THE SOBOLEV ORTHOGONALITY AND SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS {L k

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1 THE SOBOLEV ORTHOGONALITY AND SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS {L k n } FOR POSITIVE INTEGERS k W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN Abstract. For k N, we consider the analysis of the classical Laguerre differential expression 1 ³ ³ k [y](x) = x k+1 e x y (x) + rx k e x y(x) (x (, )), x k e x where r is fixed, in several non-isomorphic Hilbert and Hilbert-Sobolev spaces. In one of these spaces, specifically the Hilbert space L ((, ); x k e x ), it is well known that the Glazman-Krein-Naimark theory produces a self-adjoint operator A k, generated by k [ ], that is bounded below by ri, wherei is the identity operator on L ((, ); x k e x ). Consequently, as a result of a general theory developed by Littlejohn and Wellman, there is a continuum of left-definite Hilbert spaces {H s, k =(V s, k, (, ) s, k )} s> and left-definite self-adjoint operators {B s, k } s> associated with the pair (L ((, ); x k e x ),A k ). For A k and each of the operators B s, k, it is the case that the tail-end sequence {L k n } n=k of Laguerre polynomials form a complete set of eigenfunctions in the corresponding Hilbert spaces. In 1995, Kwon and Littlejohn introduced a Hilbert-Sobolev space W k [, ) in which the entire sequence of Laguerre polynomials is orthonormal. In this paper, we construct a self-adjoint operator in this space, generated by the second-order Laguerre differential expression k [ ], having {L k n } n= as a complete set of eigenfunctions. The key to this construction is in identifying a certain closed subspace of W k [, ) with the k th left-definite vector space V k, k. Contents 1. Introduction. Preliminaries: Properties of the Laguerre polynomials 4 3. Right-definite analysis of the Laguerre differential expression 6 4. General left-definite theory 7 5. Left-definite analysis of the Laguerre differential expression 9 6. Sobolev orthogonality of the Laguerre Polynomials The completeness of the Laguerre polynomials {L k n } n= in W k[, ) 1 8. A fundamental decomposition and identification of two inner product spaces 1 9. The self-adjoint Laguerre operator T k, The self-adjoint Laguerre operator T k, The self-adjointness of the Laguerre operator T k in W k [, ) 9 References 3 Date: July 9, 3 (Revised: October 13, 3) Mathematics Subject Classification. Primary 33C45, 34B4; Secondary 4C5, 34B4. Key words and phrases. Laguerre polynomials, orthogonal polynomials, left-definite theory, self-adjoint operators, Sobolev spaces. The authors L. L. L. and R. W. are very pleased to dedicate this paper to their co-author W. N. Everitt as a small means of large thanks for his many years of teaching, mentoring and guiding us as well as for his constant encouragement. 1

2 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN 1. Introduction For α> 1, the analysis of the classical Laguerre differential expression α [y](x) := xy (x)+(x 1 α)y (x)+ry(x) = 1 (x α+1 x α e x e x y (x)) + rx α e x y(x) (x>; r fixed) is well understood and documented from the viewpoints of differential equations, special functions, and spectral theory. Indeed, the n th Laguerre polynomial y = L α n(x) is a solution of α [y](x) =(n + r)y(x) (n N ) and classical properties of these polynomials are numerous and well-known (see [], [15], and [16]). The right-definite operator-theoretic properties and spectral analysis of this Lagrangian symmetrizable expression α [ ], whenα> 1, are also detailed in the literature (see [1], [5], [13], and [14]). More specifically, as an application of the classical Glazman-Krein-Naimark (GKN) theory, there is a self-adjoint operator A α, generated from this Laguerre differential expression, in the weighted Hilbert space L ((, ); x α e x ) having the Laguerre polynomials {L α n} n= as a complete set of eigenfunctions. However, the functional analytic theory of this expression, specifically when α := k N, is less clear and is the principle focus of this paper. One of the main differences in this case, compared to the classical situation of α> 1, stems from the fact that the Laguerre polynomials of degree <k do not belong to the Hilbert space L ((, ); x k e x ). However, the tail-end sequence of Laguerre polynomials {L k n } n=k still form a complete orthogonal set in L ((, ); x k e x ) (see Section ). In this space, we construct, again with the aid of the GKN theory, a self-adjoint operator A k, bounded below by ri in L ((, ); x k e x ), having these Laguerre polynomials {L α n} n=k as eigenfunctions. Consequently, a general left-definite theory, recently developed by Littlejohn and Wellman [11], may be applied to this operator to assert the existence of a continuum of (left-definite) Hilbert- Sobolev spaces {H s, k =(V s, k, (, ) s, k )} s> and (left-definite) self-adjoint operators {B s, k } s>, generated from A k. We explicitly construct these function spaces V s, k, their associated inner products (, ) s, k, and operators B s, k for all s N. Furthermore, we show that the Laguerre polynomials {L k n } n=k form a complete set of eigenfunctions for each of these operators. The main portion of this paper, however, is to study the entire sequence of Laguerre polynomials {L k n } n= in a Sobolev space W k[, ) that was recently discovered by Kwon and Littlejohn [8] (see also [9]). We show that these Laguerre polynomials form a complete orthonormal set in W k [, ). Furthermore, we construct a self-adjoint operator T k in W k [, ) having these polynomials as eigenfunctions. Interestingly, the key to the construction of T k is in identifying a certain subspace of W k [, ) with the k th left-definite function space V k, k and the k th left-definite operator B k, k associated with the pair (L ((, ); x k e x ),A k ). This paper is an extension of results given in [5] where the authors develop the spectral properties of the Laguerre differential expression k [ ] in various Hilbert and Hilbert-Sobolev spaces but only for the cases k =1and k =. At the time of publication of [5], the general left-definite theory, developed in [11], which is instrumental in the results of this paper, was not fully developed. Consequently, the analytic methods used in [5] were not readily applicable to the general case of k being an arbitrary positive integer. The contents of this paper are as follows. In Section, we review some important properties of the Laguerre polynomials, including a remarkable identity when the parameter α is a negative integer. Section 3 summarizes various properties of the Laguerre differential expression in the

