TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 4, April 2007, Pages 1829–1849 S 0002-9947(06)03950-X Article electronically published on November 22, 2006 COMPLETENESS OF SYSTEMS OF COMPLEX EXPONENTIALS AND THE LAMBERT W FUNCTIONS ANDRE´ BOIVIN AND HUALIANG ZHONG Abstract. We study some of the properties of the solution system {eiλnt} of the delay-differential equation y(t)=ay(t − 1). We first establish some general results on the stability of the completeness of exponential systems in L2 and then show that the solution system above is always complete, but is not an unconditional basis in L2(−1/2, 1/2). 1. Introduction It is well known that the solutions of differential-difference equations can be iλnt expressed as (an infinite) sum of exponentials cne where each iλn is a char- acteristic root of the equation. For a “nice” initial condition g(t), say, belonging to C0,orC1, the summation is known to converge pointwise to the solution it represents (see, for example, Bellman and Cooke [2]). In an attempt to better un- derstand this representation, we were led to study the structure (e.g. completeness, iλ t basis or frame properties) of the exponential systems {e n } when the iλn have a distribution similar to that of the characteristic roots of y(t)=ay(t − 1). The study of exponential systems, often referred to as the theory of nonharmonic Fourier series (see [23, 26]), has its origin in the classical works of R. Paley and N. Wiener [15] and N. Levinson [13]. One of the famous early results in the theory is int ∞ 2 that the basis property of the trigonometric system {e }−∞ is stable in L (−π,π) iλ t ∞ 2 in the sense that the system {e n }−∞ will always form a Riesz basis for L (−π,π) | − |≤ 1 if λn n L< 4 . M.I. Kadec’ [9], and R.M. Redheffer and R.M. Young [17] 1 have shown 4 to be optimal. Many (but not all) of the subsequent results on the completeness, frames, basis or interpolation properties of exponential systems required that the λn’s be located in a strip parallel to the real axis (e.g. [26]) or nearby the zeroes of a function of sine type (e.g. [1]). These results did not apply to our case since for the sequences we wish to consider, the λn’s are located in a curvilinear strip. The few known results that allowed for a (slow) growth of the imaginary part of the λn did not apply directly to our setting either and first needed to be extended. One of the main purposes of this paper is to provide such an extension. The paper is organized as follows. Section 2 contains basic definitions and nota- tions, and a discussion of the characteristic roots of the equation y(t)=ay(t − 1) Received by the editors July 4, 2003 and, in revised form, February 4, 2005. 2000 Mathematics Subject Classification. Primary 42C15, 42C30, 34K07; Secondary 30B50. The first author was partially supported by a grant from NSERC of Canada. c 2006 American Mathematical Society Reverts to public domain 28 years from publication 1829 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1830 A. BOIVIN AND H. ZHONG in terms of the Lambert W functions, including a result on the asymptotic dis- tributions of the roots. In Sections 3 and 4, we present and prove two theorems regarding the stability of the completeness property in L2 of exponential systems under perturbations within a curvilinear strip. Previously imposed conditions in theorems of A.M. Sedletskii [18], and N. Fujii, A. Nakamura and R. Redheffer [6] and A.M. Sedletskii [23, §5.4, Theorem 5] have been relaxed. In Section 5, we apply Theorem 3.2 to special sequences related to the Lambert W functions Wn(a) 2 − 1 1 introduced in Section 2. In particular, we show the completeness in L ( 2 , 2 )of the system {eWn(a)t}. In Section 6, we show that the radius of completeness of {− } 1 the sequence iWn(a) is equal to 2 , and thus that the main results of Section 5 cannot be derived from the powerful theorem of Beurling and Malliavin [3, 10]. 