Sampling measures

Publicacions Matem`atiques, Vol 42 (1998), 559–566. SAMPLING MEASURES Joaquim Ortega-Cerd` a Abstract We give a description of all measures such that for any function in a weighted Fock spaces the Lpnorm with respect to the measure is equivalent to the usual norm in the space. We do so by a process of discretization that reduces the problem to the description of sampling sequences. The same kind of result holds for weighted Bergman spaces and the Paley-Wiener space. 1. Introduction In this note, we address the following problem. Given a subharmonic function φ:C→R, under the restriction m<∆φ<M, consider the weighted Fock space Fp φ(1 ≤p<∞) consisting of entire functions f such that kfkp φ,p =ZC |f(z)|pe−pφ(z)dm(z)<∞ equipped with its natural norm k·k φ,p. The case of p= 2 and φ=|z|2 corresponds to the classical Bargmann-Fock space. We want to describe the measures µsuch that the norm of a function f∈F p φis equivalent to the original norm. That is, there is a constant K>0 such that, 1 Kkfkp φ,p ≤ZC |f(z)|pe−pφ(z)dµ(z)≤Kkfkp φ,p ∀f∈F p φ . The inequality: ZC |f(z)|pe−pφ(z)dµ(z)≤Kkfkp φ,p ∀f∈F p φ Supported by the DGICYT grant PB95-0956-C02-02 and the CIRIT grant 1996SGR00026. 560 J. Ortega-Cerd` a is usually called a Carleson-type inequality (the original Carleson inequality was for Hpfunctions in the disk) and the measures that satisfy such inequalities are called Carleson measures. We are interested in measures that satisfy both a direct and a reverse Carleson inequality. Such measures will be termed sampling measures. The description has already been achieved in the classical Bargmann-Fock space when µconsists of a sum of Dirac deltas in a remarkable work by Seip and Seip-Wallsten [Sei92], [SW92]. See also [BO95] and [OS98] for an extension to the case of weighted Fock spaces. Our work will be, in fact, a reduction of the general problem to the case of discrete measures. Note that one can prove an analogous result for the weighted Bergman space since there is also a characterization of the sampling sequences in that setting (see [Sei93] for the usual Bergman space and [BO95] and [OS98] for the weighted case). Independently of this work, D. Luecking [Lue98] has reached also a different characterization of the sampling measures for the Bergman space. His characterization is in terms of properties of the support of the measures which are weak limits of translates of the original sampling measure. This is in the spirit of the work of Beurling [Beu89, p. 345] on the description of sampling sequences in the Bernstein space, which is also in the core of all of Seip’s theorems mentioned above. It is also possible to adapt the previous ideas to the Paley-Wiener space. We can describe all sampling measures in terms of the sampling sequences in a completely analogous way. In this setting there is an obvious extra difficulty. The sampling sequences are not characterized for 1<p<∞. Nevertheless, there are necessary and sufficient conditions which in some sense are very close. They are essentially due to Beurling, although he never explicitly stated them in the Lpsetting. The original Beurling theorem was stated for the Bernstein space PW∞ τof entire functions of exponential type τwhich are bounded on the real line. The details on how to get the Lpvariant can be found in [Sei95]. Also for a complete description of the sampling sequences in the Paley-Wiener space in the range 0 <p≤1 see [Flo98]. There is nothing special about dimension one. All our arguments can be reformulated in several variables. It is possible to describe all sampling measures in terms of sampling sequences. Unfortunately very little is known about sampling sequences in higher dimensions, thus this reduction is not very useful in that case. Finally a word on notation. We write f.gif there is a constant K such that f≤Kg, and f≃gif both f.gand g.f. Sampling Measures 561 2. Sampling measures in Fock space The description for an arbitrary measure will use the characterization of sampling sequences (i.e. sequences Γ such that µ=Pγ∈Γδγis a sampling measure) that was found in [OS98]: Theorem (A).A sequence Γis a sampling sequence for the space Fp φ (1 ≤p<∞)if and only if the following two conditions