3 SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 3 right-definite setting L ((, ); x α e x ) for α> 1and α = k N. A review of the general left-definite theory is given in Section 4 and this theory is applied to the Laguerre expression in Section 5. We remark that in [11], the authors give a detailed left-definite analysis of the Laguerre expression when α > 1; some care must be exercised in extending these results to thecasewhen α = k N but the results in this case are very similar and details will be omitted in this paper. In Section 5, we also establish some important properties of functions in the k th left-definite space H k, k =(V k, k, (, ) k, k ); a key to developing these properties is an important integral inequality established by Chisholm and Everitt [3]. Section 6 reviews the Kwon- Littlejohn discovery of a Sobolev space W k [, ) where the entire sequence of Laguerre polynomials {L k n } n= is orthonormal. The completeness of {L k n } n= in W k[, ) is established in Section 7. A fundamental decomposition of W k [, ) is developed in Section 8; this decomposition is both important and necessary in the construction of three self-adjoint operators generated by k [ ]. Sections 9 and 1 are concerned with explicitly constructing two of these self-adjoint operators T k,1 and T k, in certain closed subspaces W k,1 [, ) =(S k,1 [, ), (, ) k ) and its orthogonal complement W k, [, ), respectively, of W k [, ). The fundamental decomposition, obtained in Section 8, as well as the important equality S k,1 [, ) =V k, k, which we establish in Section 5, plays a key role in the construction of T k,1. Lastly, in Section 11, the self-adjoint operator T k = T k,1 T k,, generated by the Laguerre differential expression k [ ], is constructed and various properties of this operator are developed, including the fact that the Laguerre polynomials {L k n } n= are a (complete) set of eigenfunctions of T k. Notation: In this paper, we fix k N. For α R, let L α(, ) :=L ((, ); x α e x ) (we use both notations in this paper) denote the Hilbert function space defined by (1.1) L α(, ) :={f :(, ) C f is Lebesgue measurable and with inner product and norm, respectively, given by (f,g) L (α) := f(x)g(x)x α e x dx and kfk L (α) =(f,f)1/ L (α) f(x) x α e x dx < } (f,g L α(, )). Occasionally, we shall refer to the Hilbert space L (I) in this paper; this is the usual Lebesgue square integrable space consisting of all complex-valued (Lebesgue) measurable functions that are (Lebesgue) square integrable on the real interval I. Let P[, ) denote the vector space of all complex-valued polynomials p :[, ) C of the real variable x. The set N will denote the positive integers, N = N {}, while R and C will denote, respectively, the real and complex number fields. The term AC will denote absolute continuity; for an open interval I R, the notation AC loc (I) will denote those functions f : I C that are absolutely continuous on all compact subintervals of I. If A is a linear operator, D(A) will denote its domain. The identity operator will be denoted by I and will be used in several Hilbert spaces in this paper. Lastly, a word is in order regarding displayed, bracketed information. For example, f(t) has property P (t I), and g m has property Q (m N ) mean, respectively, that f(t) has property P for all t I and g m has property Q for all m N. Further notations are introduced as needed throughout the paper.

4 4 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN. Preliminaries: Properties of the Laguerre polynomials For any α R, the Laguerre polynomials {L α n} n= are defined by nx µ n + α x L α n(x) = ( 1) j j (n N ); n j j! j= observe that L α n(x) is a polynomial of degree exactly n for any choice of α R. Moreover, in this case, y = L α n(x) is a solution of the differential equation α [y](x) =(n + r)y(x) (n N ), where (.1) α [y](x) := xy +(x 1 α)y (x)+ry(x) = 1 (x α+1 x α e x e x y (x)) + rx α e x y(x). The parameter r in (.1), which can be viewed as a spectral shift parameter, is a fixed non-negative constant and is usually presented in the literature as zero. However we can assume, without loss of generality, that r>; as we will see this assumption is critical for many of the results in this paper. When α> 1, the Laguerre polynomials {L α n} n= form a complete orthogonal set in the Hilbert space L α(, ) (see (1.1)), with inner product (.) (f,g) L (α) := f(x)g(x)x α e x dx (f,g L α(, )), and norm (.3) kfk L (α) := (f,f)1/ L (α) (f L α(, )); see [16, Chapter V] for an in-depth discussion of these orthogonal polynomials. In fact, in this case, we have (.4) (L α n,l α Γ(n + α +1) m) L (α) = δ n,m n! (n, m N ). When α< 1, α / N, the Laguerre polynomials {L α n} n= are also orthogonal on the real line but, in this case, with respect to a signed measure; this is a consequence of Favard s theorem [, Theorem 6.4, p. 75]. Inthecasewhere α := k N, the Laguerre polynomials {L k n } n= cannot be orthogonal with respect to any Lebesgue-Stieltjes bilinear form of the type Z (.5) f(x)g(x)dµ, R where µ is a (signed) Borel measure. Indeed, the three-term recurrence relation for the Laguerre polynomials {L α n} n= is L α 1(x) =; L α (x) =1 (n +1)L α n+1(x)+(x α n 1)L α n(x)+(n + α)l α n 1(x) = (n N ); observe that the coefficient of L α n 1 (x) in this recurrence relation vanishes when α = n. Consequently, Favard s theorem says that the full sequence of Laguerre polynomials {L α n} n=,whenα is a negative integer, cannot be orthogonal on the real line with respect to a bilinear form of the type

5 SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 5 (.5). However, as we will demonstrate shortly, the tail-end sequence {L k n the Hilbert space L k (, ) with inner product (.6) (f,g) L ( k) := f(x)g(x)x k e x dx and norm (.7) kfk L ( k) =(f,f)1/ L ( k) } k 1 n= / L k (, ). } n=k (f,g L k (, )), (f L k (, )); moreover, {L k n One of the more remarkable properties of the Laguerre polynomials {L k the formula (see [16, p. 1]): n is orthogonal in } n=k,whenk N, is (.8) L k k (n k)! n (x) =( 1) x k L k n k n! (x) (k N; n k). This formula (.8) plays a key role throughout this paper in our analysis of the second-order Laguerre differential expression µ 1 ³ (.9) k [y](x) := x k e x x k+1 e x y (x) + rx k e x y(x) = xy (x)+(x 1+k)y (x)+ry(x) (x>). We now establish the following result. Theorem.1. The Laguerre polynomials {L k n } n=k form a complete orthogonal set in the space L k (, ). Equivalently, the set P k[, ) of all polynomials p of degree at least k satisfying is dense in L k (, ). Proof. Observe that p() = p () =...= p (k 1) () = f(x) x k e x dx = so that x (.1) kfk L ( k) = k f, L (k) x k f(x) x k e x dx, where k k L (k) and k k L ( k) are the norms defined in (.3) and (.7), respectively. Hence f L k (, ) ifandonlyifx k f L k (, ). Let f L k (, ) and let ε>. Hence x k f L k (, ). Since the space P[, ) of polynomials is dense in L k (, ) (see [16, Theorem 5.7.]), there exists q P[, ) such that (.11) x k f q <ε. L (k) Let p(x) =x k q(x) so p P k (, ); by(.1), we see that x kf pk L ( k) = k (f p) = x k f q <εby (.11). L (k) L (k)

