arXiv:math/0104130v1 [math.FA] 12 Apr 2001 INTERPOLATION OF SUBSPACES AND APPLICATIONS TO EXPONENTIAL BASES IN SOBOLEV SPACES SERGEI IVANOV AND NIGEL KALTON Abstract. We give precise conditions under which the real interpolation space [Y0 , X1 ]θ,p coincides with a closed subspace of [X0 , X1 ]θ,p when Y0 is a closed subspace of codimension one. We then apply this result to nonharmonic Fourier series in Sobolev spaces H s (−π, π) when 0 < s < 1. The main result: let E be a family of exponentials exp(iλn t) and E forms an unconditional basis in L2 (−π, π). Then there exist two number s0 , s1 such that E forms an unconditional basis in H s for s < s0 , E forms an unconditional basis in its span with codimension 1 in H s for s1 < s. For s0 ≤ s ≤ s1 the exponential family is not an unconditional basis in its span. 1. Introduction In this paper we will apply a result on interpolation of subspaces to the study of exponential Riesz bases in Sobolev spaces. In section 2 we consider the comparison of the interpolation spaces Xθ := [X0 , X1 ]θ,p and Yθ := [Y0 , X1 ]θ,p for 1 ≤ p < ∞, where Y0 is a subspace of X0 with codimension one, say Y0 = ker ψ where ψ ∈ X0∗ . This problem, as far as we know, was first formulated in [16], v.1, Ch.1.18 in 1968. As we show in Theorem 2.1 there are two indices 0 ≤ σ0 ≤ σ1 ≤ 1 which may be explicitly evaluated in terms of the K-functional of ψ so that: 1. If 0 < θ < σ0 then Yθ is a closed subspace of codimension one in Xθ . 2. If σ1 < θ < 1 then Yθ = Xθ with equivalence of norm and 3. If σ0 ≤ θ ≤ σ1 then the norm on Yθ is not equivalent to the norm on Xθ . Let us discuss the history of this theorem. The special case of a Hilbert space of Sobolev type connected with elliptical boundary data was considered in [16], and in this case the critical indices σ0 and σ1 coincide. In the well known case [16] X1 = L2 (0, ∞), X0 = W21 (0, ∞) and Y0 is the subspace of W21 of functions vanishing at the origin, this critical value is σ0 = σ1 = 1/2. Later R. Wallsten [26] gave an example where the critical indices satisfy σ0 < σ1 . The general problem was considered by J. Löfström [17], where some special cases of Theorem 2.1 are obtained. Later, Löfström in an unpublished (but The first author was partially supported by the Russian Basic Research Foundation (grant # 99-01-00744) and the second author was supported by NSF-grant DMS-9870027. The first author is grateful to S.Avdonin for fruitful discussions. 1 web-posted) preprint from 1997, obtained most of the conclusion of Theorem 2.1: specifically he obtained the same result except he did not treat the critical values θ = σ0 , σ1 . The authors were not aware of Löfström’s earlier work during the initial preparation of this article and our approach is rather different. A more general but closely related problem on interpolating subspaces of codimension one has been recently considered in [12] and [10]. For general results on subcouples we refer to [9]. Let us recall next that a sequence (en )n∈Z in a Hilbert space H is called a Riesz basic sequence if there is a constant C so that for any finitely non-zero sequence (an )n∈Z we have 1 C X n∈Z |an |2 ! 21 ≤k X n∈Z an en k ≤ C X n∈Z |an |2 ! 12 . A Riesz basis for H is a Riesz basic sequence whose closed linear span [en ]n∈Z = H. A sequence (en ) is an unconditional basis, respectively unconditional basic sequence if (en /ken k)n∈Z is a Riesz basis, respectively, a Riesz basic sequence. In the second part of the paper we apply our interpolation result to study the basis properties of exponential families {eiλn t } in Sobolev spaces. These families appear in such fields of mathematics as the theory of dissipative operators (the Sz.–Nagy–Foias model) , the Regge problem for resonance scattering, the theory of initial boundary value problems, control theory for distributed parameter systems, and signal processing, see, e.g., [22], [8], [13], [2], [25]. One of the most important problems arising in all of these applications is the question of the Riesz basis property od these families. In the space L2 (−π, π) this problem has been studied for the first time in the classical work of Paley and Wiener [23]. The problem has now a complete solution [11], [20] on the basis of an approach suggested by B. S. Pavlov. The principal result for Riesz bases can be formulated as follows [11]. Proposition 1.1. The sequence (eiλn t )n∈Z is a Riesz basis for L2 (−π, π) if and only if sup |ℑλn | < ∞, (1.1) inf |λk − λj | > 0. k6=j and there is an entire function F of exponential type π (the generating function) with simple zeros at (λn )n∈Z and such that for some y |F (x + iy)|2 satisfies the Muckenhoupt condition (A2 ) (we shall write this as |F |2 ∈ (A2 )): Z n1 Z o 1 2 sup |F (x + iy)| dx |F (x + iy)|−2 dx < ∞, |I| I I∈J |I| I where J is the set of all intervals of the real axis. 2 In [20] a corresponding characterization is given for exponential families which form an unconditional basis of L2 (−π, π) when ℑλn can be unbounded both from above and below. Let us describe known results concerning exponential bases in Sobolev spaces. The first result in this direction has been obtained by D. L. Russell in [24] Russell studied the unconditional basis property for exponential families