THE CONSTANT OF INTERPOLATION 3 0 ARTUR NICOLAU, JOAQUIM ORTEGA-CERDA`, AND KRISTIAN SEIP 0 2 Abstract. We prove that a suitably adjusted version of Peter n a Jones’formula for interpolationin H∞ givesa sharpupper bound J for what is known as the constant of interpolation. We show how 8 this leads to precise and computable numerical bounds for this 2 constant. ] V With each finite or infinite sequence Z = (z ) (j = 1,2,...)of distinct C j points z = x +iy in the upper half-plane of the complex plane, we . j j j h associate a number M(Z) ∈ R+ ∪ {+∞} which we call the constant t a of interpolation. We may define it in two equivalent ways. The first m is related to Carleson’s interpolation theorem for H∞ [Car58]. We say [ that Z is an interpolating sequence if the interpolation problem 1 v (1) f(z ) = w , j = 1,2,... 4 j j 3 has a solution f ∈ H∞ for each bounded sequence (w ) of complex 3 j 1 numbers. Using the open mapping theorem, we find that if Z is an 0 interpolating sequence, then we can always solve (1) with a function f 3 such that 0 / h kfk ≤ Ck(w )k ∞ j ∞ t a for some C < ∞ depending only on Z. The constant of interpolation m M(Z) is declared to be the smallest such C. We set M(Z) = +∞ if Z : v is not an interpolating sequence. i X By a classical theorem of Pick (see [Gar81, p. 2]), we may alterna- r tively define M(Z) as follows. Let M (Z) be the smallest number C n a such that the matrices 1−w w j k (cid:18) z −z (cid:19) j k j,k=1,2,...,n Date: January 28, 2003. Key words and phrases. Interpolating sequences, constant of interpolation. The authors are supported by the European Commission Research Train- ing Network HPRN-CT-2000-00116. The first two authors are supported by DGICYT grants: BFM2002-00571, BFM2002-04072-C02-01 and by the CIRIT: 2001SGR00172, 2001SGR00431. The third author is partly supported by a grant from the Research Council of Norway. 1 2 ARTURNICOLAU,JOAQUIMORTEGA-CERDA`,AND KRISTIANSEIP are positive semi-definite whenever k(w )k ≤ 1/C. The constant of j ∞ interpolation is then M(Z) = M (Z) if Z is a finite sequence consisting n of n points and M(Z) = lim M (Z) if Z is infinite. We will make n→∞ n no use of this definition, but have stated it to make the reader aware of the relevance of M(Z) for the classical Nevanlinna-Pick problem. Carleson’s interpolation theorem [Car58] states that Z is an inter- polating sequence (or alternatively M(Z) < ∞) if and only if z −z j k δ(Z) = inf > 0. j6=kYk6=j(cid:12)(cid:12)zj −zk(cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Clearly, an interpolating sequence satisfies the Blaschke condition. We let B be the associated Blaschke product and set z −z j B (z) = B(z), j z −z j so that we may write δ(Z) = inf |B (z )|. j j j An interesting result related to Carleson’s theorem is thatif M(Z) < ∞, then the interpolation may be obtained by means of a linear opera- tor. In fact, P. Beurling [Car63] proved that there exist f ∈ H∞ with j f (z ) = 1 and f (z ) = 0 if k 6= j, such that j j j k M(Z) = sup |f (z)|. j z Xj The functions f have the form j B (z) 2iy 2 G(z ) j j j f (z) = , j B (z ) z −z G(z) j j (cid:16) j(cid:17) where G is a bounded analytic function solving a certain nonlinear extremalproblem. Unfortunately,Gisnotgivenexplicitly, anditseems verydifficult togetmuch further. TheproblemoffindingGcanbeseen as a version of the Nevanlinna-Pick interpolation problem, where one is interested in computing M(Z) and finding solutions of minimal norm. There are classical results of R. Nevanlinna describing these solutions, but they are very implicit and give little help in concrete situations. It is therefore of interest to find more explicit solution operators, along with good estimates for M(Z). A remarkably simple formula was found by P. Jones [Jon83]. He showed that the series B (z) 2iy 2 y y j j k