6 6 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN This completes the proof of the theorem. 3. Right-definite analysis of the Laguerre differential expression When α> 1, the Laguerre polynomials {L α n} n= form a complete set of eigenfunctions of the self-adjoint operator A α : D(A α ) L α(, ) L α(, ) defined by (3.1) ½ Aα f := α [f] f D(A α ), where α [ ] is the Laguerre differential expression, defined in (.1). Here, the domain D(A α ) of A α is given by ½ α if α 1 (3.) D(A α ):= {f α lim x + x α+1 f (x) =} if 1 <α<1, where α is the maximal domain in L α(, ), defined by α := {f :(, ) C f,f AC loc (, ); f, α [f] L α(, )}. Moreover, A α is bounded below by ri in L α(, ); thatis, (A α f,f) L (α) r(f,f) L (α) (f D(A α )), where (, ) L (α) is the inner product defined in (.). Furthermore, the spectrum of A α is discrete and given by σ(a α )={m + r m N }. We recommend the sources [11, Section 1], [14], and [19] for explicit and further details concerning both analytic and algebraic properties of the operator A α. In the case α< 1 and α / N, we recommend the contribution [7], where a spectral analysis of the Laguerre expression (.1) is carried out in a Krein space setting. Turningtothecaseα = k, wherek N, the authors in [5] show that the differential operator A k : D(A k ) L k (, ) L k (, ) defined by ½ A (3.3) k := k [f] f D(A k ):={f :(, ) C f,f AC loc (, ); f, k [f] L k (, )}, where k [ ] is the Laguerre differential expression defined in (.9), is self-adjoint and bounded below by ri in L k (, ); that is, (3.4) (A k f,f) L ( k) r(f,f) L ( k) (f D(A k )), where (, ) L ( k) is the inner product definedin(.6). NotethatD(A k ) is, in fact, the maximal domain k of k [ ] in L k (, ). This is a consequence of the expression k[ ] being strong limit-point and Dirichlet at both x =and x = ; see [5, Theorem.]. We note the spectrum of A k is discrete and given by σ(a k )={m + r m k}. Moreover, the Laguerre polynomials {L k n } n=k form a complete orthogonal set of eigenfunctions of A k in L k (, ); further details can be found in [5, Theorem.].

7 SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 7 4. General left-definite theory Let V denote a vector space (over the complex field C) and suppose that (, ) is an inner product with norm k k, generated from (, ), such that H =(V,(, )) is a Hilbert space. Suppose V r (the subscripts will be made clear shortly) is a linear manifold (subspace) of the vector space V and let (, ) r and k k r denote an inner product and associated norm, respectively, over V r (quite possibly different from (, ) and k k). We denote the resulting inner product space by W r =(V r, (, ) r ). Throughout this section, we assume that A : D(A) H H is a self-adjoint operator that is bounded below by ri, for some r>; that is, (Ax, x) r(x, x) (x D(A)). It follows that A s, for each s>, is a self-adjoint operator that is bounded below in H by r s I. We now define an s th left-definite space associated with (H, A). Definition 4.1. Let s> and suppose V s is a linear manifold of the Hilbert space H =(V,(, )) and (, ) s is an inner product on V s V s. Let W s =(V s, (, ) s ). We say that W s is an s th leftdefinite space associated with the pair (H, A) if each of the following conditions hold: (1) W s is a Hilbert space, () D(A s ) is a linear manifold of V s, (3) D(A s ) is dense in W s, (4) (x, x) s r s (x, x) (x V s ), and (5) (x, y) s =(A s x, y) (x D(A s ),y V s ). It is not clear, from the definition, if such a self-adjoint operator A generates a left-definite space for a given s>. However, in [11], the authors prove the following theorem; the Hilbert space spectral theorem plays a prominent role in establishing this result. Theorem 4.1. (see [11, Theorem 3.1]) Suppose A : D(A) H H is a self-adjoint operator that is bounded below by ri, for some r>. Let s>. Define W s =(V s, (, ) s ) by (4.1) V s = D(A s/ ), and (4.) (x, y) s =(A s/ x, A s/ y) (x, y V s ). Then W s is a left-definite space associated with the pair (H, A). Moreover, suppose Ws := (Vs, (, ) s) is another s th left-definite space associated with the pair (H, A). Then V s = Vs and (x, y) s =(x, y) s for all x, y V s = Vs; i.e. W s = Ws. That is to say, W s =(V s, (, ) s ) is the unique left-definite space associated with (H, A). Remark 4.1. Although all five conditions in Definition 4.1 are necessary in the proof of Theorem 4.1, the most important property, in a sense, is the one given in (5). Indeed, this property asserts that the s th left-definite inner product is generated from the s th power of A. In particular, if A is generated from a Lagrangian symmetric differential expression [ ], the s th left-definite inner product (, ) s is determined by the s th power of [ ]. Consequently, even though these left-definite spaces and left-definite inner products exist for all s>, we can only explicitly obtain these spaces and inner products when s is a positive integer. We refer the reader to [11] where an example is discussed in which the entire continum of left-definite spaces and inner products are explicitly obtained.

8 8 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN Definition 4.. For s>, let W s =(V s, (, ) s ) denote the s th left-definite space associated with (H, A). If there exists a self-adjoint operator B s : D(B s ) W s W s satisfying B s f = Af (f D(B s ) D(A)), we call such an operator an s th left-definite operator associated with (H, A). Again, it is not immediately clear that such an B s exists for a given s>; in fact, however, as the next theorem shows, B s exists and is unique. Theorem 4.. (see [11, Theorem 3.]) Suppose A is a self-adjoint operator in a Hilbert space H that is bounded below by ri, for some r>. For any s>, let W s =(V s, (, ) s ) be the s th left-definite space associated with (H, A). Then there exists a unique left-definite operator B s in W s associated with (H,A). Moreover, D(B s )=V s+ D(A). The next theorem gives further explicit information regarding the left-definite spaces and leftdefinite operators associated with (H, A). Theorem 4.3. (see [11, Theorem 3.4]) Suppose A is a self-adjoint operator in a Hilbert space H that is bounded below by ri, for some r>. Let {H s =(V s, (, ) s )} s> and {B s } s> be the leftdefinite spaces and left-definite operators, respectively, associated with (H,A). Then the following results are true. (1) Suppose A is bounded. Then, for each s>, (i) V = V s ; (ii) the inner products (, ) and (, ) s are equivalent; (iii) A = B s. () Suppose A is unbounded. Then (i) V s is a proper subspace of V ; (ii) V s is a proper subspace of V t whenever <t<s; (iii) the inner products (, ) and (, ) s are not equivalent for any s>; (iv) the inner products (, ) t and (, ) s are not equivalent for any s, t >, s6= t; (v) D(B s ) is a proper subspace of D(A) for each s>; (vi) D(B t ) is a proper subspace of D(B s ) whenever <s<t. Remark 4.. A statement is in order regarding the apparent ambiguity between part (v) of Definition 4.1 and the definition of (, ) s given in (4.) of Theorem 4.1. From part ()(ii) of Theorem 4.3, we see that D(A s )=V s V s. Consequently, if x D(A s ) and y V s, we see from the self-adjointness of A s/ that (x, y) s =(A s x, y) =(A s/ (A s/ x),y)=(a s/ x, A s/ y). The fact that A s/ x D(A s/ ) follows from [11, Theorem 4.3, equation (4.3) and Lemma 5.3, equations (5.8) and (5.9)]. The last theorem that we state in this section shows that the point spectrum, continuous spectrum, and resolvent set of a self-adjoint, bounded below operator A and each of its associated left-definite operators B s (s>) are identical; see [1, Section 7.] for the definitions concerning the various components of the spectrum listed below and the resolvent set of a general linear operator. Theorem 4.4. (see [11, Theorem 3.6]) For each s>, let B s denote the s th left-definite operator associated with the self-adjoint operator A that is bounded below by ri in H, for some r>. Then

