Outline of Fourier Series: Math 201B
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1 Outline of Fourier Series: Math 201B February 24, Functions and convolutions 1.1 Periodic functions Periodic functions. Let = R/(2πZ) denote the circle, or onedimensional torus. A function f : C is equivalent to a 2π-periodic function f : R C. (he real line R is the universal cover of the circle and f is the lift of f from to R.) We identify f with f. Continuous functions. he space of continuous functions f : C is denoted by C(). It is a Banach space when equipped with the maximum (sup) norm. Smooth functions. If k N, the space k-times continuously differentiable functions is denoted by C k (). his is a Banach space with the C k -norm (the sum of the maximum values of a function and its derivatives of order less than or equal to k.) he space of smooth functions (functions with continuous derivatives of all orders) is denoted by C (). his is a Fréchet space with the metric d(φ, ψ) = k=0 ( ) 1 φ ψ C k 2 k. 1 + φ ψ C k L p -spaces. For 1 p <, the Banach space L p () consists of all Lebesgue measurable functions f : C such that ( 1/p f p = f(x) dx) p <. he space L () consists of essentially bounded functions. We identify functions that are equal almost everywhere. 1
2 L 2 -Hilbert space. product he space L 2 () is a Hilbert space with inner f, g = f(x)g(x) dx Density. he space C () is dense in L p () for 1 p < and in C(), but it is not dense in L (). More specifically, according to the Weierstrass approximation theorem, the same density results are true for the space P() of trigonometric polynomials of the form n N c ne inx. 1.2 Convolutions and approximate identities Convolution. If f, g L 1 (), the convolution f g L 1 () is defined by (f g)(x) = f(x y)g(y) dy. Young s inequality. If 1 p, q, r satisfy 1 p + 1 q = r and f L p (), g L q (), then f g L r (). Moreover, f g r f p g q. his follows from Fubini s theorem. In particular, convolution with an L 1 -function is a bounded map on L p, f g p f 1 g p ; and the convolution of L 2 -functions is bounded (and therefore continuous by a density argument) f g f 2 g 2. Approximate identity. A sequence of functions {φ n L 1 () : n N} 2
3 is an approximate identity if there exists a constant M such that φ n dx = 1 for every n, φ n dx M for all n, φ n dx = 0 for every δ > 0. lim n δ< x <π (Analogous definitions apply to a family of functions that depend on a continuous parameter.) Mollification. If {φ n } is an approximate identity and f C() then φ n f C() and φ n f f uniformly as n. If f L p (), and 1 p <, then φ n f f in L p as n. If φ n C () and f L 1 (), then φ n f C (). (he Lebesgue dominated convergence theorem justifies differentiating under the integral sign. ) 2 Fourier Series 2.1 L 1 -theory Definition of Fourier coefficients. For f L 1 () define the Fourier coefficients ˆf : Z C by ˆf(n) = 1 f(x)e inx dx. 2π Fourier coefficients determine a function. If f, g L 1 () and ˆf(n) = ĝ(n) for all n Z then f = g (up to pointwise a.e.-equivalence). hus follows from approximation of a function by convolution with an approximate identity that consists of trigonometric polynomials (e.g. the Féjer kernel.) Riemann-Lebesgue lemma. If f L 1 (), then ˆf(n) 0 as n. his result follows from the estimate ˆf 1 l 2π f L 1 3
4 and the density of trigonometric polynomials (or smooth functions) in L 1 (). Convolution theorem. If f, g L 1 (), then (f g)(n) = 2π ˆf(n)ĝ(n). hat is, the Fourier transform maps the convolution product of functions to the pointwise product of their Fourier coefficients. 2.2 L 2 -theory Fourier basis. he functions { } 1 e inx : n Z 2π form an orthonormal basis of L 2 (). he orthonormality is easy to verify; the completeness follow by the use of convolution with an approximate identity that consists of trigonometric polynomials to approximate a general f L 2 (). Fourier series of an L 2 -function. A function f L 1 () belongs to L 2 () if and only if ˆf(n) 2 < and then (with the above normalization of the Fourier coefficients) f(x) = ˆf(n)e inx where the series converges unconditionally with respect to the L 2 - norm. Parseval s theorem. If f, g L 2 () then f, g = 2π ˆf(n)ĝ(n). 4