k f(z) = w exp −ai − j B (z ) z −z z −z z −z Xj j j (cid:16) j(cid:17) (cid:16) yXk≤yj(cid:0) k j k(cid:1)(cid:17) THE CONSTANT OF INTERPOLATION 3 defines afunction f ∈ H∞ such that f(z ) = w with kfk ≤ Ck(w )k . j j j ∞ Here a can be chosen freely and C is a constant depending on a and the sequence Z. The purpose of this note is to show that this explicit operator, conveniently adjusted, is close to optimal. By considering a certainextremeconfigurationofpoints, weareinfactabletoprovethat it yields a sharp upper bound for M(Z). As a result, M(Z) may be bounded from above and below by fairly explicit numerical constants. We begin by showing how to “optimize” Jones’ formula. Take an analytic function g such that g(i) = 1. We need |g| to have a harmonic majorant, so we require (z+i)−2g(z) ∈ H1 [Gar81, p. 60]. Let u denote the least harmonic majorant of |g| and set g (z) = g((z −x )/y ), u (z) = u((z −x )/y ). j j j j j j We assume further that g is such that u (z) j U (z) = k |B (z )| yXj≤yk j j defines a harmonic function; let V (z) be a harmonic conjugate of U , k k and set G = U + iV . This leads us to the following interpolation k k k formula: B (z) j f(z) = w g (z)exp −a(G (z)−G (z )) j j j j j B (z ) Xj j j (cid:16) (cid:17) with a some constant which may be chosen freely. Clearly, f(z ) = w . j j We define c (Z,g) = supU (z ), J j j j so that for arbitrary z we get the estimate exp(ac (Z,g)) a|g (z)| J j |f(z)| ≤ k(w )k exp(−aU (z)). j ∞ j a |B (z )| Xj j j Replacing |g | by u , we find that the latter sum is a lower Riemann j j sum for the integral ∞ e−tdt Z 0 so that we arrive at the estimate exp(ac (Z,g)) J |f(z)| ≤ k(w )k . j ∞ a We see that the optimal choice of a is 1/c (Z,g), and this leads us to J the bound M(Z) ≤ ec (Z,g). J 4 ARTURNICOLAU,JOAQUIMORTEGA-CERDA`,AND KRISTIANSEIP We may finally minimize c (Z,g) and define J c (Z) = infc (Z,g) J J g so that M(Z) ≤ ec (Z). J We have then proved one part of the following theorem. Theorem 1. For every sequence Z in the upper half-plane, (2) M(Z) ≤ ec (Z). J The inequality is best possible in the sense that the constant e on the right side of (2) cannot be replaced by any smaller number. We postpone for the moment the proof of the sharpness of (2); it will be established by means of an explicit example at the end of this note. It may be argued that finding the g minimizing c (Z,g) is not much J easier than solving for the function G in P. Beurling’s formula. How- ever, we will now point out that c (Z) relates nicely to more com- J putable characteristics. An immediate observation is that if we choose g(z) = −4/(z + i)2, then u(z) = 4(y +1)/|z +i|2 so that c (Z,g) becomes J 4y (y +y ) 1 j j n c (Z) = sup . HJ |z −z |2 |B (z )| n yXj≤yn j n j j This choice of g corresponds to the original version of Jones’ formula. (The letter ‘H’ in c (Z) stands for Havin; see below.) For this char- HJ acteristic we have the following result. Theorem 2. For every sequence Z in the upper half-plane, M(Z) ≤ kc (Z) HJ for some universal constant k. The best possible k lies in the interval [π/log4,e] = [2.2662...,2.7183...]. Wehavealreadyestablished theupperboundfork. Thelower bound will again follow from the example to be considered below. Our third and final characteristic was introduced by V. Havin in the first appendix of [Koo98]. We get it from the expression for M(Z) obtained from Carleson’s duality argument (see [Gar81, p. 135]): y |h(z )| M(Z) = sup 4π j j : h ∈ H1,khk ≤ 1 . 