in the Sobolev spaces H m (−π, π) with m ∈ Z. Proposition 1.2. [24] Suppose (eiλn t )n∈Z is a Riesz basis for L2 (−π, π). Suppose m ∈ N and suppose µ1 , · · · , µm ∈ C \ {λn : n ∈ Z} are distinct. Then m (eiλn t )n∈Z ∪ (eiµk t )m k=1 is an unconditional basis of H (−π, π). In particular (eiλn t )n∈Z is an unconditional basic sequence whose closed linear span has codimension m in H m (−π, π). In [21] the unconditional basis property for an exponential family was studied in H s (−π, π) for noninteger s for the case λn being the eigenvalues of a Sturm–Liouville operator with a smooth potential. Note that the generalization of the Levin–Golovin theorem for Sobolev spaces has been obtained [3] using ‘classical methods’ of the entire function theory. Suppose {λn }n∈Z are the zeros of an entire function F of exponential type π, (λn ) is separated (1.1), and on some line {x + iy}x∈IR we have C −1 (1 + |x|)s ≤ |F (x + iy)| ≤ C(1 + |x|)s . Then the family {eiλk t /(1 + |λk |)s } forms a Riesz basis in H s (−π, π). Notice that this result was applied to several controllability problems for the wave type equation [4]. Recently Yu. Lyubarskii and K. Seip [19] have established a necessary and sufficient criterion for sampling/interpolation problem for weighted PaleyWiener spaces, which gives a criterion for a sequence to be an unconditional basis in H s . For the case, sup |ℑλn | < ∞, the main result is the following: Theorem 1.3. (eiλn t )n∈Z forms an unconditional basis in H s (−π, π) if and only if (λn ) is separated (i.e. (1.1) holds) and for the generating function F we have |F (x + iy)|2 /(1 + |x|2s ) ∈ (A2 ) for some y. The main idea of the present paper is that if (eiλn t )n∈Z forms a Riesz basis in L2 (−π, π) then it also forms an unconditional basis of a subspace Y0 of H 1 (−π, π) of codimension one. Then, by interpolation, one obtains that (eiλn t )n∈Z is an unconditional basis of the intermediate spaces [Y0 , L2 ]θ,2 for 0 < θ < 1. This approach was suggested in [7] by the first author. The main result of [7] is incorrect in the general case because a mistake connected with interpolation of subspaces. Here we correct this mistake. Let us describe the results concerning unconditional bases in Sobolev spaces. One of our main results for Riesz bases is as follows: 3 Theorem 1.4. Suppose (eiλn t )n∈Z forms a Riesz basis of L2 (−π, π). Suppose (λn − n)n∈Z is bounded and let δn = ℜλn − n. Then there exist critical indices 0 < s0 ≤ s1 < 1 given by: X δn 1 1 s1 = − lim inf 2 τ →∞ t≥1 log τ n t<|n|≤τ t and s0 = 1 1 − lim sup 2 τ →∞ t≥1 log τ X δn n t<|n|≤τ t such that: (1) (eλn )n∈Z is an unconditional basis of the Sobolev space H s if and only if 0 ≤ s < s0 . (2) (eλn )n∈Z is an unconditional basis of a closed subspace of H s of codimension one if and only if s1 < s ≤ 1. (3) If s0 ≤ s ≤ s1 then (eλn ) is not an unconditional basic sequence. This result is deduced from results in Sections 3 and 4. In Section 4 we in fact consider the more general situation for unconditional bases and give rather more technical results. The above Theorem 1.4 however is the simplest case and follows by combining Theorem 4.2, Theorem 4.9 and Theorem 4.10. Our approach is based on estimates of the K-functional for the continuous linear functional on H 1 (−π, π) which annihilates each eiλn x whose existence is guaranteed by the result of Russell (Proposition 1.2). The estimates are in terms of the generating function F. Once one has Theorem 1.4 then it is easy to construct real sequences (λn ) to show that s0 , s1 can take any values in (0, 1) such that s0 ≤ s1 . In the case of regular power behavior of F i.e. for some y ≥ 0, |F (x + iy)| ∼ (1 + |x|)s one has s1 = s0 = s + 21 . The results for the whole scale H s (−π, π) can then be obtained by ‘shift’ using the fact that the differentiation operator with appropriate conditions is an isomorphism between a one-codimensional subspace of H m and H m−1 ; we will not pursue this extension. 2. Interpolation of subspaces Let (X0 , X1 ) be a Banach couple with X0 ∩ X1 dense in X0 , X1 . If 0 < θ < 1 and 1 ≤ p < ∞ the real interpolation space Xθ = [X0 , X1 ]θ,p is defined, see, e.g., [5], to be the set of all x ∈ X0 + X1 such that Z ∞  p1 θp−1 p < ∞, kxkXθ = t K(t, x) dt 0 4 where K(t, x) is the K–functional. An equivalent definition [5] p. 314 (yielding an equivalent norm) can be given by using the J-method:   ! p1  X  X k p θk max{kxk k0 , 2 kxk k1 } 2 xk , kxkXθ = inf : x=   k∈Z k∈Z where the series converges in X0 + X1 . Now suppose 0 6= ψ ∈ X0∗ and let Y0 be its kernel. We suppose also (only this case is interesting) that Y0 ∩ X1 is dense in X1 , i.e., ψ is not bounded in X1 . Let Yθ be the corresponding spaces obtained by interpolating Y0 and X1 . Clearly Yθ ⊂ Xθ and the inclusion has norm one. It is easy to show that the closure of Yθ in Xθ is either a subspace of codimension one when ψ is continuous on Xθ or the whole of Xθ when ψ is not continuous. Let us now introduce two important indices. 