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Bounded Rank-one Perturbations in Sampling Theory∗†‡ 7 0 0 Luis O. Silva and Julio H. Toloza 2 n a Departamento de Métodos Matemáticos y Numéricos J 1 Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas 3 Universidad Nacional Autónoma de México ] A C.P. 04510, México D.F. C . [email protected] h t a [email protected] m [ 2 v Abstract 2 9 4 Sampling theory concerns the problem of reconstruction of functions 1 0 from the knowledge of their values at some discrete set of points. In 7 this paper we derive an orthogonal sampling theory and associated 0 Lagrange interpolation formulae from a family of bounded rank-one / h perturbations of a self-adjoint operator that has only discrete spectrum t a of multiplicity one. m : v i X r a ∗Mathematics Subject Classification(2000): 41A05, 46E22,47A55, 47B25, 47N99, 94A20 †Keywords: rank-one perturbations, sampling expansions ‡Research partially supported by CONACYT under Project P42553F. 1 1 Introduction Sampling theory is concerned with the problem of reconstruction of functions, in a point- wise manner, from the knowledge of their values at a prescribed discrete set of points. Resolution of concrete situations in this theory generally involves the characterization of a class (usually a linear set) of functions to be interpolated, the specification of a set of sampling points to be used for all the functions in the given class, and the derivation of an interpolation formula. The cornerstone for many works on sampling theory is the Kramer sampling theorem [10] and its analytic extension [6]. Orthogonal sampling formulae often arise as realizations of this theorem. The celebrated Whittaker-Shannon-Kotel’nikov sampling theorem [9, 17, 22] is also a particular case of the Kramer theorem, although historically the former came first and motivated the latter. Orthogonal sampling formulae have been obtained in connection with differential and difference self-adjoint boundary value problems (see for instance [7, 24] and, of course, the paper due to Kramer himself [10]), and also by resorting to Green’s functions methods [4, 23], among other ODE’s techniques. These results suggest that the spectral theory of operators should provide a unifying approach to sampling theory. Following this idea, a general method for obtaining analytic, orthogonal sampling formulae has been derived in [15] on the basis of the theory of representation of simple symmetric operators due to M. G. Krein [11, 12, 13, 14]. Roughly speaking, the technique given in [15] consists in the following: By [11, 12, 13], every closed simple symmetric operator A in a Hilbert space H generates a bijective isomorphism between H and a space of functions H with certain analytic properties. If the operator A satisfies some additional conditions, all of its self-adjoint extensions have discrete spectrum and every function f in H can be unbiquely reconstructed, as long as one knows the value of f at the spectrum of any self-adjoint extension of A. b Loosely speaking, the self-adjoint extensions of a symmetric operator with deficiency indices (1,1) constitute a family of singular rank-one perturbations of one of these self- adjoint extensions [3, Sec. 1.1–1.3]. Therefore, the methods developed in [15] also holds for singular rank-one perturbations, provided that these operators correspond to a family of self-adjoint extensions of some simple symmetric (hence densely defined) operator. We cannot use, however, a family of bounded rank-one perturbations in applications to sampling theory without making substantial changes to