Fourier Series, Integrals, and, Sampling From Basic Complex Analysis Jeffrey RAUCH Outline. The Fourier series representation of analytic functions is derived from Laurent expan- sions. Elementary complex analysis is used to derive additional fundamental results in harmonic analysis including the representation of C∞ periodic functions by Fourier series, the representation of rapidly decreasing functions by Fourier integrals, and Shannon’s sampling theorem. The ideas are classical and of transcendent beauty. 1. Laurent series yield Fourier series. § A difficult thing to understand and/or motivate is the fact that arbitrary periodic functions have Fourier series representations. In this section we prove that periodic analytic functions have such a representation using Laurent expansions. Definition. A function f(z) defined on a strip z : Im z < a , a> 0 , (1.1) { | | } is 2π periodic if for all such z, f(z +2π)= f(z) . (1.2) Examples of periodic analytic functions. The elementary functions sin nz, cos nz, and e±inz are the building blocks. Any finite linear combination is an example. Nonlinear functions too, for example 1 1+sin2 z is analytic in any strip on which sin z = i. An entire function h = ∞ a zn yields the entire ± 0 n example P ∞ iz inz h(e ) = an e . X0 An analysis related to the last example yields the general case. Consider the mapping z w = eiz . (1.3) → It maps the strip (1.1) onto the annulus w : e−a < w <ea . (1.4) { | | } It maps the real axis infinitely often around the unit circle in the w plane. The preimages of a point w = eiθ are the points z = θ +2πn with n Z. Since the derivative dw/dz is nowhere zero, the mapping is locally invertible with analytic inver∈ se. The local inverses are branches of the function z = (ln w)/i. Theorem 1.1. The correspondence g(w) f(z), → f(z) = g(eiz) (1.5) 1 establishes a one to one correspondence between the analytic functions g(w) on the annulus (1.4) the 2π periodic analytic functions f(z) in the strip (1.1). Proof. That each such g yields an analytic periodic f on the strip and that distinct functions g yield distinct f is clear. It suffices to show that every f has such a representation. Suppose that f is analytic and periodic in the strip. For each point w in the annulus, the preimages z lie in the strip and differ by integer multiples of 2π. Thus, the function f has the same value at all the preimages. It follows that a function g on the annulus is well defined by the formula g(w)= f(z) since it does not matter which z one takes. For any w choose a preimage z. The Inverse Function Theorem implies that w has a local inverse z = F (w) analytic on a neighborhood of w and satisfying F (w) = z. Near w, g(w) = f(F (w)) is therefore analytic. Thus g provides the desired representation of f. Theorem 1.2. If f(z) is a 2π periodic analytic function in the strip (1.2) then f has a Fourier series representation ∞ inz f(z) = cn e , (1.6) n=X−∞ uniformly convergent on each thinner strip. The coefficients are given by the formulas 2π 1 −inθ cn = f(θ) e dθ . (1.7) 2π Z0 Proof. Choose g so that (1.5) holds. Then use the Laurent expansion of g ∞ n 1 g(w) g(w) = cn w , cn = dw , (1.8) 2πi I wn+1 X−∞ |w|=1 uniformly convergent on each subannulus. Since f(z)= g(eiz), one has ∞ iz n f(z) = cn (e ) X−∞ which is formula (1.6). Parameterizing the curve w = 1 by w = eiθ with 0 θ 2π, one has dw = iwdθ and the formula | | ≤ ≤ for cn becomes 1 2π g(eiθ) 1 2π f(θ) cn = n+1 iwdθ = n dθ , 2πi Z0 w 2π Z0 w which proves (1.7). Fourier series were discovered before Laurent expansions. If history were more logical they might have been found this way. 2. Paley-Weiner for Fourier series. § Every 2π periodic function that is analytic in a neighborhood of the real axis has a Fourier series representation (1.6)-(1.7). If (1.6) holds, multiplying by e−imx then integrating over [ π, π] yields the formula (1.6) for the coefficients since − π 2π when m = n e−imx einx dx = Z 0 when m = n . −π 2 Periodic functions that need not be analytic have Fourier expansion of the same form. The smoother are the functions, the more rapidly decreasing are the coefficients cn and the faster is the conver- gence in (1.6). Analytic periodic