1 K(τ t, ψ) log τ →∞ 0<τ t≤1 log τ K(t, ψ) σ1 = lim sup and K(τ t, ψ) 1 log , τ →∞ 0<τ t≤1 log τ K(t, ψ) σ0 = lim inf where K(t, ψ) = K(t, ψ; X0∗ , X1∗ ). ¿From the multiplicative properties of the function K(τ t, ψ)/K(t, ψ) it is clear that these limits exist and 0 ≤ σ0 ≤ σ1 ≤ 1. Since K(t, ψ) is bounded as t → ∞ we can also write: 1 K(τ t, ψ) log . τ →∞ 0<t<∞ log τ K(t, ψ) σ1 = lim sup Let us observe that: sup{|ψ(x)| : max{kxk0 , tkxk1 } ≤ 1} = K(t−1 , ψ) We define a sequence (wn )n∈Z by wn = K(2−n , ψ)−1 . Notice that inf n∈Z wn ≥ kψk−1 X0∗ > 0 and that in general wn ≤ wn+1 ≤ 2wn . Now it is easy to see that σ1 wn+k wn wn+k 1 limk→∞ inf n≥0 k log2 wn . = limk→∞ supn k1 log2 σ0 = As mentioned in the introduction the following result is a slight improvement of a result of Löfström [18], who obtains the same result by quite different arguments except for the critical indices θ = σ0 , σ1 . 5 Theorem 2.1. 1. Yθ = Xθ (with equivalence of norm) if and only if θ > σ1 . 2. Yθ is a closed subspace of codimension one in Xθ if and only if θ < σ0 . 3.If σ0 ≤ θ ≤ σ1 then Yθ is not closed in Xθ . We shall consider the weighted ℓp space ℓp (w) of all sequences (αn )n∈Z such that ! p1 X kαk = . wnp |αn |p k∈Z We shall use ζn for the standard basis vectors. On ℓp (w) we consider the shift operator S((αn )) = (αn−1 ). From the above remarks it is clear that S, S −1 are both bounded and kSk ≤ 2, kS −1 k = 1. Furthermore the spectral radius formula shows that 2σ1 is the spectral radius r(S) of S. Now let P+ be the projection P+ (α) = (δn αn ) where δn = 1 if n ≥ 0 and 0 otherwise. It is easy to calculate wk kP+ S −n k = sup k≥0 wn+k and so this implies that r(P+ S −1 ) = 2−σ0 . We will need the following key Lemma: Lemma 2.2. Let 0 < θ < 1 and let Tθ = S − 2θ I. Then 1. Tθ is an isomorphism onto ℓp (w) if and only if σ1 < θ. 2. T is an isomorphism onto a proper closed subspace if and only if θ < σ0 . In this case P the range of T is the subspace of codimension one of all α such that n∈Z 2nθ αn = 0. Proof. First observe that if θ > σ1 then Tθ must be an isomorphism onto ℓp (w) since 2θ exceeds the spectral radius of S. Furthermore since the spectrum of S is invariant under rotations it is clear that Tσ1 cannot be an isomorphism onto ℓp (w). Also note that Tθ is always injective and that if fθ is a linear functional its range then fθ (ζn ) = c2nθ for some constant c, i.e., Pannihilating fθ (α) = n∈Z 2nθ αn = 0. This implies that the closure ofPthe range is either the whole space or the subspace of codimension one when n∈Z 2nθq wn−q < ∞. Here p1 + 1q = 1 and the formula must be modified if p = 1. We next show that if θ < σ0 then Tθ is an isomorphism onto a closed subspace of codimension one. Next let E = [{ζn : n ≤ −1}] and F = [{ζn : n ≥ 1}]. We remark that Tθ (E) is easily seen to be closed because Tθ is an isomorphism on the unweighted ℓp and wn is bounded for n ≤ −1. If we show Tθ (F ) is closed then we are done, since it is clear this will imply that Tθ (E + F ) is closed and this is a subspace of co-dimension one in the range. However 2−θ > r(P+ S −1 ) so that 2−θ − P+ S −1 is an isomorphism. Restricting to F this implies (2−θ − S −1 )F and hence Tθ (F ) is closed. 6 The proof is completed by showing that if θ ≤ σ1 then if Tθ if has closed range it must satisfy θ < σ0 . Note first that it is enough to establish this for θ < σ1 since the set of operators with Fredholm index one is open. Suppose σ0 < θ < σ1 and Tθ is closed. Then Tθ has a lower estimate kTθ αk ≥ ckαk for all α where c > 0. Assume wn+k > 2nθ wk for some n ∈ N and k ∈ Z. Then consider α = (I + 2−θ S + · · · + 2−nθ S n )2 ζk . Note that kαk ≥ n2−nθ wn+k . However kTθ2 αk =22θ wk + 2 · 2(−n+1)θ wn+k+1 + 2−2nθ w2n+k+2 ≤ 8 max{wk , 2−nθ wn+k , 2−2nθ w2n+k }. Let vn = 2−nθ wn . Then we have if nc2 > 8, (nc2 − 8)vk+n ≤ 8 max{vk , vk+2n }. In particular if nc2 > 16, vk+n < max{vk , vk+2n }. Now since θ < σ1 , we can find k ∈ Z, n > 16c−2 so that wn+k < 2nθ wk or vn+k < vk . Iterating gives us that (vk+rn )∞ r=0 is monotone increasing. Now for any large N and any j ≥ 0 we have wj+N wk+r2n ≥ ≥ 2n(r2 −r1 )θ wj wk+r1n where r1 , r2 such that k + (r1 − 1)n ≤ j ≤ k + r1 n and k + r2 n ≤ j + N ≤ k + (r2 + 1)n. This gives us wj+N ≥ 2(N −2n)θ . wj Hence 1 2n wj+N inf log2 ≥ (1 − )θ. j≥0 N wj N Letting N → ∞ gives σ0 ≥ θ. To show that in fact θ < σ0 needs only the observation again that the set of θ where Tθ has Fredholm index one is open. We now use Lemma 2.2 to establish our main result Theorem 2.1 on interpolating subspaces: Proof. Let us suppose next that either (a) θ < σ0 or (b) θ > σ1 . This implies there exists a constant D so that kαk ≤ DkTθ αk for allP α ∈ ℓp (w); in case (a) Tθ maps onto the subspace of ℓp (w) defined by fθ (α) = n∈Z 2nθ αn = 0, while in case (b) Tθ is an isomorphism onto the whole space (see Lemma 2.2). We observe that in case (a) the linear functional ψ extends to a continuous linear functional on Xθ as X 2nθ K(2n , ψ) < ∞. n∈Z 7 Now suppose x ∈ Xθ with kxkXθ = 1 with the additional in case P assumption θn (a) that ψ(x) = 0. Then we may find (xn )n∈Z such that n∈Z 2 xn = x and ! p1 X max{kxk k0 , 2k kxk k1 }p ≤ 2. k∈Z Then X n∈Z since |ψ(xn )|p wnp ! p1 ≤ 2, |ψ(x)| ≤ wn−1 max{kxk0 , 2n kxk1 }. In case (a) we additionally have X 2nθ ψ(xn ) = 0. n∈Z Thus we can find α ∈ ℓp (w) with Tθ (α) = (ψ(xn )) and kαk ≤ 2D. Then we can find un ∈ X0 ∩ X1 such that max{kun k0 , 2n kun k1 } ≤ 2|αn |wn and ψ(un ) = αn . Let vn = un−1 − 2θ un . Then ! p1 X max{kvk k, 2k kvk k1 }p ≤ 16DkxkXθ . k∈Z P Now ψ(vn ) = αn−1 − 2θ αn = ψ(xn ) and n∈Z 2nθ vn = 0. Hence X x= 2θn (xn − vn ) n∈Z and so x ∈ Yθ with kxkYθ ≤ (16D + 2)kxkXθ . From this it follows that in case (a) we have Yθ = {x : ψ(x) = 0, x ∈ Xθ } and in case (b) Yθ = Xθ . Next we consider the converse directions. Assume either (aa) ψ is continuous on Xθ and Yθ = {x : ψ(x) = 0, x ∈ Xθ } or (bb) Yθ = Xθ . In either case there is a constant D so that if x ∈ Yθ then kxkYθ ≤ DkxkXθ . Observe that in case (a) the linear functional fθ is continuous on ℓp (w) and so the range of Tθ is contained in its kernel; in case (bb) its range is dense. Assume α = (αn )n∈Z ∈ ℓp (w) with kαk = 1; in case (aa) we also assume fθ (α) = 0. We first find xn ∈ X0 ∩ X1 withPψ(xn ) = αn and so that max{kxn k0 , 2n kxn k} ≤ 2|αn |wn for n ∈ Z. Let x = n∈Z 2nθ xn so that x ∈ Xθ with kxkXθ ≤ 2. In case (aa) we have P additionally that ψ(x) = fθ (α) = 0. Now we can find yn ∈ Y0 ∩ X1 so that n∈Z 2nθ yn = x and ! p1 X max{kyk k, 2k kyk k1 }p ≤ 4D. k∈Z 8 Now let un = xn − yn and vn = X k∈Z We argue that X (2.1) k∈Z where P∞ k=n+1 max{kuk k, 2k kuk k1 }p max{kvk k0 , 2k kvk k1 }p Cθ = ( X 2kθ + n 2 kvn k1 ≤ and (since ∞ X k=n+1 nθ 2 un = 0) kvn k0 ≤ ! p1 X ≤ 4D + 2. ≤ Cθ (4D + 2), 2k(θ−1) ). k≥0 k<0 To show (2.1) we note that P 2(k−n−1)θ uk . Then ) p1 n X 2(k−n−1)(θ−1) 2k kuk k1 k=−∞ 2(k−n−1)θ kuk k0 . Let βn = ψ(vn ). Then β ∈ ℓp (w) and kβk ≤ Cθ (4D +2). But now (Tθ (β))n = ψ(un ) = ψ(xn ) = αn so that Tθ is an isomorphism onto the kernel of fθ in case (aa) or onto ℓp (w) in case (bb). These two cases combined with the observation that Yθ can only be a proper closed subspace of Xθ if ψ is continuous on Xθ complete the proof of the Theorem. 3. Sobolev spaces In this section we investigate a special case of the results of the previous section for Sobolev spaces. These results are preparatory for Section 4 where we apply them to exponential bases. Let L2 = L2 (−π, π) and let us denote the standard inner-product on L2 (−π, π) by Z π (f, g) = f (x)g(x)dx. −π We denote by kf k the standard norm on L2 . For s > 0 we define the Sobolev space H s (R) to be the space of all f ∈ L2 (R) so that Z ∞ 2 kf kH s := |fˆ(ξ)|2(1 + |ξ|2s )dξ < ∞ −∞ (fˆ is the Fourier transform). We then define the Sobolev space H s = H s (−π, π) to be the space of restrictions of H s (R)−functions to the interval (−π, π) (with 9 the obvious induced quotient norm). When s = 1 the space H 1 reduces to the space of f ∈ L2 (−π, π) so that f ′ ∈ L2 under the (equivalent) norm: Z π 2 kf k1 = |f (t)|2 + |f ′(t)|2 dt < ∞. −π s Then if 0 < s < 1 we have H = [H 1 , L2 ]1−s = [H 1 , L2 ]1−s,2 [16]. For z ∈ C we define ez (x) = eizx ∈ L2 (−π, π). Now suppose ψ ∈ (H 1 )∗ ; we define its Fourier transform F = ψ̂ to be the entire function F (z) := ψ(ez ) for z ∈ C. Let us first identify (H 1 )∗ via its Fourier transform: Proposition 3.1. Let F be an entire function. In order that there exists ψ ∈ (H 1 )∗ with F = ψ̂ it is necessary and sufficient that: (3.1) (3.2) F is of exponential type ≤ π. Z ∞ −∞ |F (x)|2 dx < ∞. 1 + x2 These conditions imply the estimate: (3.3) sup z∈C |F (z)| < ∞. (1 + |z|)eπ|ℑz| Proof. These results follow immediately from the Paley-Wiener theorem once one observes that ψ ∈ (H 1 )∗ if and only if ψ is of the form ψ(f ) = αf (0) + ϕ(f ′ ) where ϕ ∈ (L2 )∗ . Consider H 1 with the inner product: hf, git = (f ′ , g ′) + t2 (f, g) where t > 0. Let us denote by kψkt the norm of ψ with respect to k · kt where kf k2t = hf, f it, i. e. kψkt := sup{|ψ(f )| : kf kt ≤ 1}. Set (3.4) s0 = 1 − lim sup τ →∞ t≥1 1 kψkt log log τ kψkτ t and (3.5) kψkt 1 . log τ →∞ t≥1 log τ kψkτ t s1 = 1 − lim inf We can specialize Theorem (2.1) to the this special case of interpolating between L2 and H 1 . 10 Proposition 3.2. Suppose ψ ∈ (H 1 )∗ and let Y0 = {f ∈ H 1 : ψ(f ) = 0}. Then: (1) (L2 , Y0)s,2 = H s if and only if 0 ≤ s < s0 . (2) (L2 , Y0 )s,2 is a closed subspace of codimension one in H s if and only if s1 < s ≤ 1. Proof. We can then apply Theorem 2.1 with X0 = H 1 and X1 = L2 . To estimate K(t, ψ) we note that if f ∈ H 1 and t ≥ 1 then √ max(kf k1 , tkf k) ≤ kf kt ≤ 2 max(kf k1, tkf k). and so, for t ≥ 1, √ kψkt ≤ K(t−1 , ψ) ≤ 2kψkt . Hence we can describe the numbers σ0 , σ1 of Theorem 2.1 by σ1 = lim sup τ →∞ t≥1 and kψkt 1 log log τ kψkτ t kψkt 1 log . τ →∞ t≥1 log τ kψkτ t Since σ1 = 1 − s0 and σ0 = 1 − s1 this proves the Proposition. σ0 = lim inf We next turn to the problem of estimating kψkt . The following lemma will be useful: Lemma 3.3. Suppose F satisfies (3.1) and (3.2). Then for any real t we have:  12  Z ∞ 1 |F (x)|2 1 π|t| |F (it)| ≤ |t| 2 e (3.6) dx . π −∞ t2 + x2 Proof. It suffices to consider t > 0. Then by (3.3) F (z)eiπz (z +it)−1 is bounded and analytic in the upper-half plane and so we have: Z teπt ∞ F (x) eiπx F (it) = dx π −∞ x + it x − it Applying the Cauchy–Bunyakowski inequality we prove the lemma. We can now give an estimate for kψkt which essentially solve the problem of determining s0 and s1 . Theorem 3.4. There exist a constant C so that for t ≥ 2 we have Z ∞  21  21 Z ∞ |F (x)|2 1 |F (x)|2 dx dx (3.7) ≤ kψkt ≤ C 2 2 2 2 C −∞ x + t −∞ x + t . 