5 2.3 Absolutely convergent Fourier series Absolutely convergent Fourier series. If f L 1 () has absolutely convergent Fourier coefficients ˆf l 1 (Z), meaning that ˆf(n) <, then f C(). his 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(). Pointwise divergence of Fourier series. here 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 f L p () for 1 < p, then the Fourier series of f converges pointwise a.e. to f (Carlson, Hunt) but there exist functions f L 1 () whose Fourier series diverge pointwise a.e. (Kolmogorov). Convolution theorem. If f, g A(), then fg A() and (fg)(n) = 2π ˆf(n k)ĝ(k). k Z hat 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 Weak derivative. A function f L 1 () has weak derivative g = f L 1 () if fφ dx = gφ dx for all φ C (). hat is, weak derivatives are defined by integration by parts. If f C 1 (), then the weak derivative agrees with the pointwise derivative (up to pointwise a.e.-equivalence). Fourier coefficients of derivatives. If f L 1 () has weak derivative f L 1 (), then f (n) = in ˆf(n). 5
6 Decay of Fourier coefficients. If k N and f W k,1 (), meaning that f has weak derivatives f, f,..., f (k) L 1 () of order less than or equal to k, then n k ˆf(n) 0 as n. his follows from an application of the Riemann-Lebesgue lemma to f (k). L 2 -Sobolev spaces. If 0 s <, the Sobolev space H s () consists of all functions f L 2 () such that ( 1 + n 2 ) s 2 ˆf(n) <. his is a Hilbert space with inner product f, g H s = 2π ( 1 + n 2 )s ˆf(n)ĝ(n). If k N is an integer, then H k () consists of all functions f L 2 () that have weak derivatives of order less than or equal to k belonging to L 2 (). Sobolev embedding. If f H s () and s > 1/2, then f C() and there is a constant C > 0, depending only on s, such that f C f H s for all f H s (). his theorem follows by showing that ˆf l 1 (Z) is absolutely convergent so f A(). Roughly speaking, for functions of a single variable, more than one-half an L 2 -derivative implies continuity. 2.5 Periodic distributions est functions. he space D() of periodic test functions consists of all smooth functions φ C () with the following notion of convergence of test functions: φ n φ in D() if φ (k) n φ (k) uniformly as n for every k = 0, 1, 2,.... Here φ (k) denotes the kth derivative of φ. 6
7 Fourier series of test functions. A function φ L 1 () belongs to D() if and only if its Fourier coefficients are rapidly decreasing, n k ˆφ(n) 0 as n for every k N. his follows from the decay estimates for the Fourier coefficients of smooth functions and the Sobolev embedding theorem. (Note that φ H k for every k N if and only if φ C k for every k N.) he Fourier series of φ D() converges to φ in the sense of test functions. Distributions. A distribution is a continuous linear functional : D() C. he space of distributions is denoted D () and the action of D on φ D by, φ. (his duality paring is linear in both arguments, not anti-linear in the first argument like the inner product on a Hilbert space.) Convergence of distributions. A sequence of of distributions { n } converges to a distribution, written n, if n, φ, φ as n for every φ D(). Order of a distribution. If D (), there is a non-negative integer k and a constant C such that, φ C φ C k for all φ D(). he minimal such integer k is called the order of. Regular distributions. If f L 1 (), we define f D () by f, φ = fφ dx. Any distribution of this form is called a regular distribution. We identify f with f and regard L 1 () as a subspace of D (). Distributional derivative. Every D() has a distributional derivative D() defined by, φ =, φ A function f L 1 () has a weak derivative f L 1 () if and only if its distributional derivative is regular, and then ( f ) = f. 7
8 Fourier series of distributions. he Fourier coefficients ˆ : Z C of a distribution D() are defined by ˆ (n) = 1 2π, e inx A linear functional on D() is a distribution if and only if its Fourier coefficients have slow growth, meaning that the exists a non-negative integer k and a constant C such that ˆ (n) C ( 1 + n 2) k/2 for all n Z. In that case, the Fourier series of, ˆ (n)e inx converges to in the sense of distributions. he delta function. he periodic δ-function supported at 0 is the distribution δ D () defined by δ, φ = φ(0) his is a distribution of order zero, but it is not a regular distribution. (It is, in fact, a measure.) Its Fourier series is δ = 1 2π e inx. 8
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