9 SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 9 (a) the point spectra of A and B s coincide; that is, σ p (B s )=σ p (A); (b) the continuous spectra of A and B s coincide; that is, σ c (B s )=σ c (A); (c) the resolvent sets of A and B s are equal; that is, ρ(b s )=ρ(a). We refer the reader to [11] for other theorems, and examples, associated with the general leftdefinite theory of self-adjoint operators A that are bounded below. 5. Left-definite analysis of the Laguerre differential expression Since, for α> 1, the Laguerre differential operator, defined in (3.1) and (3.), is self-adjoint and bounded below by ri, there exists a continuuum of left-definite spaces and left-definite operators associated with (L α(, ),A α ). Indeed, this is an immediate consequence of the results in the previous section. We remind the reader of the definition of the space L α(, ) in (1.1). In [11], the authors show that the n th left-definite space associated with (L α(, ),A α ), when α> 1, is given by H n,α := (V n,α, (, ) n,α ) (n N), where (5.1) V n,α := {f :(, ) C f AC (n 1) loc (, ); f (j) L α+j(, ) (j =, 1,...n)}, and where the inner product (, ) n,α is given by nx (5.) (f,g) n,α := b j (n, r) f (j) (t)g (j) (t)t α+j e t dt (f,g V n,α ). j= Here, the numbers b j (n, r) (j =, 1,...,n) are defined to be ½ if r = (5.3) b (n, r) := r n if r>, and, for j {1,,...n}, ( S n (j) if r = (5.4) b j (n, r) := P n 1 n (j) m= m S n m rm if r>, where {S (j) n (5.5) S (j) n := i= } are the classical Stirling numbers of the second kind, defined by jx ( 1) i+j µ j i n (n, j N ). j! i We note that each b j (n, r) is positive for r>and b n (n, r) =1for n N. The Laguerre polynomials {L α n} n= form a complete orthogonal set in each H n,α; in fact, (L α j,l α m) n,α = (j + r)n Γ(j + α +1) δ j,m (j, m ). j! Moreover, for each n N, the n th left-definite operator B n,α : D(B n,α ) H n,α H n,α associated with (A α,l α(, )) is given explicitly by B n,α f := α [f] f D(B n,α ):=V n+,α,

10 1 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN where α [ ] is the Laguerre differential expression defined in (.1); the Laguerre polynomials {L α n} n= are a complete set of eigenfunctions of each operator B n,α and the spectrum of each B n,α is discrete and given by σ(b n,α )=σ(a α )={m + r m N }. When α = k, for some k N, the self-adjoint operator A k, defined in (3.3), is bounded below in L k (, ) by ri (see (3.4)). In this case, the above results extend mutatis mutandis so we will not prove these results here. The following theorem summarizes various properties of the left-definite spaces and operators associated with (L k (, ),A k). Theorem 5.1. Let k N and let A k denote the self-adjoint operator, defined in (3.3), that is bounded below by ri in L k (, ). Then the sequence of left-definite spaces associated with (L k (, ),A k) is given by (5.6) {H n, k := (V n, k, (, ) n, k )} n=1, where (5.7) and (5.8) (f,g) n, k := V n, k := {f :(, ) C f (j) AC loc (, ) (j =, 1,...,n 1); f (j) L j k (, ) (j =, 1,...,n)}, nx b j (n, r) j= f (j) (x)g (j) (x)x j k e x dx (f,g V n, k ). In particular, the k th left-definite space H k, k =(V k, k, (, ) k, k ) associated with (L k (, ),A k) is given by (5.9) and (5.1) (f,g) k, k := V k, k := {f :(, ) C f (j) AC loc (, ) (j =, 1,...,k 1); f (j) L j k (, ) (j =, 1,...,k)}, kx b j (k, r) j= The Laguerre polynomials {L k H n, k ; in fact, n } n=k f (j) (x)g (j) (x)x j k e x dx (f,g V k, k ). form a complete orthogonal set in each left-definite space (5.11) (L k j,l k m ) n, k = (j + r)n (j k)! δ j,m (j, m k). j! Furthermore, the sequence {B n, k } n=1 of left-definite (self-adjoint) operators associated with the pair (L k (, ),A k) is given explicitly by where B n, k f := k [f] D(B n, k ):=V n+, k B n, k : D(B n, k ) H n, k H n, k, = {f V n, k f (j) AC loc (, ); f (j+1) L j+1 k (, ) (j = n, n +1)},

11 SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 11 and where k [ ] is the Laguerre differential expression defined in (.9).The Laguerre polynomials {L k n } n=k form a complete set of eigenfunctions of each B n, k; furthermore,thespectrumofb n, k is discrete and given by σ(b n, k )=σ(a k )={m + r m k}. In particular, we note that the k th left-definite operator B k, k : D(B k, k ) H k, k H k, k is given by ½ Bk, k f := (5.1) k [f] D(B k, k ):=V k+, k, where (5.13) V k+, k := {f :(, ) C f (j) AC loc (, ) (j =, 1,...,k+1); f (j) L j k (, ) (j =, 1,...,k+)}, or, equivalently, (5.14) V k+, k := {f V k, k f (k),f (k+1) AC loc (, ); f (k+1) L ((, ); xe x ),f (k+) L ((, ); x e x )}. As we will see in Section 9, both the operator B k, k and the k th left-definite vector space V k, k play an important role in obtaining a certain self-adjoint operator T k,1. We seek to obtain a new characterization of the k th left-definite vector space V k, k associated with (A k,l k (, )); this characterization will be important in the developments in the rest of this paper. Before obtaining this characterization, we state an important operator inequality result that will be used on several occasions in this paper. Theorem 5.. Let (a, b) R with a<b and suppose ϕ, ψ :(a, b) C satisfy ϕ L (a, c), ψ L (c, b) (c (a, b)). Define the linear operators S, T : L (a, b) L loc (a, b) by Sf(x) =ϕ(x) Tf(x) =ψ(x) Z b x a ψ(x)f(x)dx ϕ(x)f(x)dx (x (a, b)) (x (a, b)). Then S and T are bounded operators into L (a, b) if and only if there exists a positive constant K such that (5.15) Moreover, for fixed f L (a, b), and a ϕ(x) dx ϕ(x) ψ(x) Z b x Z b x a ψ(x) dx K ψ(x)f(x)dx L (a, b) ϕ(x)f(x)dx L (a, b) if and only if (5.15) holds for some positive constant K. (x (a, b)).