11 1 1 Proof. We start with the remark that the functions {(2π)− 2 (n2 + t2 )− 2 en : 1 n ∈ Z} together with ( 12 (t sinh 2πt)− 2 (eit + e−it )) form an orthonormal basis of H 1 for k · kt . Hence 1 X |F (n)|2 |F (it) + F (−it)|2 kψk2t = (3.8) + 4 . 2π n∈Z n2 + t2 t sinh 2πt By (3.6) the last term in (3.8) can be estimated by Z ∞ |F (it) + F (−it)|2 |F (x)|2 2 (3.9) ≤C dx 2 2 t sinh 2πt −∞ t + x for t ≥ 1. Now if −1 ≤ τ ≤ 1 the map Tτ : H 1 → H 1 defined by Tτ f = eτ f satisfies kTτ kt ≤ 2 provided t ≤ 1. Hence if ψτ = Tτ∗ ψ we have 21 kψτ kt ≤ kψkt ≤ 2kψτ kt . However using (3.8) and (3.9) gives: Z ∞ 1 X |F (n + τ )|2 1 X |F (n + τ )|2 |F (x + τ )|2 2 2 ≤ kψ k ≤ + C dx. τ t 2π n∈Z n2 + t2 2π n∈Z n2 + t2 t2 + x2 −∞ Now by integrating for 0 ≤ τ ≤ 1 we obtain (3.7). 4. Application to nonharmonic Fourier series At this point we turn our attention to exponential Riesz bases. Let Λ = (λn )n∈Z be a sequence of complex numbers. For convenience we shall write σn = ℜλn and τn = ℑλn . Let us suppose that (eλn )n∈Z is an unconditional basis of L2 , or equivalently, 1 ((1 + |τn |) 2 e−π|τn | eλn )n∈Z is a Riesz basis of L2 . Then this family is complete interpolating set [25]. In particular, we have sampling condition: there exists a constant D so that if f ∈ L2 then ! 12 X (4.1) ≤ Dkf k D −1 kf k ≤ (1 + |τn |)e−2π|τn | |fˆ(λn )|2 n∈Z (i.e., the latter family is a frame). We also note that it must satisfy a separation condition i.e. for some 0 < δ < 1 we have: |λm − λn | (4.2) ≥δ m 6= n. 1 + |λm − λn | Then we can define an entire function F by Y zk (4.3) F (z) = lim (1 − ). R→∞ λk |λk |≤R λ−1 k z) We replace the term (1 − by z if λk = 0. We call F the generating function for the unconditional basis (eλn ). 12 Proposition 4.1. [23, 14, 7] The product (4.3) converges to an entire function of exponential type π and satisfies the integrability conditions (3.2) and Z ∞ (4.4) |F (x)|2 dx = ∞. −∞ Let us note, that the inequality in (3.2) is necessary for minimality of the family and (4.4) for completeness of (eλn ). Note that since F satisfies (3.1) and (3.2) so that there exists ψ ∈ (H 1 )∗ with ψ̂ = F. We remark that F is a Cartwright class function and then [14] we have the Blaschke condition X |τn | <∞ |λn |2 λ 6=0 n Note that this implies that the families (eλn )ℑλn >0 , (eλP n )ℑλn <0 are minimal in 2 2 L (0, ∞), L (−∞, 0) correspondingly. Also we have λn 6=0 |λn1 |2 < ∞, (this follows from [15] p. 127). Thus, we have a strong Blaschke condition X 1 + |τn | (4.5) < ∞. 2 |λ n| λ 6=0 n Now by the result of Russell, Proposition 1.2, the functions (eλn ) form an unconditional basis of a closed subspace Y0 of H 1 of codimension one. It is clear that the kernel of ψ coincides with Y0 . Hence our above results Proposition 3.2 and Theorem 3.4 apply to this case. Theorem 4.2. Suppose (eλn )n∈Z is an unconditional basis of L2 . Then: (1) (eλn )n∈Z is an unconditional basis of the Sobolev space H s if and only if 0 ≤ s < s0 . (2) (eλn )n∈Z is an unconditional basis of a closed subspace of H s of codimension one if and only if s1 < s ≤ 1. (3) If s0 ≤ s ≤ s1 then (eλn ) is not an unconditional basic sequence. Proof. By Russell’s theorem, Proposition 1.2 above, (eλn )n∈Z is an unconditional basis for a closed subspace Y0 of codimension one which is the kernel sinh(2πτn ) of the linear functional ψ. Let vn be the weight sequence vn = = τn keλn k2L2 and let hn = (1+|λn |2 )vn = keλn k2H 1 . It follows from the basis property that the map V : ℓ2 (h) → Y0 defined by X V (α) = αn eλn n∈Z is an isomorphism (onto). Clearly V is an isomorphism of ℓ2 (v) onto L2 (−π, π) = Y1 . Hence by interpolation V is an isomorphism of ℓ2 (v 1−s hs ) onto Y1−s = 13 [Y0 , L2 ]1−s,2 . In other words, setting qn = vn1−s hsn = vn (1 + |λn |2 )s , we have X X X |αn |2 qn C −1 |αn |2 qn ≤ k αn eλn k2Y1−s ≤ C 1/2 and the almost normalized family (eλn /qn )n∈Z forms a Riesz basis in Y1−s . Thus, if Y1−s is a closed subspace in H s , (eλn ) forms an unconditional basic sequence in H s also. We next estimate keλn kH s to have the inverse implication. In fact from interpolation between L2 and H 1 we have 1 keλn kH s ≤ Ckeλn k1−s keλn ks1 = C(vn1−s hsn ) 2 , where C depends only on s. Similarly if we define φn (f ) = (f, eλn ) then the norm of φn in (H s )∗ can be estimated by 1 2 kφn k(H s )∗ ≤ C1 kφn k1−s kφn ks(H 1 )∗ = C1 (vn1−s )1/2 (vn2 /hn )s/2 = C1 (vn1+s h−s n ) . ¿From the other hand, keλn kH s ≥ |φn (eλn )|/kφn kH s , what gives keλn kH s ≥ C1−1 (vn1−s hsn )1/2 . Therefore the norms keλn kH s and keλn kY1−s are both equiv√ alent to qn . Therefore the assumption that (eλn ) is an unconditional basic sequence leads to equivalence of metrics H s and Y1−s . Remark. It is easy to have necessary and sufficient condition for an exponential family (eλn )n∈Z which is complete and minimal in L2 to be complete and/or minimal in H s [3]. To do this we connect the generating function F with the critical exponent sΛ Z ∞ |F (x)|2 sΛ := inf{s : dx < ∞ = inf{s : ψ ∈ (H s )∗ }. 