are fulfilled •Γis a finite union of uniformly separated sequences. •There is a uniformly separated subsequence Γ0⊂Γsuch that lim inf r→∞ inf z∈C #(D(z,r)∩Γ0) RD(z,r)∆φ>1 2π. The first of the two conditions implies that µis a Carleson measure for the Fock space and the second condition (the density condition) is needed in order to have a reverse Carleson inequality. In this description we could have used any compact rK+zinstead of the disk D(z,r) under very mild assumptions on the boundary of K(see [Lan67]). We could have taken a square for instance, which will be more convenient for the description in the case of general measures. In order to state the main result let us introduce some notation. We split the plane into a grid of squares of side length r. We take Na large integer and δ>0. We group together N2little squares of size rforming a big square Sof side length Nr. We denote by n(S) the number of little squares sof the original grid that are in Ssuch that µ(s)≥δ. Of course, n(S)does depend on δ,N,rand S. The theorem is the following: Theorem. The measure µis a sampling measure if and only if the following conditions are fulfilled •There is a constant Csuch that µ(D(z,1)) ≤Cfor all disks of radius 1. •There is an r>0and a grid made out of squares of side-length r, an integer N>0and a positive δ>0such that (1) inf S n(S) RS∆φ>1 2π, where the infimum is taken over all squares Smade out of N2 little squares from the original grid. 562 J. Ortega-Cerd` a Before proving the theorem, let us note that if this density condition (1) holds for some r,N,δit also holds for an arbitrary small r (eventually decreasing δand increasing N). We can also pick an arbitrary big N. Note that condition (1) controls whether the measure µis massive enough everywhere on a large scale. This control is done by taking into account averages of µover very big squares S. Thus, if we have a sampling measure µand we alter it by erasing its mass on an arbitrary big strip it will remain a sampling measure. On the other hand if we want µto be a sampling measure it must be somehow spread out. In the case of the sum of Dirac deltas this follows because one considers separated sequences. In our case this is achieved by disregarding all little squares with small contribution of mass in the definition of n(S). As a corollary of this theorem we may give an alternative description of sampling measures in the particular situation when dµ(z)=χ E (z)dm(z), where Eis any measurable set in C. This is an easier case that does not require the use of Beurling-type densities in order to be solved. It has, in fact, been considered previously by Logvinenko and Sereda in the Paley-Wiener space [VL74], by Luecking in the Bergman space [Lue81] and by Janson and Peetre in the Fock space [JPR87]. A measurable set E⊂Cis said to be relatively dense in Cif and only if there is a big R>0 such that infz∈C|E∩D(z,R)|>0. The following corollary is a direct consequence of the theorem Corollary (Janson-Peetre-Rochberg).A measurable set E⊂Csatisfies ZC |f(z)|pe−pφ(z)dm(z).ZE |f(z)|pe−pφ(z)dm(z),∀f∈F p φ , if and only if Eis relatively dense in C. Similarly we may obtain the Logvinenko-Sereda theorem for the PaleyWiener space [VL74] (in dimension 1) from Beurling description of sampling sequences in the Bernstein space PW∞ τ. Finally in the setting of the weighted Bergman space Bp αof holomorphic functions in the unit disk, it is possible to reobtain Luecking’s theorem [Lue81] from the description of the sampling sequences in Bp αthat was found in [Sei93]. In the proof of the theorem we will need the following lemma which plays the role of the Plancherel-Polya inequality in the case of PaleyWiener spaces and which is proved in [OS98]. Sampling Measures 563 Lemma. If fbelongs to Fp φ, then |f(z)|pe−pφ(z).ZD(z,1) |f(w)|pe−pφ(w)dm(w), |∇(|f|pe−pφ)(z)|.ZD(z,1) |f(w)|pe−pφ(w)dm(w) provided f(z)6=0. The first condition µ(D)≤Cfor all disks of radius 1 is equivalent to the Carleson inequality ZC |f(z)|pe−pφ(z)dµ(z).kfkp φ,p. Let us check that this condition is indeed sufficient. By the first inequality in the lemma, there is a constant Ksuch that ZC |f(z)|pe−pφ(z)dµ(z)≤ZC KÃZD(z,1) |f(w)|pe−pφ(w)dm(w)!dµ(z) ≤CK ZC |f(w)|pe−pφ(w)dm(w). We have