12 1 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN Remark 5.1. This theorem was established by Chisholm and Everitt in [3] in Results were extended to the general case of conjugate indices p and q (p, q > 1) in [4] in It recently came to our attention that this general result is contained in a result due to Muckenhoupt [1] in 197. Moreover, the contributions by Talenti [17] and Tomaselli [18], both in 1969, also contain results equivalent to Theorem 5.. Theorem 5.3. Let (5.16) S k,1 [, ) :={f :[, ) C f (j) AC loc [, ), f (j) () = (j =, 1,...,k 1); f (k) L (, )}. Then (5.17) V k, k = S k,1 [, ), where V k, k is defined in (5.9). Remark 5.. The subscript 1 in S k,1 [, ) will be made clearer in Section 8. We note that functions in V k, k are defined on the interval (, ) whereas functions in S k,1 [, ) have domain [, ). In the course of the proof of Theorem 5.3, we will see that the limits lim f (j) (x) (f V k, k ; j =, 1,...,k 1), x + exist and are finite so, in this case, we define f (j) () := lim x + f (j) (x). Using standard arguments, we then show that f (j) () = for j =, 1,...,k 1. Proof. S k,1 [, ) V k, k : Let f S k,1 [, ). We need to show that (5.18) f (j) L j k (, ) (j =, 1,...,k). By definition of S k,1 [, ), the claim in (5.18) is true for j = k. Suppose, using mathematical induction, it is the case that (5.19) f (r) L r k (, ) (r = k, k 1,...,j+1), where j {, 1,...,k 1}; we need to prove (5.) f (j) L j k (, ). Before proving (5.), we first show (5.1) f (r) L (, ) (r =, 1,...,k). Again, from the definition of S k,1 [, ), we see that so (5.1) is true for r = k. Suppose f (k) L (, ), (5.) f (r) L (, ) (r = k, k 1,...,j+1), for some j {, 1,...,k 1}. Since f (j) () = and f (j) AC loc [, ), we see that ³ (5.3) f (j) (x)e x/ = e x/ e t/ e t/ f (j+1) (t) dt. By assumption, f (j+1) L (, ) or, equivalently, e t/ f (j+1) L (, ). We apply Theorem 5. with a =,b=, ϕ(x) =e x/, and ψ(x) =e x/ to see that e x/ f (j) (x) L (, ) or,

13 SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 13 equivalently, f (j) L (, ). This completes the induction and establishes (5.1). To prove (5.), notice from our induction hypothesis in (5.19) that f (j+1) L j+1 k (, ) or, equivalently, x (j+1 k)/ e x/ f (j+1) L (, ). In particular, f (j+1) (x) x (k j 1)/ L (, 1). Since f (j) () =, we see that, for <x<1, (5.4) f (j) (x) x (k j)/ = 1 x (k j)/ t (k j 1)/ f (j+1) (t) dt. t (k j 1)/ Again, we apply Theorem 5. with a =,b=1,ϕ(x) =x (k j 1)/ 1, and ψ(x) = x Z 1 ½ t k j 1 1 x ln x dt dt = if j = k 1 x tk j 1 (k j)(j k+1) x k j x ( <x<1) if j<k 1 K for some <K<, we see that f (j) (x) x (k j)/ L (, 1) or, equivalently, µ (5.5) f (j) L (, 1); e x x k j. For x 1, x k j 1 so that f (j) e (x) x dx 1 xk j f (j) (x) e x dx <, 1 from (5.1). Consequently, we see that µ (5.6) f (j) L (1, ); e x x k j. Combining (5.5) and (5.6), we obtain f (j) L µ (, ); e x x k j = L j k (, ), and this completes the induction on (5.18). Hence S k,1 [, ) V k, k, as required. V k, k S k,1 [, ): Let f V k, k. In particular, (5.7) f (k) L (, ); that is, f (k) (x) e x dx <. (k j)/ ;since Hence f (k) L (, 1) and thus, using a standard measure theory argument, f (j) AC[, 1] for j =, 1,...,k 1. By assumption, f (j) AC loc (, ) (j =, 1,...,k 1) so it follows that (5.8) f (j) AC loc [, ) (j =, 1,...,k 1). By definition of V k, k, we see that f (j) (x) e x dx < xk j (j =, 1,...,k);

14 14 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN in particular, Z 1 f (j) (x) x k j dx < (j =, 1,...,k). If, for some j {, 1,...,k 1}, f (j) () 6=, then there exists ε (, 1) and c> such that f (j) (x) c (x [,ε]). But then > Z 1 f (j) (x) dx x k j Z ε c dx =, xk j a contradiction. Hence (5.9) f (j) () = (j =, 1,...,k 1). Combining (5.7), (5.8), and (5.9), we see that V k, k S k,1 [, ), completing the proof of the theorem. We note the following result, that will be used later in this paper, whose proof follows along the same lines as given above. Corollary 5.1. For each n N and f V n, k, where V n, k is defined in (5.7), we have f (j) () = (j =, 1,...,n 1). A key result in establishing the self-adjointness of the operator T k,1 in Section 9 is the following theorem. Theorem 5.4. Let f,g D(B k, k )=V k+, k, where V k+, k is defined in (5.13). Then (a) x 1/ f (k+1) L (, ); that is, R f (k+1) (x) xe x dx <. (b) lim x xe x f (k+1) (x)g (k) (x) =. Proof. We first prove part (a). Let f V k+, k V k, k and, without loss of generality, assume f is real-valued. Since B k, k f V k, k, we see that hence, ( k [f]) (k) f (k) L 1 (, ) satisfies Since (5.3) lim x ( k [f](t)) (k) f (k) (t)e t dt = f (k), ( k [f]) (k) L (, ); ( k [f](t)) (k) f (k) (t)e t dt <. ( k [f](t)) (k) = tf (k+) (t)+(t 1)f (k+1) (t)+(k + r)f (k) (t) = 1 ³ e t te t f (k+1) t (t) +(k + r)f (k) (t)e, we see that, for x>, ( k [f](t)) (k) f (k) (t)e t dt ³ ³ = te t f (k+1) (t) f (k) (t)+(k + r) f (k) (t) e t dt.

15 SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 15 Consequently, since R f (k) (t) e t dt <, we see that ³ (5.31) te t f (k+1) (t) f (k) (t)dt <. By integration by parts, we see that ³ te t f (k+1) (t) f (k) (t)dt + xe x f (k+1) (x)f (k) (x) ³ = f (k+1) (t) te t dt ( <x< ). Hence, from (5.31), we see that if then ³ f (k+1) (t) te t dt =, (5.3) lim x xe x f (k+1) (x)f (k) (x) =. Hence, there exists x > such that f (k+1) (x)f (k) (x) ex x (x x ). Integrate this inequality over [x,x] to obtain f (k) (x) f (k) (x ) (5.33) = Moreover, integration by parts yields hence (5.33) implies that f (k) (x) Therefore > x f (k+1) (t)f (k) (t)dt x e t t dt. x e t t dt = ex x ex e t + x x t dt ex x ex x ; ex x ex f (k) (x ) + x x f (k) (t) e t dt = x 1 x =, = ex x + c (x x ). µ e t t + c e t dt t dt + c x e t dt