2s 1 + |x| −∞ Now (eiλn t ) is complete in H s (−π, π) for s < sΛ and is minimal for s > sΛ − 1. The situation for s = sΛ or s = sΛ − 1 depends on whether ψ is bounded in H sΛ . Note that s0 ≤ sΛ ≤ s1 in general. Thus, the family (eλn ) is minimal in H s for 0 < s < 1, and for any (αn ) ∈ l2 , α 6= 0, X X X √ √ 0<k αn eλn / qn k2H s ≤ k αn eλn / qn k2Y1−s ≤ C |αn |2 . We do not know whether (eλn )n∈Z can be a conditional basis of H s , for some appropriate ordering, when s0 ≤ s ≤ sΛ . In order to get precise estimates of s0 and s1 we will need to establish an alternative formula for kψkt in this special case. Let us introduce the function Φ(z) defined by Φ(z) = |F (z)|d(z, Λ)−1 when z ∈ / Λ and Φ(λn ) = |F ′ (λn )| for n ∈ Z. The function Φ plays an important role in the known conditions for (eλn ) to be an unconditional basis (see [20] and [19]). We will call Φ the carrier function for (eλn ). The following lemma lists some useful properties: 14 Lemma 4.3. Suppose −∞ < t < ∞. Then: (i) There is at most one n ∈ Z so that |it−λn | < 21 δ|t| where δ is the separation constant in (4.2). There is also at most one n ∈ Z so that |it − λn | < 41 δ|λn |. (ii) |F (it)| ≤ (|λ0 | + |t|)Φ(it). (iii) There is a constant C independent of t, n so that for every n ∈ Z we have: |F (it)| ≤ C(|λ0 | + |t|) |it − λn | 1 (|λn |2 + t2 ) 2 Φ(it). (iv) We have for t 6= 0, Φ(it) ≤ |t| − 12 π|t| e  Z ∞  21 1 |F (x)|2 dx . π −∞ t2 + x2 Proof. (i) Suppose m, n are distinct and |it − λn |, |it − λm | < |λm − λn | < δt while 1 δ|t|. 2 Then |λm − λn | ≥ |(λm − it) − (λn − it) + 2it| ≥ (2 − δ)|t| > |t|. Hence |λm − λn | <δ 1 + |λm − λn | which contradicts (4.2). For the second part note that if |it − λn | < 41 δ|λn | then |λn | < 2|t| so that |it − λn | < 21 δ|t|. (ii) is immediate from the fact that d(it, Λ) ≤ |λ0 | + t. (iii) If |it − λn | < 21 δt then, in view of (i), |it − λn |Φ(it) = |F (it)| and t2 + |λn |2 ≤ 5t2 . Let |it − λn | ≥ 12 δt. Then |it − λn | |it − λn | |it − λn | ≥ ≥ ≥ c > 0. 2 2 1/2 (|λn | + t ) |λn | + |t| |it − λn | + 2|t| Since |λn | + |t| ≥ d(it, Λ), we have (iii). (iv) Let λn satisfy |it − λn | = d(it, Λ). If t and τn have opposite signs or if τn = 0, then d(it, Λ) ≥ t and so that Φ(it) ≤ t−1 |F (it)|. If they have the same sign define G(z) = (z − λn )(z − λn )−1 F (z) and note that Φ(it) = |G(it)| ≤ t−1 |G(it)|. |it − λn | Since |G(x)| = |F (x)| for x real, we obtain (iv) from (3.6). We next show that the Blaschke condition (4.5) can be improved for Riesz bases: 15 Proposition 4.4. If (eλn )n∈Z is an unconditional basis of L2 then there is a constant C so that for any 0 < t < ∞, X t(1 + |τn |) (4.6) ≤ C. |λn |2 + t2 λn 6=0 Proof. Let us apply (4.1) to e±it . Then X sin(π(λn ± it)) (1 + |τn |)e−2π|τn | (4.7) λn ± it n∈Z 2 ≤ 4D 2 ke±it k2 = 4D 2 sinh 2πt . t Now for each n there is a choice of sign so that: and hence | sinh(π(|τn | + t))| sin(π(λn ± it)) ≥ λn ± it |λn | + t X sinh(π(t + |τn |)2 2 sinh 2πt (1 + |τn |)e−2π|τn | ≤ 4D . 2 2 |λ t n| + t n∈Z This yields P(4.6) for−2t ≥ 1 and this extends to t ≥ 0 in view of (4.5) and the fact that n6=0 |λn | < ∞. We will also need a perturbation lemma: Lemma 4.5. Let (eλn ) and (eµn )n∈Z be two unconditional bases of L2 . Suppose further that there is a constant C so that X t|µn − λn | ≤C 1 < t < ∞. |µn ||λn | + t2 n∈Z Suppose Φ and Ψ are the carrier functions for (eλn ) and (eµn ). Then there exist constants B, T > 0 so that if t ≥ T Y |λn | 1 Ψ(it) Ψ(it) ≤ ≤B . B Φ(it) |µn | Φ(it) 0<|λn |≤t |µn |6=0 1 1 Proof. We observe that for each n we have (taking t = max(1, |µn | 2 |λn | 2 )) 1 Hence 1 |λn − µn | ≤ C max(1, 2|µn | 2 |λn | 2 ). 1 |λn | ≤ |µn | + 2C|λn |1/2 |µn |1/2 + C ≤ |µn | + |λn | + 2C 2 |µn | + C, 2 Along with a similar estimate for |µn | and setting C1 = 2 + 4C 2 > 1 we get: (4.8) |λn | ≤ C1 (|µn | + 1), 16 |µn | ≤ C1 (|λn | + 1). Now let c = 41 min(δ, δ ′ ) where δ, δ ′ are the separation constants of (λn )n∈Z and (µn )n∈Z respectively. We next make the remark that there is a constant M so that if |w|, |z| ≤ 2C1 + 1 and |1 − w|, |1 − z| ≥ c then (4.9) | log |1 − w| − log |1 − z|| ≤ M|w − z|. Let us fix T = |µ0| + |λ0 | + 2C1 . Suppose that t ≥ T, and let p = p(t), q = q(t) ∈ Z be chosen so that |it − λp | = min{|it − λn | : n ∈ Z} and |it − µq | = min{|it − µn | : n ∈ Z}. It may happen that p = q. Note that we have an automatic estimate, (4.10) |it − λp | ≤ |t| + |λ0 | ≤ 2t, |it − µq | ≤ |t| + |µ0| ≤ 2t. Then if n 6= p, q and |λn | > t we have |µn | > 21 C1−1 |λn | and so |it − µn | ≤ |t| + |µn | ≤ (2C1 + 1)|µn |. By Lemma 4.3 (i) we have |it − λn | ≥ c|λn | and |it − µn | ≥ c|µn |. Hence we have by (4.9) | log |it − µn | − log |it − λn | − log |µn | + log |λn || ≤ M t|λn − µn | |λn ||µn | ≤ (2C1 + 1)M t|λn − µn | . |λn ||µn | + t2 Next suppose n 6= p, q and |λn | ≤ t. Then |µn | ≤ C1 (t + 1) ≤ 2C1 t. We also have |it − λn |, |it − µn | ≥ c|t| and so by (4.9) | log |it − µn | − log |it − λn || ≤ M |λn − µn | t ≤ (2C1 + 1)M t|λn − µn | . |λn ||µn | + t2 Combining and summing over all n 6= p, q we have X X it − µp Ψ(it) log |µn | + γ(t) + log |λn | − log = δ(t) log Φ(it) it − λq 0<|λn |≤t 0<|λn |≤t |µn |6=0 where |γ(t)| ≤ C(2C1 + 1)M and δ(t) = 1 if p 6= q and 0 if p = q. To conclude we need only consider the case p 6= q. In this case |it − µp |, |it − λq | ≥ ct. We also have |λp |, |µq | ≤ 3t by (4.10) and so by (4.8) |λq |, |µp | ≤ C1 (3t+1) ≤ 4C1 t. Hence |it−µp |, |it−λq | ≤ 5C1 t. This concludes the proof. Lemma 4.6. Suppose (eλn ) is an unconditional basis of L2 . Then there exist constants B, T so that if t ≥ T then 1 Φ(it) ≤ Φ(−it) ≤ BΦ(it). B Proof. This follows from Lemma 4.5 taking µn = λn in view of Lemma 4.4. 