the results of [15]. Krein’s approach to symmetric operators with equal deficiency indices does not work for bounded rank-one perturbations since operators of this kind may only be seen as self-adjoint extensions of a certain not densely defined, Hermitian operator [3, Sec. 1.1]. The main motivation of the present work is to develop a method in sampling theory analogous to [15] for the case of bounded rank-one perturbations. With this purpose in mind, we begin by constructing a representation space for bounded rank-one perturbations in some sense similar to that of Krein for simple symmetric operators. Elements of this representation space are the functions to be interpolated. Then we obtain a general 2 Kramer-type analytic sampling formula which turns out to be a Lagrange interpolation formula. We also characterize the space of interpolated functions as a space of meromorphic functions with some properties resembling those of a de Brange space. Examples are discussed in the last part of this work. 2 Preliminaries Thefollowingreviewisbasedonthestandardtreatmentconcerning rank-oneperturbations ofself-adjointoperators,asdiscussed indetailbyGesztesy andSimon[8,18], andAlbeverio and Kurasov [3]. In a separable Hilbert space H, we consider a possibly unbounded, self-adjoint operator A with discrete spectrum of multiplicity one. Let µ be a cyclic vector for A, that is, {(A−zI) 1µ : z ∈ C} is a total set in H. Throughout this work we assume that kµk = 1. − Given µ, let us define the family of bounded rank-one perturbations of A, A := A+hhµ,·iµ, h ∈ R, (2.1) h where the inner product is taken, from now on, anti-linear in its first argument. Naturally, Dom(A ) = Dom(A) for any h ∈ R. Elementary perturbation theory implies that all the h elements of (2.1) have discrete spectrum. As pointed out in [3, Sec. 1.1], the operators A may be seen as self-adjoint extensions of some Hermitian operator, in the sense of [21], h with non-dense domain. Consider the family of functions F (z) := hµ,(A −zI) 1µi, z 6∈ Sp(A ), h ∈ R. (2.2) h h − h InthesequelweshalldenoteF byF. Bythespectraltheorem, F (z)istheBoreltransform 0 h of the spectral function m (t) = hµ,E (t)µi, where E (t) is the spectral resolution of the h h h identity corresponding to A . Hence, F (z) is a Herglotz meromorphic function having h h simple poles at the eigenvalues of A . h From the second resolvent identity [21, Thm. 5.13] one obtains (A −zI) 1 = (A−zI) 1 −hh(A −zI) 1µ,·i(A−zI) 1, h ∈ R. h − − h − − This equation yields the well-known Aronzajn-Krein formula [18, Eq. 1.3] F(z) F (z) = , h ∈ R. (2.3) h 1+hF(z) The spectral properties of the whole family (2.1) are contained in (2.3). Indeed, one can easily show that the function Fh/Fh′, h 6= h′, is also Herglotz and its zeros and poles are given by the poles of Fh′ and Fh respectively [16]. Thus, the spectra of any two different elements of the family (2.1) interlace, i.e., between two neighboring eigenvalues of one 3 operator there is one and only one eigenvalue of any other. Also, (2.3) implies that x ∈ R 0 is a pole of F if and only if h 1 +h = 0. (2.4) F(x ) 0 Therefore for any x ∈ R which is not a zero of F, there exists a unique h ∈ R such that x is an eigenvalue of A . One can extend this result to every x ∈ R by considering an h infinite coupling constant h = ∞ in (2.1) (see [18, Sec 1.5], [3, Sec. 1.1.2]). From the properties of F(z), it is shown that A also have simple discrete spectrum (see footnote in [1, p. 55]) and Sp(A ) = {x ∈ R : F(∞x) = 0}. Thus, for any x ∈ R, there exists a unique h ∈ R∪{∞} such th∞at x is an eigenvalue of A . h The main peculiarity of A , that separates it from the family A with finite h, is its h ∞ domain. Indeed, for the kind of perturbations considered here, the domain of A is the ∞ set {ϕ ∈ Dom(A) : hϕ,µi = 0} [3, Sec. 1.1.1], [18, Thm. 1.15]. 