16 16 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN a contradiction. Hence, we must have which proves part (a) of the theorem. ³ f (k+1) (t) te t dt <, To prove (b), we again assume that f,g V k+, k are both real-valued. Since ³ f (k+1) (t)g (k+1) (t)te t dt = xe x f (k+1) (x)g (k) (x) te t f (k+1) (t) g (k) (t)dt (x >), we see, from part (a) and the definition of V k+, k, that lim x xe x f (k+1) (x)g (k) (x) :=c exists and is finite. If this limit is not zero, we may assume that c>. Hence, there exists x > such that (5.34) xe x f (k+1) (x)g (k) (x) c with f (k+1) (x) >,g (k) (x) > (x x ), so that xe x f (k+1) (x) g (k+1) (x) c g(k+1) (x) g (k) (x x (x) ). Consequently, Z x te t f (k+1) (t) g (k+1) g (k+1) (t) (t) dt c x x g (k) dt (t) g (k+1) (t) c x g (k) (t) dt (5.35) = c ln(g (k) (t)) x x c ln(g (k) (x)) c 1 (x x ). >From part (a), we see that lim x x te t f (k+1) (t) g (k+1) (t) dt <, so we must have (5.36) lim sup ln(g (k) (x)) <. x It follows that there exists constants M 1,M > such that (5.37) M 1 <g (k) (x) <M (x x ). For if g (k) is unbounded on [x, ), there exists a sequence {x n} [x, ) such that x n and g (k) (x n ), contradicting (5.36). Furthermore, if g (k) is not bounded away from zero, then there exists a sequence {y n } [x, ) such that g(k) (y n ) and, thus, ln(g (k) (y n )) ; however, this also contradicts (5.36). From (5.34) and (5.37), we see that xe x f (k+1) (x) c := ec, M

17 SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 17 so that ³ f (k+1) (x) xe x (ec) e x x Integrating on [x, ) yields x ³ Z f (k+1) (t) te t dt (ec) x (x x ). e t dt =, t contradicting part (a). Hence, we must have lim x xe x f (k+1) (x)g (k) (x) =, and this completes the proof of the theorem. Remark 5.3. Property (a) of Theorem 5.4 says that k [ ] is Dirichlet at x = on V k+, k in the k th left-definite space H k, k, while property (b) shows that k [ ] is strong limit-point at x = on V k+, k. 6. Sobolev orthogonality of the Laguerre Polynomials In [8], the authors show that, for each k N, the entire sequence of Laguerre polynomials {L k n } n= is, remarkably, orthonormal with respect to the positive-definite inner product (, ) k, defined on P[, ) P[, ) by mx h i (6.1) (p, q) k := B m,j (k) p (m) ()q (j) () + p (j) ()q (m) () + p (k) (x)q (k) (x)e x dx m= j= = hw k,r,p (r) q (r) i + r= p (k) (x)q (k) (x)e x dx (p, q P[, )), where the numbers B m,j (k) are given by ( P j p= (6.) B m,j (k) = ( 1)m+j k 1 p k 1 p m p j p if j<m k 1 if j = m k 1, 1 P m p= k 1 p m p and where w k,r is the linear functional defined by µ k r 1 k X µ k r 1 w k,r = δ (j) ; r j j= here δ (j) is the classic Dirac delta distribution defined, in this case, on the polynomial space P[, ) through the standard formula hδ (j),pi =( 1) j p (j) () (p P[, )). That is to say, (6.3) (L k n,l k m ) k = δ n,m (n, m N ). We note that it is precisely the identity in (.8) that led the authors in [8] to constructing this inner product (, ) k. For example, the Laguerre polynomials {L 1 are orthonormal with respect to (6.4) (p, q) 1 = p()q() + n } n= p (x)q (x)e x dx (p, q P[, )),

18 18 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN while {L 3 n } n= are orthonormal with respect to (p, q) 3 = p()q() [p ()q() + p()q ()] + 5p ()q () (6.5) +[p ()q() + p()q ()] 3[p ()q () + p ()q ()] +3p ()q () + p (x)q (x)e x dx (p, q P[, )). As discussed in Section, we remark that these Laguerre polynomials {L 3 n } of degree 3 are orthogonal as well in the Hilbert space L 3 (, ); indeed, it is the case that (L 3 n,l 3 m ) L ( 3) = 1 n(n 1)(n ) δ n,m (n, m 3), where (, ) L ( 3) is the inner product definedin(.6). The discovery of the orthonormality of {L 1 n } n= with respect to the inner product (, ) 1 in (6.4) was first reported in the paper [9] by Kwon and Littlejohn. Subsequently, in [8], the authors extended this result and determined explicitly the inner product (, ) k, given in (6.1), for each k N. For k N, we define the function space S k [, ) to be (6.6) S k [, ) :={f :[, ) C f,f,...,f (k 1) AC loc [, ); f (k) L ((, ); e x )}. Observe that P[, ) S k [, ). Furthermore, notice that, for f,g S k [, ), (f,g) k is welldefined, where (, ) k isgivenin(6.1).however,eventhoughitisclearthat (, ) k : S k [, ) S k [, ) C is a bilinear form, it is not immediately obvious, for large values of k N, that it is an inner product on S k [, ) S k [, ). Indeed, the authors in [8] only showed that (, ) k is an inner product on the proper subspace P[, ) P[, ) of S k [, ) S k [, ). In fact, it is not difficult to see that (, ) 1,defined in (6.4), is an inner product on S 1 [, ) S 1 [, ). As for (, ) 3, given in (6.5), a calculation shows that (f,f) 3 = f() f () + f () + f () f () + f () + f (x) e x dx (f S 3 [, )), from which it follows that (, ) 3 is an inner product on S 3 [, ) S 3 [, ). In general, we have the following result which readily shows that (, ) k is an inner product on S k [, ) S k [, ). Lemma 6.1. Let k N. Then, for f S k [, ), (6.7) (f,f) k µ = k r 1 ( 1) j r f (j) () j r r= j=r + f (k) (x) e x dx. Proof. Expanding µ k r 1 ( 1) j r f (j) () j r, r= j=r

19 SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 19 we obtain µ µ µ k 1 k 1 f() f () + f ()...+( 1) k 1 f (k 1) () 1 µ µ µ k 1 f() f k 1 () + f ()...+( 1) k 1 f (k 1) () 1 µ µ µ k k + f () f () + f ()...+( 1) k f (k 1) () 1 µ µ µ f k () f k () + f ()...+( 1) k f (k 1) () µ µ µ k j 1 k j 1 + f (j) () f (j+1) () + f (j+) ()...+( 1) k 1 j f (k 1) () 1 µ µ µ f (j) k j 1 () f (j+1) k j 1 () + f (j+) ()...+( 1) k 1 j f (j 1) () f (k 1) ()f (k 1) (). h i f (m) ()f (j) () + f (j) ()f (m) () in this above It is straightforward to check that the coefficient of expression is given by ( P m p= c m,j (k) := ( 1)m+j k 1 p k 1 p m p j p if j<m k 1 if j = m k 1. 1 P m p= k 1 p m p But this coefficient is exactly B m,j (k), defined in (6.). By comparing (6.1) and (6.7), we see that the proof of this lemma is now complete. Let (6.8) W k [, ) :=(S k [, ), (, ) k ) be this inner product space; for each k N, we write kfk k := (f,f) 1/ k (f S k [, )) for the norm k k k obtained from (, ) k. We are now in position to prove the following theorem. Theorem 6.1. For each k N, W k [, ) is a Hilbert space. Proof. Suppose {f n } n=1 W k[, ) is a Cauchy sequence. From (6.7), we see that k kf n f m k k = X µ k r 1 ³ (6.9) ( 1) j r f n (j) () f m (j) r= j r () j=r + f n (k 1) () f m (k 1) () + f n (k) (x) f m (k) (x) e x dx, from which we see that (6.1) kf n f m k k f (k 1) n () f m (k 1) (),