thus obtained a Carleson inequality. The smallest constant that appears in the Carleson inequality is called the Carleson constant and we have proved that it goes to 0 as C= supz∈Cµ(D(z,1)) goes to 0. The condition is also necessary. If it did not hold, for any ε>0 and n>0, there would be disks D(zn,ε) with µ(D(zn,ε)) >n. This is not possible, since any single point znis an interpolating sequence by itself (see [BO95]), therefore there are functions fnsuch that kfnkp,φ .1 and |fn(zn)|pe−pφ(zn)= 1. Because of the inequality on the gradient in the lemma, the function |fn|pe−pφ is close to 1 in an εneighborhood of zn, thus RC|fn|pe−pφ dµ &nwhich is contradictory since we assumed that µwas a Carleson measure. From now on, we will assume that the measure µthat we are considering satisfies µ(D(z,1)) ≤C. We will deal now with the density condition. 564 J. Ortega-Cerd` a Let us start by proving the necessity of (1). We pick a square grid of side length rand for any δ>0, we split the measure µinto two measures µ1and µ2. The measure µ1will be the original measure µrestricted to the squares ssuch that µ(s)≥δand µ2will be the measure µ restricted to the squares ssuch that µ(s)<δ. We start by proving that if δis small enough then µis a sampling measure if and only if µ1 is a sampling measure. The non-trivial direction is to check that µ1is sampling when µis sampling. This is the case because for any square s on the grid we know that µ2(s)/|s|≤δ/r2.Thusifwepickδvery small (with respect to r) we see that µ2is a Carleson measure with a Carleson constant that goes to 0 as δ→0. Therefore, ZC |f|pe−pφ dm .ZC |f|pe−pφ dµ =ZC |f|pe−pφ dµ1+ZC |f|pe−pφ dµ2 ≤ZC |f|pe−pφ dµ1+O(δ)ZC |f|pe−pφ dm. For a very small δwe can absorb the last summand in the left hand side of the inequality and we get ZC |f|pe−pφ dm .ZC |f|pe−pφ dµ1. On the other hand, if the square grid has a very small length r>0, we may discretize the measure µ1and still get a sampling measure. We set µ∗ 1=Pnµ1(sn)δan, where snis any square of the grid and anare the centers of the squares sn. We are going to check that µ∗ 1is sampling if and only if µ1is sampling. We know that if |z−w|≤r<1/2 then ||f(z)|pe−pφ(z)−|f(w)| p e −pφ(w)|.rZD(z,1) |f(x)|pe−pφ(x)dm(x), because of the inequality for the gradient in the lemma. Thus, ZC |f|pe−pφ dm.ZC |f|pe−pφ dµ1 .ZC |f|pe−pφ dµ∗ 1+ZC rZD(z,1) |f(w)|pe−pφ(w)dm(w)dµ1(z) .ZC |f|pe−pφ dµ∗ 1+rZC |f(w)|pe−pφ(w)dm(w). Sampling Measures 565 If ris small enough, we absorb the last term on the left-hand side of the inequality. On the other direction the same is true, if µ∗ 1is sampling we get ZC |f|pe−pφ dm.ZC |f|pe−pφ dµ∗ 1 .ZC |f|pe−pφ dµ1+ZC rZD(z,1) |f(w)|pe−pφ(w)dm(w)dµ1(z) .ZC |f|pe−pφ dµ1+rZC |f(w)|pe−pφ(w)dm(w), and we conclude that µ1is sampling as well. Finally, since for any little square sof the grid δ<µ 1 (s)<Cor µ1(s) = 0 we may conclude that the sequence Γ made out of the centers of the squares where µ1has mass is a sampling sequence if and only if µ∗ 1is a sampling measure and we get the theorem thanks to the characterization of the sampling sequences (Theorem A). Acknowledgement. I am grateful to the referee for some appropriate remarks that have improved the readability of this note. References [Beu89] A. Beurling,“The collected works of Arne Beurling,” vol. 2, Birkh¨auser, Boston, 1989. [BO95] B. Berndtsson and J. Ortega-Cerd` a, Interpolation and sampling in Hilbert spaces of holomorphic functions, J. Reine Angew. Math. 464 (1995), 109–120. [Flo98] K. M. 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Seip, On the connection between exponential bases and certain related sequences in L2(−π,π), J. Funct. Anal. 130 (1995), 131–160. [SW92] K. Seip and R. Wallst´ en, Density theorems for sampling and interpolation in the Bargmann-Fock space II, J. Reine Angew. Math. 429 (1992), 107–113. [VL74] Y. F. Sereda and V. N. Logvinenko, Equivalence of norms in spaces of entire functions of exponential type, Teor. Funktsi˘ı Funktsional Anal. i Prilozhen 20 (1974), 62–78. Departament de Matem`atica Aplicada i An`alisi Universitat de Barcelona Gran Via 585 08071 Barcelona SPAIN e-mail: [email protected] Primera versi´o rebuda el 19 de maig de 1998, darrera versi´o rebuda el 9 d’octubre de 1998