3 Sampling theory Based on the theory of rank-one perturbations, we construct in this section a linear space of meromorphic functions H and derive an interpolation formula valid for all the elements µ in H . µ In our considerations beblow, the following vector-valued function of complex argument willbplay an important rôle. (A−zI) 1µ − ξ(z) := , z 6∈ Sp(A ). F(z) ∞ Notice that ξ(z) is well defined for z ∈ Sp(A). Lemma 1. For any x ∈ R\Sp(A ), there exists (a unique) h ∈ R such that ∞ ξ(x) ∈ Ker(A −xI). (3.1) h Similarly, (A−xI) 1µ ∈ Ker(A −xI) (3.2) − ∞ for every x ∈ Sp(A ). ∞ Proof. We first consider x 6∈ Sp(A) ∪ Sp(A ). Let h 6= 0 be such that x ∈ Sp(A ) (we h ∞ already know that there is always such h). We have 1 1 x A ξ(x) = A (A−xI) 1µ = +h µ+ (A−xI) 1µ. h h − − F(x) F(x) F(x) (cid:18) (cid:19) The first assertion of the lemma follows from the last expression and (2.4). When x ∈ Sp(A), the statement follows by a limiting argument based on the fact that A is closed. 4 We now prove the last assertion of the lemma. Define P := hµ,·iµ. By virtue of [18, Thm. 1.18, Rem. 2], there is a cyclic vector η that obeys 1 (A −zI) 1η = (I −P)(A−z) 1µ, z 6∈ Sp(A)∪Sp(A ). (3.3) − − ∞ F(z) ∞ Clearly, η ∈ Dom(A ). We compute the projection of η along the eigenspace associated ∞ to x ∈ Sp(A ). Using (3.3) and some ǫ > 0 sufficiently small, we obtain ∞ 1 [E (x+0)−E (x−0)]η = (A −z) 1ηdz − ∞ ∞ 2πi ∞ Zx z=ǫ | − | 1 = (A−zI) 1 −I µdz − F(z) Zx z=ǫ(cid:20) (cid:21) | − | 1 = Res (A−xI) 1µ. − z=x F(z) (cid:18) (cid:19) Since the last expression is different from zero, (3.2) is proven. Definition 1. For any ϕ ∈ H, let Φ be the mapping given by µ (Φ ϕ)(z) := hξ(z),ϕi, z ∈ C\Sp(A ). µ ∞ We sometimes shall denote Φ ϕ by ϕ. µ The mapping Φ is a linear injective operator from H onto a certain space of meromor- µ b phic functions H := Φ H. The injectivity may be verified with the aid of (3.1). Some µ µ properties that characterize the set H will be accounted for in the next section. µ b Proposition 1. Given some fixed h ∈ R, let {x } = Sp(A ). Define G (z) := 1/F (z) = b j j h h h h+1/F(z). Then, for every f(z) ∈ H , we have µ G (z) f(z) = bh f(x ), z ∈ C\Sp(A ). (3.4) j (z −xj)G′h(xj) ∞ xj∈XSp(Ah) The series is uniformly convergent on every compact subset of the domain. Proof. Because of the assertion (3.1) of Lemma 1, {ξ(x )} is a complete orthogonal set in j j H. Hence hξ(z),ξ(x )i j ϕ(z) = hξ(z),ϕi = ϕ(x ), (3.5) kξ(x )k2 j j xj∈XSp(Ah) where the series convebrges uniformly on compacts of C\Sp(Ab) by virtue of the Cauchy- ∞ Schwarz inequality. 5 Now, the first resolvent identity implies 1 1 hξ(z),ξ(w)i = (z −w) 1 − . − F(w) F(z) (cid:20) (cid:21) In conjunction with (2.4) and the convention 1/∞ = 0 when h = 0, the last equation gives rise to the identity 1 hξ(z),ξ(x )i = −(z −x ) 1 h+ . j j − F(z) (cid:20) (cid:21) Finally, notice that 1 1 G (w) = − F (w) = − µ,(A−w)2µ = −hξ(w),ξ(w)i. ′h F2(w) ′ F2(w) (cid:10) (cid:11) Evaluation of the last expression at w = x yields the desired result. j Remark 1. Equation (3.5) is an orthogonal sampling formula of Kramer-type [10]. Since the function G (z) has simple zeroes at the points of Sp(A ), expression (3.4) is indeed a h h Lagrange interpolation formula. 