1 (cid:26) |B (z )| (cid:27) X j j THE CONSTANT OF INTERPOLATION 5 If we choose h(z) = π−1y /(z −z )2, k = 1,2,..., we arrive at k k 4y y 1 k j c (Z) = sup H |z −z¯ |2|B (z )| k Xj k j j j along with the estimate M(Z) ≥ c (Z). H Since clearly c (Z) ≤ 2c (Z), we may summarize our findings as a HJ H chain of inequalities: (3) c (Z) ≤ M(Z) ≤ ec (Z) ≤ ec (Z) ≤ 2ec (Z). H J HJ H In [Koo98], Havin proves that c (Z) ≤ M(Z) ≤ kc (Z), H H with k a universal constant. To prove the right inequality, he proceeds by duality and uses the invariant Blaschke characterization of Carleson measures, which is closely related to the original proof of Carleson. By computing both c (Z) and M(Z) when Z consists of two points, he H also shows that the left inequality is best possible. In fact, it may be checked that each of the inequalities in our chain (3) is sharp. To interpret the “geometric” contents of our characteristics, it may be useful to relate them to the condition y y j k (4) sup < +∞, |z −z¯ |2 k Xj k j which is called the invariant Blaschke condition (see [Gar81, p. 239]). We see that our three characteristics are closely related to the supre- mum appearing in (4). It may also be noted that by the bound M(Z) ≤ 2ec (Z) and a calculus argument applied to the invariant H Blaschke sum, we obtain 2e+4elog(1/δ(Z)) M(Z) ≤ ; δ(Z) see [Koo98, p. 268]. We finally turn to our example which proves the sharpness of (2) and the lower bound for k in Theorem 2. In what follows the notation a(γ) ∼ b(γ) will mean that a(γ) and b(γ) are asymptotically equal, i.e., lim a(γ)/b(γ) = 1. γ→+∞ An example. Fix γ > 0 and consider the Blaschke product defined by z −iek/γ iek/γ −z B(z) = B(γ,z) = . z +iek/γ z +iek/γ Y Y k≤0 k>0 6 ARTURNICOLAU,JOAQUIMORTEGA-CERDA`,AND KRISTIANSEIP The signs have been chosen so that iB′(i) > 0, which ensures conver- gence of the product. The sequence of zeros Zγ = (iek/γ)k∈Z is clearly aninterpolatingsequence withM(Z )blowingupwhenγ tendsto+∞. γ To obtain appropriate estimates for B, we relate it to the function 2 F(z) = 2e−π2γ sin(πγlog(−iz)), where log(z) is the principal branch of the logarithm. Both B and F are bounded functions, and they have the same zeros. The quotient F(z)/B(z) is anouter function with modulus close to 1 when γ islarge. More precisely, we have |F(x)| sup log ∼ e−π2γ, (cid:12) |B(x)|(cid:12) x∈R\{0}(cid:12) (cid:12) (cid:12) (cid:12) and therefore the same asym(cid:12) ptotic rel(cid:12)ation holds in the upper half- plane. The Blaschke product B is highly symmetric. It is real on the imaginary half-axis iR+ and moreover B(e1/γz) = −B(z). We check that on iR+ the modulus of B peaks at the points {ie(k+1/2)/γ : k ∈ Z}. Again comparing it to F, we check that 2 (5) B(ie(k+1/2)/γ) = (−1)k2e−π2γt with t ∼ 1. γ γ We will now obtain a lower estimate for M(Z ) by finding a minimal γ norm solution of the interpolation problem f(iek/γ) = (−1)k, k ∈ Z. By (5), the problem is solved by the function g(z) = c B(e1/(2γ)z), γ 2 π γ with c an appropriate constant satisfying c ∼ e 2 /2. This means γ γ that if we can prove that g is a minimal norm solution, then it follows that t 2 γ π γ (6) M(Z ) ≥ e 2 with t ∼ 1. γ γ 2 We wish to prove that g is a solution of minimal norm. To this end, observe that an arbitrary minimal norm solution can expressed as f = g +hB with h a bounded analytic function. We may assume that f is real on iR+ because by symmetry we may if necessary replace f by (f(−z)+ f(z))/2. Thus h is also real on iR+. We define m−1 1 h (z) = h(e2k/γz), m m X k=0 THE CONSTANT OF INTERPOLATION 7 ˜ andchooseaconvergent subsequence h (z) → h(z)suchthatthelimit mk function satisfies h˜(e2/γz) = h˜(z), and h˜(iy) ∈ R for real y. Hence f˜ = g + h˜B is also a minimal norm solution and f˜(e2/γz) = f(z). Finally, note that 1 ϕ(z) = (f˜(z)−f˜(e1/γz)), 2 is a minimal norm solution as well such that (7) ϕ(e1/γz) = −ϕ(z). Assume now that g is not a minimal norm solution. Then kϕk < ∞ kgk . Between the points i and ie1/γ, ϕ has a zero iδ because it is real ∞ on iR. Therefore, by the periodicity expressed by (7), ϕ has zeros at iδek/γ, k ∈ Z. It follows that we may factorize ϕ as ϕ(z) = B(z/δ)ϕ (z). 