17 The next Theorem is the key step in the proof of our main result: Theorem 4.7. Suppose (eλn )n∈Z is an unconditional basis of L2 . Then there is a constant C and T > 0 so that if t ≥ T then (4.11) 1 1 C −1 t 2 e−πt Φ(it) ≤ kψkt ≤ Ct 2 e−πt Φ(it). Proof. The left-hand inequality in (4.11) is an immediate consequence of Lemma 4.3 (iv) and (3.7). We turn to the right-hand inequality. We first use Lemma 4.3(ii), (iii) and Lemma 4.6. There are constants C, T > 1 so that if |t| ≥ T we have Φ(−it) ≤ CΦ(it), |F (it)| ≤ CtΦ(it) and for every n, |λn − it| |F (it)| ≤ Ct (4.12) 1 Φ(it). (|λn |2 + t2 ) 2 Choose g ∈ H 1 so that ψ(f ) = hf, git for f ∈ H 1 . Let h be the orthogonal projection with respect to h·it of g onto the subspace H01 of all f so that f (−π) = f (π) = 0 and let k = g − h. Then kψk2t = kkk2t + khk2t . The orthogonal complement of H01 (with respect to h·it ) is a 2-dimensional space with orthonormal basis {e±it /ke±it kt }. Hence and k = keit k−2 t (F (it)eit + F (−it)e−it ) 2 2 kkk2t = keit k−2 t (|F (it)| + |F (−it)| ). Since keit k2t = 2t sinh 2πt, we deduce 1 kkkt ≤ C1 t 2 Φ(it)e−πt t≥T and a suitable constant C1 . It remains therefore only to estimate khkt . We first argue that hez , kit = (2t sinh 2πt)−1 (F (it)hez , eit it + F (−it)hez , e−it it ) i = (F (−it) sin π(z − it) − F (it) sin π(z + it)). sinh 2πt Since ψ(eλn ) = F (λn ) = 0 for n ∈ Z we deduce that i heλn , hit = (F (it) sin π(λn + it) − F (−it) sin π(λn − it)). sinh 2πt Now if we use (4.12 we get an estimate valid for t ≥ T : |heλn , hit | ≤ CΦ(it) t|λn + it||λn − it| 1 (|λn |2 + t2 ) 2 sinh 2πt Since h ∈ H01 we then have  sin(π(λn + it)) sin(π(λn − it)) + λn − it λn + it heλn , hit = (λ2n + t2 )(eλn , h) 18  . and we can then rewrite the above estimate as   Φ(it) 1 sin(π(λn − it)) sin(π(λn + it)) |(eλn , h)| ≤ C + . sinh 2πt (|λn |2 + t2 )1/2 λn − it λn + it Now (eλn , th + h′ ) = (t − iλn )(eλn , h). We next use the sampling inequality (4.1): X (1 + |τn |)e−2π|τn | |t − iλn ||(eλn , h)|2 . khk2t = kth + h′ k2L2 ≤ D 2 n∈Z However we can combine with (4.7) to deduce that 1 1 kth + h′ kL2 ≤ 4C 2 D 2 t 2 Φ(it)(sinh 2πt)− 2 for t ≥ T which gives the conclusion. We now consider the case when (λn ) is a small perturbation of the sequence µn = n. For convenience we shall assume that λn = 0 can only occur when n = 0. Theorem 4.8. Suppose (eλn )n∈Z is an unconditional basis of L2 and for some constant C and all t ≥ 1 X t|λn − n| (4.13) < C. 2 + t2 n n6=0 Then s1 = 1 1 + lim sup 2 τ →∞ t≥1 log τ and s0 = 1 1 + lim inf 2 τ →∞ t≥1 log τ X log |n| |λn | X log |n| . |λn | t<|n|≤τ t t<|n|≤τ t Proof. In this case we compare the carrier function Φ for the basis (eλn ) with the carrier function Ψ for the basis (en ). Clearly Ψ(it) = | sin πit|/πt. We can next use Lemma 4.5 to estimate Φ(it) and then the theorem follows directly from Theorem 4.7 together with (3.5) and (3.4). Let us specialize to some important cases. Let δn = ℜλn − n = σn − n. Theorem 4.9. Suppose (eλn )n∈Z is an unconditional basis of L2 such that P 2 −2 sup |δn | < ∞ and n6=0 τn n < ∞. Then X δn 1 1 (4.14) s1 = − lim inf 2 τ →∞ t≥1 log τ n t<|n|≤τ t 19 and (4.15) s0 = 1 1 − lim sup 2 τ →∞ t≥1 log τ X δn . n t<|n|≤τ t Remark. In particular (4.14) and (4.15) hold if |λn − n| is bounded. Proof. Combining Proposition 4.4 and the boundedness of (δn ) gives us (4.13). Note that if n 6= 0, |λn | − |n| |n| = − log(1 + ). log |λn | |n| Now  1 2δn δn2 + τn2 2 |λn | − |n| = 1+ + |n| n n2 δn = + αn n where 1 + τn2 |αn | ≤ C n2 for a suitablePconstant C. By (4.5) and the assumption of the theorem, this implies that n6=0 |αn | < ∞ and yields the Theorem. Before discussing examples we observe one more property of s0 and s1 in this case, which uses recent results of [19] and the theory of A2 −weights. Theorem 4.10. If (eλn )n∈Z is an unconditional basis of L2 then s0 > 0 and s1 < 1. Proof. We will use the connections between Riesz basis property and sampling/interpolation in the spaces of entire functions of exponential type. These connections in the case of L2 and the Paley–Wiener space may be found in [25]. Let L2π,s consisting of all entire functions of exponential type at most π and satisfying Z ∞ |f (ξ)|2 dξ < ∞. 2s −∞ (1 + |ξ|) (Note that the Fourier transform of L2π,s is the set of all distributions from H −s (R) supported on [−π, π].) Now the formal adjoint of the map from ℓ2 (Z) to P 1 H s defined by (αn ) → n∈Z αn (1 + |τn |)− 2 (1 + |λn |)−s eλn is the map from L2π,s