20 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN and (6.11) kf n f m k k f n (k) (x) f m (k) (x) e x dx. >From the completeness of C and L ((, ); e x ), weseethatthereexistsa k 1 C and g L ((, ); e x ) such that f n (k 1) () A k 1 in C, and f n (k) g in L ((, ); e x ). It follows then that g L 1 loc (, ). Returning to (6.9), we see that k 3 kf n f m k k = X µ k r 1 ³ ( 1) j r f n (j) () f m (j) j r () r= ³ + + j=r f (k ) n f n (k 1) () f m (k ) () () f m (k 1) () ³ f n (k 1) + f n (k) () f m (k 1) () (x) f m (k) (x) e x dx. >From this it follows that {f n (k ) ()} n=1 is Cauchy in C and hence there exists A k C such that f n (k ) () A k in C. Continuing, by induction, we see that, for j =, 1,...,k 1, there exists A j C such that (6.1) f n (j) () A j in C (j =, 1,...,k 1). Define f :[, ) C by (6.13) f(x) = j= A j x j j! + Z t1 Z tk 1 g(u)dudt k 1...dt 1. Then f satisfies the following properties: (i) f,f,...,f (k 1) AC loc [, ); (ii) f (j) () = A j (j =, 1,,...,k 1); (iii) f (k) (x) =g(x) for a.e. x> and f (k) = g L ((, ); e x ). Hence f S k [, ) and k 1 kf n fk k = X µ k r 1 r= j r j=r + (x) f (k) (x) e x dx f n (k) as n. ³ ( 1) j r f n (j) () f (j) () This completes the proof of the theorem.

21 SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 1 7. The completeness of the Laguerre polynomials {L k n } n= in W k[, ) We now prove Theorem 7.1. P[, ) is dense in the Hilbert space W k [, ) for each k N. Equivalently, the Laguerre polynomials {L k n } n= form a complete orthonormal set in W k[, ). Proof. We remind the reader that the orthonormality of {L k n } n= (see (6.3)) is established in [8]. Let f W k [, ) and let ε>. Since P[, ) is dense in L ((, ); e x ) and f (k) L ((, ); e x ), there exists q P[, ) such that Z (7.1) f (k) q = f (k) (x) q(x) e x dx < ε, L () where k k L () is the norm defined in (.3). Define p P[, ) by (7.) p(x) = j= f (j) ()x j j! + Z t1 Z tk 1 q(u)dudt k 1...dt 1. Note that (7.3) p() = f(), p () = f (),...,p (k 1) () = f (k 1) () and (7.4) p (k) (x) =q(x) (x [, )). >From the identity (6.7), together with the properties in (7.1), (7.3), and (7.4), we see that k 1 kf pk k = X µ k r 1 ( 1) j r f (j) () p (j) r= j r () j=r + f (k) (x) p (k) (x) e x dx = f (k) (x) p (k) (x) e x dx = f (k) (x) q(x) e x dx < ε ; i.e. kf pk k <ε.this completes the proof of this theorem. 8. A fundamental decomposition and identification of two inner product spaces Webeginwiththefollowingdefinition. Definition 8.1. Let S k,1 [, ) be the function space defined in (5.16) so that (8.1) S k,1 [, ) ={f S k [, ) f (j) () = (j =, 1,...,k 1)}. and let S k, [, ) be the function space defined by (8.) S k, [, ) :={f S k [, ) f (k) (x) =(x [, ))}. Define the inner product spaces W k,1 [, ) and W k, [, ) by (8.3) W k,1 [, ) :=(S k,1 [, ), (, ) k ),

22 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN and (8.4) W k, [, ) :=(S k, [, ), (, ) k ). We remind the reader that, by Theorem 5.3, S k,1 [, ) =V k, k, where V k, k is the k th leftdefinite vector space, defined in (5.9), associated with the pair (L k (, ),A k). Remark 8.1. It follows from (.8) that {L k n } n=k S k,1[, ). It is precisely this remarkable property (.8) of the Laguerre polynomials that prompts our definition of S k,1 [, ). Theorem 8.1. W k,1 [, ) and W k, [, ) are closed, orthogonal subspaces of W k [, ), where W k [, ) is defined in (6.8), with (8.5) W k [, ) =W k,1 [, ) W k, [, ). are complete orthonormal se- Furthermore, the Laguerre polynomials {L k n } n=k quences in W k,1 [, ) and W k, [, ), respectively. and {L k n } n= k 1 Proof. Since W k, [, ) is k-dimensional, it is a closed subspace of W k [, ) and it is straightforward to check that W k,1 [, ) Wk, [, ), where Wk, [, ) is the orthogonal complement of W k,[, ). Let f W k [, ). Define f 1 (x) :=f(x) f (j) () xj j! j= j= f (x) := f (j) () xj j!. A calculation shows that f (j) 1 () = for j =, 1,...k 1 so that f 1 W k,1 [, ); similarly, it is clear that f W k, [, ). Furthermore, (f 1,f ) k = =, m= j= mx h i B m,j f (m) 1 ()f (j) () + f (j) (m) 1 ()f () + f (k) 1 (x)f (k) (x)e x dx since f (j) (k) 1 () = for j =, 1,...,k 1 and f (x). This establishes (8.5). Aproofthatthe Laguerre polynomials {L k n } n=k form a complete orthonormal sequence in W k,1[, ) is identical to that given in Theorem 7.1; the proof that {L k n } k 1 n= is complete in W k,[, ) is obvious since W k, [, ) is k dimensional. Observe that, for f,g S k,1 [, ), we have (8.6) (f,g) k = f (k) (x)g (k) (x)e x dx; indeed, this follows since f (j) () = for j =, 1,...,k 1. Interestingly, the two inner products (, ) k and the k th left-definite inner product (, ) k, k are equivalent on V k, k = S k,1 [, ), as we show.

23 SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 3 Theorem 8.. The two inner products (, ) k, k and (, ) k, defined in (5.1) and (6.1) respectively, are equivalent on V k, k = S k,1 [, ). Proof. Let f V k, k = S k,1 [, ). Since b j (k, r) (j =, 1,...k) and b k (k, r) =1(recall the definition of these numbers in (5.4)), we see from (8.6) that (f,f) k = f (k) (x) e x dx (8.7) kx b j (k, r) j= =(f,f) k, k. f (j) (x x j k e x dx Recall (see Theorem 5.1) that the Laguerre polynomials {L k n } n=k are a complete orthogonal set in H k, k with (L k j,l k m ) k, k = (j + r)k (j k)! δ j,m (j, m k). j! Consequently, ( ) (j!) 1/ (8.8) (j + r) k/ L k ((j k)!) 1/ j is a complete orthonormal set in H k, k. Furthermore, since (j + r)k (j k)! 1 for j k and j! (j + r) k (j k)! lim =1, j j! there exists L = L(k, r) satisfying (8.9) <L<1 and (j + r)k (j k)! < 1 (j k). j! L Let {ξ j } j=k be the Fourier coefficients of f in H k, k relative to the orthonormal basis given in (8.8); that is, (j!) 1/ (8.1) ξ j = (j + r) k/ (f,l k ((j k)!) 1/ j ) k, k (j k). Then, from the classical theory, as n, we have f n := nx j=k j=k (j!) 1/ ξ j (j + r) k/ L k ((j k)!) 1/ j f in H k, k. Observe, from (8.7), it is also the case that f n f in W k,1 [, ); indeed, kf n fk k =(f n f,f n f) k (f n f,f n f) k, k = kf n fk k, k as n.