4 Spaces of interpolated functions The present section is devoted to the characterization of the space of functions H intro- µ duced by means of the mapping Φ of Definition 1. Notice that H depends on both the µ µ operator A and the cyclic vector µ. b The following statement gives a quite explicit description of Hb. µ Proposition 2. b c H = f(z) = c+ n : c,c ∈ C, |c |2F (x ) < ∞ , µ  z −x n n ′ n  n  xn∈XSp(A∞) xn∈XSp(A∞)  b where the series above converge uniformly on compact subsets of C\Sp(A ).  ∞ Proof. Let G denotes the set defined by the right-hand side of the statement. Given x ∈ Sp(A ), it follows from (3.2) of Lemma 1 that ω(x ) := (A−x I) 1µ is n n n − ∞ the associated eigenvector (up to normalization). Taking into account the first resolvent identity, we get \ 1 1 F(z) 1 ω(x )(z) = h(A−z) 1µ,(A−x ) 1µi = +F(x ) = . n − n − n F(z) F(z) z −x z −x (cid:20) n (cid:21) n 6 Theexpression afterthefirst equality abovealso implies thatkω(x )k2 = F (x ). Recalling n ′ n the definition of Dom(A ), it follows that the set ∞ B := {µ}∪ kω(xn)k−1ω(xn) xn Sp(A∞) ∈ (cid:8) (cid:9) is an orthonormal basis in H. Now, take an arbitrary element ϕ of H and expand it on the basis B. By applying Φ to ϕ and using the Cauchy-Schwarz inequality, we conclude µ that H ⊂ G. µ The inclusion G ⊂ H follows from noticing that any f(z) ∈ G is the image under Φ µ µ of anbelement in H of the form cµ+ c ω(x ). xn Sp(A∞) n n b ∈ P Notice that, by virtue of Proposition 2, the only entire functions in H are the constant µ functions. Also, one easily verifies that any constant function in H is the image under Φ µ µ of a vector in Span{µ}. b The following straightforward result shows that the functionsbin H share some prop- µ erties with those in a de Branges space. b Lemma 2. The space H has the following properties: µ (i). Assume that f(z) ∈ H has a non-real zero w. Then g(z) := z wf(z) also belongs to b µ z−w H . − µ b (ii). The evaluation functional f(·) 7→ f(z) is continuous for every z ∈ C\Sp(A ). ∞ (iii). For every f(z) ∈ H , g(z) := f(z) belongs to H . µ µ Proof. We have f(z) = hξ(z),ϕi for some ϕ ∈ H. Given w such that f(w) = 0, consider b η = (A−wI)(A−wI) 1ϕ. A short computation yields g(z) = hξ(z),ηi, thus showing (i). − Assertion (ii) is rather obvious so the proof is omitted. On the basis of Proposition 2 one verifies (iii). In what follows we show that H can be endowed with several Hilbert space structures, µ each one determined by the spectral functions m (x), h ∈ R. h b Lemma 3. Let h ∈ R and {x } = Sp(A ), arranged in non-decreasing order. Then the j j h spectral function m (x) is given by h mh(x) = kξ(xj)k−2 . xXj≤x Proof. Let us recall first the following well-known results [18, Thm. 1.6] limǫReF (x+iǫ) = 0, h ǫ 0 → 7 ǫ2 dm (y) h limǫImF (x+iǫ) = lim = m ({x}), ǫ 0 h ǫ 0 R (y −x)2 +ǫ2 h → → Z and also the identity 1 (A −zI) 1µ = (A−zI) 1µ. h − − 1+hF(z) Consider xj ∈ Sp(Ah). It suffices to verify that mh({xj}) = kξ(xj)k−2. By resorting to the equalities mentioned above, a straightforward computation shows that 1 hξ(x −iǫ),ξ(x −iǫ)i = (A−(x +iǫ)I) 1µ,(A−(x +iǫ)I) 1µ j j |F(x +iǫ)|2 j − j − j 1 (cid:10) (cid:11) = (A −(x +iǫ)I) 1µ,(A −(x +iǫ)I) 1µ |F (x +iǫ)|2 h j − h j − h j (cid:10) 1 ǫ2 dm (y) (cid:11) h = [ǫReF (x +iǫ)]2 +[ǫImF (x +iǫ)]2 (y −x )2 +ǫ2 h j h j R j Z 1 → , ǫ → 0. m ({x }) h j The proof is now complete. Remark 2. For h 6= 0, this result is in fact statement (ii) of [18, Thm. 2.2] in disguise. Proposition 3. For arbitrary h ∈ R, the map Φ is a unitary transformation from H onto µ L2(R,dm ). h Proof. Φ is a linear isometry from H into L2(R,dm ). Indeed, µ h ϕ(·),ψ(·) := hϕ,ξ(x)ihξ(x),ψidm (x) h h R D E Z hϕ,ξ(x )ihξ(x ),ψi b b = j j = hϕ,ψi. kξ(x )k2 j xj∈XSp(Ah) Now consider f(x) ∈ L2(R,dm ). This means that h |f(x )|2 kf(·)k2 = j < ∞. h kξ(x )k2 j xj∈XSp(Ah) Define f(x ) j η = ξ(x ), kξ(x )k2 j j xj∈XSp(Ah) 8 which is clearly an element in H. It is not difficult to verify that kf(·)−η(·)k = 0. h b Define C := Φ 1Cφ , where (Cf)(z) = f(z) for f ∈ H . By (iii) of Lemma 2 and −µ µ µ Proposition 3, it follows that C is a complex conjugation with respect to which both A and µ are real. b b b We conclude this section with a comment about the representation of the operators A h as operators on H . A simple computation shows that, for every h ∈ R, A is transformed µ h by Φ into a quasi-multiplication operator, in the sense that µ b 1 A ϕ(z) = hµ,ϕi+zϕ(z) (4.1) h F (z) h for every ϕ ∈ Dom(A). Thisdis obviously the multiplibcation operator in Φ Dom(A ). µ ∞ Moreover, (4.1) reduces to the multiplication operator in a weak sense; indeed, ϕ(·),A ψ(·) = ϕ(·),(·)ψ(·) , h h h D E D E for every ϕ,ψ ∈ Dom(A). b d b b 5 Examples Rank-one perturbations of a Jacobi matrix. Consider the following semi-infinite Jacobi matrix q b 0 0 ··· 1 1 b q b 0 ···  1 2 2  0 b2 q3 b3 (5.1) 0 0 b q ...  3 4   ... ... ... ...     with q ∈ R and b > 0 for n ∈ N, and define in the Hilbert space l2(N) the operator J n n in such a way that its matrix representation with respect to the canonical basis {δ } n ∞n=1 in l2(N) is (5.1). By this definition, J (cf. [2, Sec.47]) is the minimal closed symmetric operator satisfying hδ ,Jδ i = q , hδ ,Jδ i = hδ ,Jδ i = b , ∀n ∈ N. n n n n+1 n n n+1 n The Jacobi operator J may have deficiency indices (1,1) or (0,0) [1, Chap.4 Sec.1.2], [19, Cor.2.9]. For this example we consider J to be self-adjoint, i.e., the case of deficiency indices (0,0). We also assume that J has only discrete spectrum. Our family of self-adjoint operators is given by J := J +hhδ ,·iδ , h ∈ R. (5.2) h 1 1 9 It is relevant to note that δ is a cyclic vector for J since the matrix elements b are always 1 n assumed to be different from zero. One can study J through the following second order difference system b f +q f +b f = zf n > 1, z ∈ C. (5.3) n 1 n 1 n n n n+1 n − − with boundary condition q f +b f = zf . (5.4) 1 1 1 2 1 If one sets f = 1, then f is completely determined by (5.4). Having f and f , the 1 2 1 2 equation (5.3) gives all the other elements of a sequence {f } that formally satisfies n ∞n=1 (5.3) and (5.4). f is a polynomial of z of degree n − 1, so we denote f =: P (z). n n n 1 − The polynomials P (z), n = 0,1,2,... are referred to as the polynomials of the first kind n associated with thematrix (5.1). The polynomials ofthe second kind Q (z), n = 0,1,2,... n associated with (5.1) are defined as the solutions of b f +q f +b f = zf n ∈ N\{1} n 1 n 1 n n n n+1 n − − under the assumption that f = 0 and f = b 1. Then 1 2 −1 Q (z) := f , ∀n ∈ N. n 1 n − Q (z) is a polynomial of degree n−1. n Let P(z) = {P (z)} and Q(z) = {Q (z)} . Then, classical results in the theory n ∞n=0 n ∞n=0 of Jacobi matrices [1] give us the following expression for ξ(z) defined in Section 3: 1 ξ(z) = P(z)+ Q(z), F(z) where F(z) is the function given by (2.2) with h = 0. In this context F(z) is referred to as the Weyl function of J and may be determined by 1 P (z) n F(z) = − lim , w (z) := , (5.5) n n→∞ wn(z) Qn(z) where the convergence is uniform on any compact subset of C\Sp(J) [1, Secs.2.4,4.2]. The operator J corresponds in this case to the operator in l2(2,∞) whose matrix ∞ representation is (5.1) with the first column and row removed. For any f ∈ H there is a sequence {ϕ } ∈ l2(N) such that δ1 k ∞k=1 b ∞ ϕk f(z) = P (z)ϕ + Q (z) . k 1 k k 1 − F(z) − k=1(cid:18) (cid:19) X Notice that the poles of f are the eigenvalues of J . ∞ 10