to ℓ2 (Z) given by f → f (λn )(1 + |τn |)1/2 (1 + |λn |)s e−πτn n∈Z . Hence (eλn )n∈Z is an unconditional basic sequence (resp. unconditional basis) if and only if (λn )n∈Z is an interpolating sequence (resp. complete interpolating sequence) in L2π,s . 20 Note that if (λn ) is interpolating for L2π,s−1 then it is interpolating for L2π,s by the simple device of considering functions of the form f (z) = (z − µ)g(z) where µ ∈ / Λ = {λn }n∈Z and g ∈ L2π,s−1 . It follows that our result can be proved by showing that (λn )n∈Z is a complete interpolating sequence for L2π,s for all |s| < ǫ for some ǫ > 0. To do this we use the results of [19] that this is equivalent to requiring that (1 + |ξ|)2s Φ(ξ)2 is an A2 -weight for |s| < ǫ. Now Φ2 is an A2 -weight ([19] or [20]) and so there exists η > 0 so that Φ2(1+η) is an A2 −weight (cf. [6] p. 262, Corollary 6.10). Hence the Hilbert transform is bounded on both L2 (R, Φ2(1+η) ) on L2 (R, (1 + |ξ|)2θ ) for 0 < θ < 12 . It then follows by complex interpolation that Φ(ξ)2 (1 + |ξ|)2s is an A2 −weight when |s| < η(1 + η)−1 . Note that these results now imply Theorem 1.4. Examples. We recall the classical theorem of Kadets, see, e.g., [11] or [15], that if (λn ) are real then supn |δn | < 41 is a sufficient condition for (eλn )n∈Z to be a Riesz basis. First, we consider the case of regular behavior. For example, we can set δn = − 12 q sign n, see [1]. Then we obtain s1 = s0 = 21 + q. More generally if for some y > 0 we have C −1 (1 + |x|)2q ≤ |F (x + iy)| ≤ C(1 + |x|)2q , we obtain s1 = s0 = 12 +q (if we use the integral estimates of kψkt , i.e. Theorem 3.4). One can easily make sequences (δn ) with sup |δn | < 14 to exhibit any required behavior. In fact if we put X δk 1 bn = log 2 n k n+1 2 <|k|≤2 then and n+N X 1 1 s0 = − lim inf bk 2 N →∞ N n≥1 k=n+1 s1 = n+N X 1 1 bk . − lim sup 2 N →∞ N n≥1 k=n+1 To be more specific if − 12 < p ≤ q < 21 , set 2k ) δn = 12 q sign n for 2(2 δn = 21 p sign n for 2(2 Then for 22k < m ≤ 22k+1 2k−1 ) m+1 2k+1 ) < |n| ≤ 2(2 2 q X 1 = q + o(1) bm = log 2 2m +1 k 21 2k ) < |n| ≤ 2(2 and for 22k−1 < m ≤ 22k bm = p + o(1). Thus 1 1 − q, s1 = − p 2 2 (note that an example of irregular behavior is given in [1]) s0 = References [1] S.A. Avdonin, On Riesz bases of exponentials in L2 , Vestnik Leningrad Univ., Ser. Mat., Mekh., Astron.,(1974) (13):5–12, (Russian); English transl. in Vestnik Leningrad Univ. Math., v. 7 (1979), 203-211. [2] S. Avdonin and S.A. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, Cambridge, 1995. [3] S. Avdonin and S. Ivanov. Levin–Golovin theorem for the Sobolev spaces. Matemat. Zametki, 68(2):163–172, 2000. (Russian); English transl. in Math. Notes. [4] S. A. Avdonin, S. A. Ivanov, and D. L. Russell, Exponential bases in Sobolev spaces in control and observation problems for the wave equation, Proceedings of the Royal Society of Edinburgh, to appear. [5] C. Bennett and R. Sharpley, Interpolation of operators, Academic Press, Orlando, 1988. [6] J.B. Garnett, Bounded analytic functions, Academic Press, Orlando, 1981. [7] S.A. Ivanov, Nonharmonic Fourier series in the Sobolev spaces of positive fractional orders, New Zealand Journal of Mathematics, 25 (1996) 36–46, [8] S.A. Ivanov and B.S. Pavlov, Carleson series of resonances in the Regge problem, Izv. Akad. Nauk SSSR, Ser. Mat., 42(1) (1978) 26–55, (Russian); English transl. in Math. USSR Izvestija., v. 12, no. 1 (1978), 21–51. [9] S. Janson, Interpolation of subcouples and quotient couples, Ark. Mat., 31 (1993) 306– 338. [10] S. Kaijser and P. Sunehag, Interpolation of subspaces of codimension one, in preparation. [11] S. V. Khrushchev, N. K. Nikol’skii, and B. S. Pavlov, Unconditional bases of exponentials and reproducing kernels, volume 864 of Lecture Notes in Math.“Complex Analysis and Spectral Theory”. Springer–Verlag, Berlin/Heidelberg, 1981. pp. 214–335. [12] N. Krugliak, L. Maligranda and L.E. Persson,The failure of the Hardy inequailty and interpolation of intersections,Ark. Mat. 37 (1999) 323–344. [13] I. Lasiecka, J.-L. Lions, and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures et Appl., 65(2) (1986) 149–192. [14] B.Ya Levin, Distribution of zeros of entire functions, GITTL, Moscow, 1956, English transl., Amer.Math.Soc., Providence, RI,1964. [15] B. Ya. Levin, Lectures on entire functions, Translations of Mathematical Monographs, Vol. 150, Amer. Math. Soc., Providence 1996. [16] J.-L. Lions and E. Magenes, Problèmes aux limites nonhomogénes et applications, volume I,II. Dunod, Paris, 1968. [17] J. Löfström, Real interpolation with constraints, J. Approx. Theory 82 (1995), 30–53. [18] J. Löfström, Interpolation of subspaces, Preprint, University of Goteborg, 1997 http://www.math.chalmers.se/˜jorgen. [19] Yu. Lyubarskii and K. Seip, Weighted Paley-Wiener spaces, submitted 22 [20] A.M. Minkin, Reflection of exponents, and unconditional bases of exponentials, Algebra i Anal., 3(5) (1992) 110–135. (Russian); English transl. in St. Petersburg Math. J., v. 3, no. 5 (1992), 1043–1068. [21] K. Narukawa and T. Suzuki, Nonharmonic Fourier series and its applications, Appl. Math. Optim., 14 (1986) 249–264. [22] N. K. Nikol’skiı̆, A Treatise on the Shift Operator, Springer, Berlin, 1986. [23] R.E.A.C. Paley and N. Wiener, Fourier Transforms in the Complex Domain, volume XIX, AMS Coll. Publ., New York, 1934. [24] D.L. Russell, On exponential bases for the Sobolev spaces over an interval, J. Math. Anal. Appl., 87(2) (1982) 528–550. [25] K. Seip, On the connection between exponential bases and certain related sequences in L2 (−π, π), J. of Functional Analysis, 130(1) (1995) 131–160. [26] R. Wallsten, Remarks on interpolation of subspaces, Lecture Notes in Math., 1302 (1988) 410–419. Russian Center of Laser Physics, St.Petersburg State University, Ul’yanovskaya 1,198904 St.Petersburg, Russia E-mail address: Sergei.Ivanov@pobox.spbu.ru Department of Mathematics University of Missouri Columbia Mo. 65211 USA E-mail address: nigel@math.missouri.edu 23