norm. 2 • Parseval's theorem. If f; g 2 L (T) then X hf; gi = 2π f^(n)^g(n): n2Z 4 2.3 Absolutely convergent Fourier series 1 • Absolutely convergent Fourier series. If f 2 L (T) has absolutely 1 convergent Fourier coefficients f^ 2 ` (Z), meaning that X ^ f(n) < 1; n2Z then f 2 C(T). This follows from the fact that the Fourier series of f converges uniformly to f by the Weierstrass M-test. We denote the space of functions with absolutely convergent Fourier series by A(T). • Pointwise divergence of Fourier series. There are continuous functions whose Fourier series converge uniformly but not absolutely, and continuous function whose Fourier series do not do not converge uniformly; in fact, there are continuous functions whose Fourier series diverge pointwise on an arbitrary set of Lebesgue measure zero. If p f 2 L (T) for 1 < p ≤ 1, then the Fourier series of f converges 1 pointwise a.e. to f (Carlson, Hunt) but there exist functions f 2 L (T) whose Fourier series diverge pointwise a.e. (Kolmogorov). • Convolution theorem. If f; g 2 A(T), then fg 2 A(T) and X (dfg)(n) = 2π f^(n − k)^g(k): k2Z That is, the Fourier transform maps the pointwise product of functions to the discrete convolution product of their Fourier coefficients. 2.4 Weak derivatives and Sobolev spaces 1 • Weak derivative. A function f 2 L (T) has weak derivative g = 0 1 f 2 L (T) if Z Z 0 1 fφ dx = − gφ dx for all φ 2 C (T): T T That is, weak derivatives are defined by integration by parts. If f 2 1 C (T), then the weak derivative agrees with the pointwise derivative (up to pointwise a.e.-equivalence). 1 • Fourier coefficients of derivatives. If f 2 L (T) has weak deriva- 0 1 tive f 2 L (T), then fb0(n) = inf^(n): 5 k;1 • Decay of Fourier coefficients. If k 2 N and f 2 W (T), meaning 0 00 (k) 1 that f has weak derivatives f ; f ; : : : ; f 2 L (T) of order less than or equal to k, then jnjkf^(n) ! 0 as jnj ! 1: This follows from an application of the Riemann-Lebesgue lemma to f (k). 2 s • L -Sobolev spaces. If 0 ≤ s < 1, the Sobolev space H (T) consists 2 of all functions f 2 L (T) such that 2 X 2s ^ 1 + n f(n) < 1: n2Z This is a Hilbert space with inner product X 2s ^ hf; giHs = 2π 1 + n f(n)^g(n): n2Z k 2 If k 2 N is an integer, then H (T) consists of all functions f 2 L (T) that have weak derivatives of order less than or equal to k belonging 2 to L (T). s • Sobolev embedding. If f 2 H (T) and s > 1=2, then f 2 C(T) and there is a constant C > 0, depending only on s, such that s kfk1 ≤ C kfkHs for all f 2 H (T): 1 This theorem follows by showing that f^ 2 ` (Z) is absolutely conver- gent so f 2 A(T). Roughly speaking, for functions of a single variable, more than one-half an L2-derivative implies continuity. 2.5 Periodic distributions • Test functions. The space D(T) of periodic test functions consists 1 of all smooth functions φ 2 C (T) with the following notion of con- vergence of test functions: φn ! φ in D(T) if (k) (k) φn ! φ uniformly as n ! 1 for every k = 0; 1; 2;:::: Here φ(k) denotes the kth derivative of φ. 6 1 • Fourier series of test functions. A function φ 2 L (T) belongs to D(T) if and only if its Fourier coefficients are rapidly decreasing, k jnj φ^(n) ! 0 as jnj ! 1 for every k 2 N: This follows from the decay estimates for the Fourier coefficients of smooth functions and the Sobolev embedding theorem. (Note that k k φ 2 H for every k 2 N if and only if φ 2 C for every k 2 N.) The Fourier series of φ 2 D(T) converges to φ in the sense of test functions. • Distributions. A distribution T is a continuous linear functional T : D(T) ! C: 0 0 The space of distributions is denoted D (T) and the action of T 2 D on φ 2 D by hT; φi.