Banach Spaces of analytic functions (MAGIC 082) Jonathan R. Partington University of Leeds, School of Mathematics [email protected] March 25, 2014 1 LECTURE 1 Books: K. Hoffman, Banach spaces of analytic functions. P. Koosis, Introduction to Hp spaces. W. Rudin, Real and complex analysis. N. Nikolski, Operators, functions and systems, an easy reading, Vol. 1. 1 Introduction Recall that a Banach space is a complete normed space, and a Hilbert space is a special kind of Banach space, where the norm is given by an inner prod- uct. You should have met Lp and `p for 1 ≤ p ≤ 1 and C(K) (the space of continuous functions on K). We are going to work with complex Banach and Hilbert spaces whose ele- ments are functions. 1.1 Examples (treated informally for the moment) 1. The Hardy spaces Hp (1 ≤ p ≤ 1) are Banach spaces consisting of analytic functions in the unit disc D = fz 2 C : jzj < 1g; whose boundary values are in Lp(T), where T = fz 2 C : jzj = 1g is the unit circle. Thus 1 X n f(z) = anz n=0 for jzj < 1, and on jzj = 1, f corresponds to an Lp function (pointwise radial limits almost everywhere, in fact). Recall that functions of period 2π have Fourier series, that is, 1 X int g(t) ∼ cne ; n=−∞ 2 where Z 2π 1 −int cn = g(t)e dt: 2π 0 If g 2 Lp(0; 2π) for some 1 ≤ p ≤ 1, then g 2 L1(0; 2π), and Z 2π 1 int jcnj ≤ jg(t)jje j dt = kgkL1 ; 2π 0 so the sequence (cn) is bounded. Now if cn = 0 for all n < 0, we can define 1 X n f(z) = cnz ; n=0 and it converges for jzj < 1. So we think of Hp either as a space of functions defined on the disc, or as functions on the circle, putting f(eit) = g(t), or indeed as functions of period 2π. In the special case p = 1, H1 consists of all bounded analytic functions on D, and it forms a Banach algebra. 2. The harmonic Hardy spaces. We may proceed similarly for the set of all Lp P1 int functions on the circle. Now the natural extension to the disc of −∞ cne is going to be harmonic rather than analytic, i.e., we take 1 1 X n X n cnz + c−nz : n=0 n=1 (Recall that harmonic functions satisfy @2f @2f + = 0 @x2 @y2 and include analytic functions such as zn and anti-analytic ones such as zn.) p We define hp to be the space of harmonic functions in the disc with L bound- ary values. 3. The disc algebra A(D). These are the functions continuous on the closed unit disc D, and analytic on the open unit disc. We use the supremum norm. 3 This is a closed subalgebra of H1. 4. The Bergman spaces. These are again spaces of analytic functions in the disc, but now in Lp on the disc, i.e., f 2 Ap for 1 ≤ p < 1 if and only if f is analytic in D and Z 1=p 1 p kfkp = jf(z)j dA(z) < 1; π D where dA(z) is two-dimensional Lebesgue measure, i.e., Z 1 Z 2π p 1 iθ p kfkp = jf(re )j r dr dθ: π r=0 θ=0 In the case p = 2, if f has a Taylor series 1 X n f(z) = anz ; n=0 then 1 2 X janj kfk2 = : n + 1 n=0 5. The Wiener algebra, W . Take all functions of period 2π that can be written as 1 X int g(t) = cne ; n=−∞ P1 with kgkW = −∞ jcnj < 1. Here the series converges uniformly so g is actually continuous. Transferring to the circle we can write 1 X n f(z) = cnz ; −∞ P1 with −∞ jcnj < 1. The positive or analytic Wiener algebra W+ consists of all analytic functions P1 n f(z) = n=0 cnz in the disc with 1 X kfkW = jcnj < 1: 0 4 These functions all lie in A(D). As the name suggests, both W and W+ are Banach algebras. Elementary inclusions: 1 2 1 W+ ⊂ A(D) ⊂ H ⊂ H ⊂ H (and many others). 