24 4 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN Furthermore, from the orthonormality of {L k j } j=k in W k,1[, ), we see from (8.9) that nx (f n,f n ) k = ξ j j! (j + r) k (j k)! j=k nx (8.11) >L ξ j j=k = L(f n,f n ) k, k (n k). Letting n in (8.11) yields (8.1) (f,f) k L(f,f) k, k. Together, (8.7) and (8.1) complete the proof of the theorem. Remark 8.. We note that the equivalence of the inner products (, ) k and (, ) k, k on V k, k = S k,1 [, ) also follows from the inequality in (8.7), the completeness of the two inner product spaces W k,1 [, ) =(S k,1 [, ), (, ) k ) and H k, k =(V k, k, (, ) k, k ), and an application of the Open Mapping Theorem. Indeed, if (X, k k 1 ) and (X, k k ) are Banach spaces and there exists a positive constant c such that kxk 1 c kxk (x X), then, from the Open Mapping Theorem, the norms k k 1 and k k are equivalent on X; see [1, Chapter 4, page 91, Problem 8]. 9. Theself-adjointLaguerreoperatorT k,1 Definition 9.1. The operator T k,1 : D(T k,1 ) W k,1 [, ) W k,1 [, ) is given by ½ T (9.1) k,1 f := k [f] f D(T k,1 ):=V k+, k, where k [ ] is the Laguerre expression defined in (.9) and where V k+, k is given in (5.13). Since the k th left-definite operator B k, k, defined in (5.1), has the same form and domain as does T k,1 and since B k, k f V k, k = S k,1 [, ) for f V k+, k V k, k, it is clear that T k,1 does indeed map V k+, k W k,1 [, ) into W k,1 [, ). We remind the reader that, from Theorem 5.3, V k+, k, defined in (5.14), consists of precisely those functions f :[, ) C satisfying (9.) (i) f (j) AC loc [, ) for j =, 1,...,k 1; (ii) f (k+j) AC loc (, ) for j =, 1; (iii) f (j) () = for j =, 1,...,k 1; (iv) f (k+j) L ((, ); x j e x ) for j =, 1,. Lemma 9.1. The space V k+, k is dense in W k,1 [, ); that is to say, T k,1 is a densely defined operator. Proof. Let f W k,1 [, ) =(S k,1 [, ), (, ) k ), where S k,1 [, ) =V k, k by Theorem 5.3, and let ε>. Since the k th left-definite operator B k, k, with domain D(B k, k )=V k+, k is densely defined in H k, k =(V k, k, (, ) k, k ), there exists g V k+, k such that kf gk k, k <ε,

25 where SPECTRAL ANALYSIS OF THE LAGUERRE POLYNOMIALS 5 kx kf gk k, k = b j (k, r) j= f (j) (x) g (j) (x) x j k e x dx However, since each b j (k, r) and b k (k, r) =1, we see that kx ε> b j (k, r) f (j) (x) g (j) (x) x j k e x dx (9.3) j= µ b 1/ k (k, r) f (k) (x) g (k) (x) e x dx µ 1/ = f (k) (x) g (k) (x) e dx x. 1/ On the other hand, since f S k [, ) =V k, k and g V k+, k V k, k, we see that Consequently, f (j) () = g (j) () = (j =, 1,...,k 1). k 1 kf gk k = X µ k r 1 ³ ( 1) j r f (j) () g (j) r= j r () j=r + f (k) (x) g (k) (x) e x dx = f (k) (x) g (k) (x) e x dx < ε by (9.3). This shows that T k,1 is densely defined in W k,1 [, ) and completes the proof of the lemma. We can now show, among other results, that T k,1 is symmetric in W k,1 [, ). Theorem 9.1. The operator T k,1, defined in (9.1), is symmetric and bounded below by (r + k)i in W k,1 [, ); that is, (9.4) (T k,1 f,f) k (k + r)(f,f) k (f D(T k,1 )). Furthermore, the Laguerre polynomials {L k n } n=k form a complete orthonormal set of eigenfunctions of T k,1 with y = L k n (x) corresponding to the simple eigenvalue λ n = n + r for each integer n k. Proof. >From Lemma 9.1, to show symmetry of T k,1, it suffices to show that T k,1 is Hermitian; that is (T k,1 f,g) k =(f,t k,1 g) k (f,g D(T k,1 )). Let f,g D(T k,1 )=V k+, k. Then, from Corollary 5.1, (9.5) f (j) () = g (j) () = (j =, 1,...,k+1); moreover, since T k,1 [f] = k [f] =B k, k f V k, k = S k,1 [, ), we see from the definition of V k, k that (9.6) ( k [f]) (j) () = (j =, 1,...,k 1). 1/ 1/.

26 6 W. N. EVERITT, L. L. LITTLEJOHN, AND R. WELLMAN Consequently, (T k,1 f,g) k mx i = B m,j h( k [f]) (m) ()g (j) () + ( k [f]) (j) ()g (m) () = = m= j= + ( k [f]) (k) (x)g (k) (x)e x dx ( k [f]) (k) (x)g (k) (x)e x dx by (9.5) and (9.6) ³ xe x f (k+1) x (x) g (k) (x)+(k + r)f (k) (x)g (k) (x)e dx (see (5.3)). Furthermore, integration by parts yields (T k,1 f,g) k ³ = xe x f (k+1) x (x) g (k) (x)+(k + r)f (k) (x)g (k) (x)e dx h i = xe x f (k) (x)g (k+1) (x) xe x f (k+1) (x)g (k) (x) ³ + xe x g (k+1) x (x) f (k) (x)+(k + r)g (k) (x)f (k) (x)e dx ³ = xe x g (k+1) x (x) f (k) (x)+(k + r)g (k) (x)f (k) (x)e dx since, by Theorem 5.4, part (b), we have (9.7) lim x xe x f (k) (x)g (k+1) (x) = lim x xe x f (k+1) (x)g (k) (x) = while the definition of V k+, k and Corollary 5.1 gives That is, (9.8) (T k,1 f,g) k = On the other hand, a similar computation yields f (k) () = f (k+1) () = g (k) () = g (k+1) () =. ³ xe x g (k+1) x (x) f (k) (x)+(k + r)g (k) (x)f (k) (x)e dx. (f,t k,1 g) k mx i = B m,j hf (m) () ( k [g]) (j) () + f (j) () ( k [g]) (m) () = m= j= + =(T k,1 f,g) k, f (k) (x)( k [g]) (k) (x)e x dx ³ xe x g (k+1) x (x) f (k) (x)+(k + r)g (k) (x)f (k) (x)e dx by Corollary 5.1