2. Preliminaries In this paper, we denote by C (respectively by R) the set of all complex (re- spectively real) numbers. Z denotes the set of all integers, and and mean summation and multiplication, respectively, through all the integers except 0. Un- less otherwise specified, all sequences considered in this paper will be indexed by the integers from −∞ to ∞. We say that an entire function f(z)isofexponential type γ if there is a constant A>0 such that |f(z)|≤Aeγ|z|. The totality of all entire functions of exponential type at most π that are square integrable on the real axis is known as the Paley-Wiener space (see [26]) which ∞ is a Hilbert space with respect to the inner product (f,g)= −∞ f(x)g(x)dx. Asystem{eiλnt} of complex exponentials is closed in Lp(−γ,γ), 1 ≤ p<∞,if every f ∈ Lp(−γ,γ) can be approximated in Lp norm by (finite) linear combinations of the functions eiλnt. Asystem{eiλnt} of complex exponentials is complete in Lp(−γ,γ), 1 ≤ p<∞, if the relations γ f(t)eiλntdt =0 −γ p for all n with f ∈ L imply that f = 0 a.e. In this case {λn} is called a complete sequence. q p 1 1 Duality shows that closure in L is equivalent to completeness in L if p + q =1, and 1 <p<∞. Especially, when p = q = 2, completeness is equivalent to closure in L2(−γ,γ) (see [13] or [26]). The following important result of N. Levinson establishes the deep connection that exists between completeness of sets of exponentials and entire functions. Proposition A (N. Levinson, [13]). For the system {eiλnt} to be incomplete in C(−γ,γ) (or in Lp(−γ,γ), 1 <p<∞), it is necessary and sufficient that there ex- ists a non-trivial entire function f(z), which vanishes at every λn and is expressible License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use COMPLETENESS OF SYSTEMS OF COMPLEX EXPONENTIALS 1831 in the form γ (2.1) f(z)= eiztdω(t), −γ where ω(t) is of bounded variation on (−γ,γ) (or ω(t) ∈ Lq(−γ,γ), 1/p+1/q =1). Note 2.1. Equation (2.1) defines a function of exponential type at most γ and thus, when p =2andγ = π, the Plancherel Theorem shows that f in fact belongs to the Paley-Wiener space. Definition 2.2. Acompletesystem{eiλnt} in Lp that ceases to be complete when any one of its element is removed is said to be an exact complete system. When exactly m elements have to be removed (or added) in order that the new system be exact, then the excess Ep(λ) of the system is m (or −m). Definition 2.3. A sequence {f1,f2, ...} in an infinite-dimensional Banach space X is said to be a Schauder basis for Xif for each f ∈ X, there is a unique sequence { } ∞ of scalars c1,c2,... such that f = n=1 cnfn, i.e. n f − cifi → 0asn →∞. i=1 Henceforth, the term basis will always mean a Schauder basis. To study the completeness of the solution system of the delay-differential equa- tion y(t)=ay(t−1), where a is a nonzero fixed real number, we need to understand the zeroes of its characteristic function g(z)=z −ae−z as a function of the variable a as well as their asymptotic distribution. − 1 − 1 Note that when a = e , all the zeroes of g(z)aresimpleandthatwhena = e , − − 1 there is a double zero at z = 1. Assuming first that a = e and that g(z)=0, iφ and writing z as z = η1 + iη2 = re (−π<φ≤ π), we get that (2.2) η1 =log|a|−log r and η2 =arga +2πn − φ for some n ∈ Z. Now assume that a is real. It is known (see [25] or [24]) that: − 1 ≤ ≤ (1) If a does not satisfy e a 0, then there is a zero zn corresponding to each integer n ∈ Z. − 1 (2) If a satisfies e <a<0, then there is a zero zn corresponding to each integer n in Z \{−1, 0},andton = 0 there correspond two zeroes z0 and z−1 satisfying −1 <z0 < 0andz−1 < −1. Noting that from (2.2) we cannot have n = −1whena<0andz<0, we still have a one-to-one correspondence between the zeroes of g and all the integers. − 1 − (3) If a = e , there is a zero of order two at 1, and thus after setting z0 = z−1 = −1, the zeroes of g are again in exact correspondence with all the integers. To each integer n, there thus corresponds a unique zero of g which depends on the variable a. This defines a function Wn(a) called the Lambert W function (see [5]). The following asymptotic properties of these functions were obtained by S. Verblunsky in 1961. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1832 A. BOIVIN AND H. ZHONG Proposition B (S. Verblunsky, [24]). Suppose Wn(a) is the Lambert W function ∈ |a| for n Z,andα = 2π .Then i) when a>0, for all n =0 , we have α 1 log |n| 2 W (a)={log + sign(n)+O } n |n| 4n n α 1 log | | log |n| + i{2π(n − sign(n)) + n + O }, 4 2πn n2 and W0(a) > 0; − 1 ∈ \{− } ii) when e <a<0, for all n Z 1, 0 , we have α 1 1 log |n| 2 W (a)={log − ( − sign(n)) + O } n |n| 2n 4n n (2.3) α 1 1 log | | log |n| + i{2π(n + − sign(n)) + n + O }, 2 4 2πn n2 and −1 <W0(a) < 0, W−1(a) < −1; − 1 ∈ \{− } iii) when a = e , we have (2.3) for all n Z 1, 0 ,andW0(a)=W−1(a)= −1; − 1 ∈ \{ } iv) when a< e , we have (2.3) for all n Z 0 ,andW0(a) is not real.