0 We evaluate ϕ at the point i and get 1 kϕk kϕk ∞ ∞ 1 = |ϕ(i)| = |B(i/δ)||ϕ (i)| ≤ kϕ k = = < 1, 0 0 ∞ c c kgk γ γ ∞ which is a contradiction. We conclude that g has minimal norm so that (6) holds. The next step is to compute c (Z ). Since B(e1/γz) = −B(z), we J γ have that |ek/γB′(iek/γ)| = |B′(i)| for each integer k. Hence |B (iek/γ)| = 2ek/γ|B′(iek/γ)| = 2|B′(i)|. k The derivative B′(i) can be estimated in terms of F′(i), which gives us 2 iB′(i)eπ2γ/(2πγ) → 1 as γ → +∞. Thus 2 (8) c (Z ) ∼ (4πγ)−1eπ2γ inf sup u(iy /y ) J γ k j g(i)=1 k∈Z X yj≤yk with u denoting as before the least harmonic majorant of |g|. Using the explicit expression for this majorant, we get 1 1 inf sup u(iy /y ) = inf |g(ek/γt)|dt. g(i)=1 k∈ZyXj≤yk k j g(i)=1Xk≥0 π ZR 1+t2 8 ARTURNICOLAU,JOAQUIMORTEGA-CERDA`,AND KRISTIANSEIP We interpret the sum on the right as a Riemann sum, so that γ 1 ∞ u(iy /y ) ∼ |g(tex)|dxdt = X k j π ZR 1+t2 Z0 yj≤yk γ 1 ∞ |g(u)| dudt. π ZR 1+t2 Zt u Integrating by parts, we get γ arctant u(iy /y ) ∼ |g(t)|dt. k j X π ZR t yj≤yk We want to minimize the latter integral over all functions g such that (z + i)−1g ∈ H1 and g(i) = 1. This can be restated as an extremal problem in the weighted Hardy space with norm arctant khk2 = |h(t)|2 dt. ZR t Inturn,wecanreducethisproblemtooneforthestandardHardyspace H2, and we find that our original problem is solved by the function 2 2i ψ(z) g (z) = , 0 (cid:18)z +i(cid:19) ψ(i) where ψ(z) is the outer function whose modulus is t/arctant on R. Since arctant 4 1 4π |g (t)| dt = = , ZR 0 t ZR (t2 +1)|ψ(i)| |ψ(i)| we get 2 π γ e 2 (9) c (Z ) ∼ J γ π|ψ(i)| when plugging our extremal function g into (8). 0 We are left with the computation of |ψ(i)|. We first note that i 1 t ψ(i) = iexp − + log|arctan(t)|dt . (cid:16) π ZR(cid:16)i−t t2 +1(cid:17) (cid:17) Since 1 1−it |arctan(t)| = log , 2 1+it (cid:12) (cid:16) (cid:17)(cid:12) (cid:12) (cid:12) the change of variables (cid:12) (cid:12) 1−it eiθ = 1+it THE CONSTANT OF INTERPOLATION 9 brings us to the explicit expression 1 π 2ie ψ(i) = 2iexp − log|logeiθ|dθ = . 2π Z π (cid:16) −π (cid:17) Combining (6) and (9), we conclude that 1 2 t π γ γ c (Z ) ∼ e 2 ≤ M(Z ) with t ∼ 1, J γ γ γ 2e e which proves the sharpness of (2) of Theorem 1. The computation of c (Z ) is straightforward. Indeed, HJ γ 2 4e−k/γ c (Z ) ∼ (4πγ)−1eπ2γ . HJ γ (1+e−k/γ) X k≥0 The sum is again regarded as a Riemann sum, i.e., e−k/γ ∞ e−x ∼ γ dx = γ log2 1+e−k/γ Z 1+e−x X 0 k≥0 so that we arrive at the relation log2 2 t 2log2 π γ γ c (Z ) ∼ e 2 ≤ M(Z ) with t ∼ 1. HJ γ γ γ π π This proves the lower bound for k in Theorem 2. References [Car58] LennartCarleson,Aninterpolationproblemforboundedanalyticfunctions, Amer. J. Math. 80 (1958), 921–930. MR 22 #8129 [Car63] , Interpolations by bounded analytic functions and the Corona problem, Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963,pp. 314–316. MR 31 #549 [Gar81] John B. Garnett, Bounded analytic functions, Pure and Applied Mathe- matics, vol. 96, Academic Press Inc. [HarcourtBrace JovanovichPublish- ers], New York, 1981. MR 83g:30037 [Jon83] Peter W. Jones, L∞ estimates for the ∂¯ problem in a half-plane, Acta Math. 150 (1983), no. 1-2, 137–152. MR 84g:35135 [Koo98] Paul Koosis, Introduction to H spaces, second ed., Cambridge Tracts p in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998, With two appendices by V. P. Havin [Viktor Petrovich Khavin]. MR 2000b:30052 10 ARTURNICOLAU,JOAQUIMORTEGA-CERDA`,AND KRISTIANSEIP Dept. Matema`tiques, Universitat Auto`noma de Barcelona, 08193 Bellaterra, Spain E-mail address: [email protected] Dept. Matema`tica Aplicada i Ana`lisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain E-mail address: [email protected] Dept. of Mathematical Sciences, Norwegian University of Science and Technology, N–7491 Trondheim, Norway E-mail address: [email protected]