f are characterized by the fact that the cn are exponentially decreasing as n . | | → ∞ Paley-Weiner Theorem 2.1. i. If f is an analytic periodic function in the strip (1.1), then its Fourier coefficients cn satisfty for any ǫ> 0 there is C(ǫ) so that c C(ǫ) e(−a+ǫ)|n| . (2.1) | n| ≤ ii. Conversely, if f is given by (1.6) with cn satisfying (2.1) then f has an analytic continuation to the strip (1.1). a Exercise 2.1. Prove i. Hint. For n > 0 start from the formula for cn in (1.8). For 1 <b<e move the contour to z = b using Cauchy’s Theorem. On that contour, 1/wn+1 is exponentially small as n . Perform| | an analogous estimate to treat n < 0. Alternatively use the fact that the Laurent→ expansion ∞ is convergent to estimate the Laurent coefficients. Exercise 2.2. Prove ii. Hint. The Fourier series is uniformly convergent on any strip thinner than (1.1) 3. Fourier series for nonanalytic periodic functions. § For infinitely differentiable periodic f the cn decrease faster than any negative power of n , that is, | | C N, C , n, c N , n := (1+ n 2)1/2 . (3.1) ∀ ∃ N ∀ | n| ≤ n N | | This is a consequence of the formula for the Fourier coefficients of the derivative, ′ cn(f ) = in cn(f) , (3.2) valid for example if f is continuously differentiable. The proof of (3.2) is by integrating by parts with boundary terms cancelling by periodicity to give, π π −inx ′ ′ −inx de 2πcn(f ) = f (x) e dx = f(x) dx = 2πincn(f) . (2.3) Z−π − Z−π dx Therefore if f CN , ∈ π N N (N) 1 d f ( in) cn(f) = cn(f ) N (x) dx (2.4) − ≤ 2π Z−π dx which implies (2.1). For smooth periodic f the Fourier series and each differentiated series converges uniformly. For less regular f the convergence is less strong. For example, for f which are merely square integrable, one has only c 2 < , the convergence is in the root mean square sense. For periodic | n| ∞ distributions inP the sense of Schwartz, the convergence is in the sense of distributions. The C∞ result implies the others (see remarks below). We prove the C∞ result using complex analysis. Theorem 3.1. If f is an infinitely differentiable 2π periodic function on the real line, then the representation (1.6)–(1.7) is valid. The Fourier coefficients (1.7) satisfy the rapid decay estimate (3.1) so the series and all differentiated series converge uniformly on R. 3 Historically, many examples of expansions (1.6) were discovered before it was realized how general was the phenomenon. For example if a < 1, one has | | ∞ 1 n = a sin θ . 1 a sin θ − nX=0 It was Fourier who uncovered the fact that the representations were general and their utility in analysing differential equations. This preceded the flowering of complex analysis. 4. The Fourier transform. § Our treatment of Fourier series is intimately entangled with the Fourier transform representation ∞ g(x) = gˆ(ξ) eixξ dξ , (4.1) Z−∞ 1 ∞ gˆ(ξ) = gˆ(x) e−ixξ dx := (g)(ξ) , (4.2) 2π Z−∞ F for functions g defined on R so that g andg ˆ tending to zero sufficiently fast at . Using contour integration, this reciprocal relation is verified in the two concrete cases, ±∞ 2 a g(x)= e−a|x|, a> 0 , gˆ(ω)= , (4.3) rπ a2 + ξ2 and 2 2 g(x)= e−x /2 , gˆ(ω)= e−ξ /2 . (4.4) The standard proofs of (4.1), (4.2) rely on one of these two examples. A convenient class of functions for studying the Fourier transform is the Schwartz class consisting of those g so that for all 0 < n,m there is a C(n, m) so that S dmg C x −n . dxm ≤ For such g, an integration by parts as in (3.3) shows that (g′) = iξ gˆ . (4.5) F Differentiating the definition ofg ˆ yields d gˆ = ( ix g) . (4.6) dξ F − It follows that the Fourier transform of a function in belongs to so that in (4.1), (4.2) the integrals are very rapidly convergent. S S Exercise 4.1. Define the inverse transform h(ξ) ∗h by → F 1 ∞ ∗h(x) := e−ixξ h(ξ) dx . F 2π Z−∞ 4 Derive formulas analogous to the preceding two for ∗h′ and ∗(ξh). From those prove that the operator ∗ from to itself commutes with multiplicalionF byF x and also with d/dx. F F S The analysis requires the following fundamental result. Riemann-Lebesgue Lemma 4.1. If g(x) is an absolutely integrable function on R, then lim gˆ(ξ) = 0 . |ξ|→∞ Proof. For ǫ> 0, choose ψ so that ∈ S ∞ g(x) ψ(x) dx < ǫ.