6. The Paley{Wiener space PW (K), where K ⊂ R is compact. Let f be a function on R which can be written as 1 Z 1 f(t) = f^(w)eiwt dw; 2π −∞ ^ ^ 2 ^ where f(w) = 0 for w 62 K, and fjK 2 L (K). Here f is the Fourier transform of f, and can be defined (for f 2 L1(R)) by Z 1 f^(w) = f(t)e−iwtdt: −∞ It turns out that such f can be extended to be analytic on the whole of C, and that Z 1 1 Z 1 kfk2 = jf(t)j2 dt = jf^(w)j2 dw; −∞ 2π −∞ makes PW (K) into a Banach space, indeed even a Hilbert space. An important special case is K = [−a; a], where a > 0. Then PW (K) is the space of band-limited signals (no high-frequency components). 7. The Segal{Bargmann (or Fock) space SB2 consisting of all functions f analytic on the whole of C such that 1 Z 2 kfk2 = jf(z)j2e−|zj dA(z) < 1: π C This has applications in quantum mechanics. 5 2 Hardy spaces on the disc We begin with H2, which is a Hilbert space. Write iθ T = fz 2 C : jzj = 1g = fe : 0 ≤ θ ≤ 2πg: dθ Normalize it to have measure 1, so all integrals will be with respect to . 2π 2.1 Proposition The set of functions n inθ fz : n 2 Zg = fe : n 2 Zg forms an orthonormal basis for L2(T), and its linear span is dense in Lp(T) for all 1 ≤ p < 1. Here we use the inner product on L2(T) given by 1 Z 2π hf; gi = f(eiθ)g(eiθ) dθ: 2π 0 Proof: We see that ( 1 Z 2π 0 if m 6= n; hzn; zmi = einθe−imθ dθ = 2π 0 1 if m = n; and hence we have the orthonormality. Recall now that if f 2 Lp(0; 2π) and " > 0, then there is a continuous periodic P1 inθ function g with kf − gkLp < "=2. Next, g has a Fourier series n=−∞ cne , whose Fej´ersums m k 1 X X G (θ) = c einθ m m + 1 n k=0 n=−k converge uniformly to g on [0; 2π]. Hence there is a trigonometric polynomial Gm with kg − GmkLp < "=2; since uniform convergence implies Lp convergence. Now kf − Gmk ≤ kf − gk + kg − Gmk < ": 6 Translating this to the circle with z = eiθ we see that for f 2 Lp(T) we have N X n kf − anz kp < " −N for suitable a−N ; : : : ; aN 2 C. 7 LECTURE 2 2.2 Definition The Hardy space H2 consists of all functions 1 X n f(z) = anz n=0 analytic in the unit disc D such that 1 2 X 2 kfk = janj < 1: n=0 It can naturally be identified with the closed subspace of L2(T) consisting of P1 inθ 2 all functions n=0 ane (convergence in the L sense) with 1 X 2 janj < 1; n=0 i.e., the closed linear span of feinθ : n ≥ 0g. Note that if jzj < 1 then 1=2 1=2 X n X 2 X 2n janz j ≤ janj jz j < 1; by Cauchy{Schwarz, so all these power series have radius of convergence at least 1, and define analytic functions in D. 2.3 Proposition H2 is a Hilbert space with inner product * 1 1 + X n X n X anz ; bnz = anbn; 0 0 or equivalently 1 Z 2π hf; gi = f(eiθ)g(eiθ) dθ: 2π 0 8 Proof: This is clear, since it is a closed subspace of L2(T). Alternatively, it's clear since the correspondence 1 1 X n (an)n=0 ! anz n=0 2 2 identifies H with ` (Z+). Analytic and harmonic extensions Recall the Dirichlet problem: given f continuous on the unit circle, find g harmonic on the interior which extends it to a continuous function on the closed disc. These can be real or complex functions. Recall also that if g is analytic, then its real and imaginary parts are both harmonic, so g itself is. but if g = u + iv, then g = u − iv is also harmonic. So if f(eiθ) = einθ for some n 2 Z, then a solution is ( rneinθ = zn if n ≥ 0; g(reiθ) = rme−imθ = zm if n = −m < 0; inθ P1 inθ so that if f(e ) ∼ n=−∞ ane , then 1 iθ X jnj inθ g(re ) = anr e n=−∞ is the `natural' extension to D. 2.4 Definition The Poisson kernel P (z; w), defined for 0 ≤ jwj < jzj ≤ 1, is the function jzj2 − jwj2 P (z; w) = : jz − wj2 1−|wj2 When jzj = 1, this equals j1−wzj2 , since jz − wj = jzz − wzj = j1 − wzj. This equals 1 − ww 1 1 = + − 1 (1 − wz)(1 − wz) 1 − wz 1 − wz 1 1 X X = wnzn + wnzn: n=0 n=1 9 Thus 1 it iθ X jnj in(t−θ) P (e ; re ) = r e = Pr(t − θ); n=−∞ say. Clearly the Poisson kernel is a positive continuous function of z and w for 0 ≤ jwj < jzj ≤ 1. Note that 1 1 X X rjnjein(t−θ) = rjnjein(θ−t); n=−∞ n=−∞ so Pr(t − θ) = Pr(θ − t).