Non-asymptotic Analysis of Bandlimited Functions. Andrei Osipov Research Report YALEU/DCS/TR-1449 Yale University January 12, 2012

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1 Prolate spheroidal wave functions PSWFs play an important role in various areas, from physics e.g. wave phenomena, fluid dynamics to engineering e.g. signal processing, filter design. Even though the significance of PSWFs was realized at least half a century ago, and they frequently occur in applications, their analytical properties have not been investigated as much as those of many other special functions. In particular, despite some recent progress, the gap between asymptotic expansions and numerical experience, on the one hand, and rigorously proven explicit bounds and estimates, on the other hand, is still rather wide. This paper attempts to improve the current situation. We analyze the differential operator associated with PSWFs, to derive fairly tight estimates on its eigenvalues. By combining these inequalities with a number of standard techniques, we also obtain several other properties of the PSFWs. The results are illustrated via numerical experiments. Non-asymptotic Analysis of Bandlimited Functions Andrei Osipov Research Report YALEU/DCS/TR-1449 Yale University January 12, 212 This author s research was supported in part by the AFOSR grant #FA Approved for public release: distribution is unlimited. Keywords: bandlimited functions, prolate spheroidal wave functions 1

2 Contents 1 Introduction 2 2 Mathematical and Numerical Preliminaries Prolate Spheroidal Wave Functions Elliptic Integrals Oscillation Properties of Second Order ODEs Prüfer Transformations Summary 9 4 Analytical Apparatus Oscillation Properties of PSWFs Special Points of ψ n A Sharper Inequality for χ n Elimination of the First-Order Term of the Prolate ODE Growth Properties of PSWFs Numerical Results 37 1 Introduction The principal goal of this paper is non-asymptotic analysis of bandlimited functions. A function f : R R is bandlimited of band limit c >, if there exists a function σ L 2 [ 1, 1] such that fx = 1 1 σte icxt dt. 1 In other words, the Fourier transform of a bandlimited function is compactly supported. While 1 defines f for all real x, one is often interested in bandlimited functions, whose argument is confined to an interval, e.g. 1 x 1. Such functions are encountered in physics wave phenomena, fluid dynamics, engineering signal processing, etc. see e.g. [14], [19], [2]. About 5 years ago it was observed that the eigenfunctions of the integral operator F c : L 2 [ 1, 1] L 2 [ 1, 1], defined via the formula F c [ϕ] x = 1 1 ϕte icxt dt, 2 provide a natural tool for dealing with bandlimited functions, defined on the interval [ 1, 1]. Moreover, it was observed see [9], [1], [12] that the eigenfunctions of F c are precisely the prolate spheroidal wave functions PSWFs, well known from the mathematical physics [16], [19]. The PSWFs are the eigenfunctions of the differential operator L c, defined via the formula L c [ϕ] x = d 1 x 2 dϕ dx dx x + c 2 x

3 In other words, the integral operator F c commutes with the differential operator L c [9], [18]. This property, being remarkable by itself, also plays an important role in both the analysis of PSWFs and the associated numerical algorithms [3], [4]. It is perhaps surprising, however, that the analytical properties of PSWFs have not been investigated as thoroughly as those of several other classes of special functions. In particular, when one reads through the classical works about the PSWFs [9], [1], [11], [12], [13], one is amazed by the number of properties stated without rigorous proofs. Some other properties are only supported by analysis of an asymptotic nature; see, for example, [6], [7], [15], [17]. This problem has been addressed in a number of recently published papers, for example, [2], [4], [5]. Still, the gap between numerical experience and asymptotic expansions, on the one hand, and rigorously proven explicit bounds and estimates, on the other hand, is rather wide; this paper offers a partial remedy for this deficiency. This paper is mostly devoted to the analysis of the differential operator L c, defined via 3. In particular, several explicit bounds for the eigenvalues of L c are derived. These bounds turn out to be fairly tight, and the resulting inequalities lead to rigorous proofs of several other properties of PSWFs. The analysis is supported by and is illustrated through several numerical experiments. The analysis of the eigenvalues of the integral operator F c, defined via 2, requires tools different from those used in this paper; it will be published at a later date. The implications of the analysis of both L c and F c to numerical algorithms involving PSWFs are being currently investigated. This paper is organized as follows. In Section 2, we summarize a number of well known mathematical facts to be used in the rest of this paper. In Section 3, we provide a summary of the principal results of this paper. In Section 4, we introduce the necessary analytical apparatus and carry out the analysis. In Section 5, we illustrate the analysis via several numerical examples. 2 Mathematical and Numerical Preliminaries In this section, we introduce notation and summarize several facts to be used in the rest of the paper. 2.1 Prolate Spheroidal Wave Functions In this subsection, we summarize several facts about the PSWFs. Unless stated otherwise, all these facts can be found in [4], [5], [7], [9], [1]. Given a real number c >, we define the operator F c : L 2 [ 1, 1] L 2 [ 1, 1] via the formula F c [ϕ] x = 1 1 ϕte icxt dt. 4 Obviously, F c is compact. We denote its eigenvalues by λ, λ 1,...,λ n,... and assume that they are ordered such that λ n λ n+1 for all natural n. We denote by ψ n the eigenfunction corresponding to λ n. In other words, the following identity holds for all 3

4 integer n and all real 1 x 1: λ n ψ n x = 1 1 ψ n te icxt dt. 5 We adopt the convention 1 that ψ n L 2 [ 1,1] = 1. The following theorem describes the eigenvalues and eigenfunctions of F c. Theorem 1. Suppose that c > is a real number, and that the operator F c is defined via 4 above. Then, the eigenfunctions ψ, ψ 1,... of F c are purely real, are orthonormal and are complete in L 2 [ 1, 1]. The even-numbered functions are even, the odd-numbered ones are odd. Each function ψ n has exactly n simple roots in 1, 1. All eigenvalues λ n of F c are non-zero and simple; the even-numbered ones are purely real and the odd-numbered ones are purely imaginary; in particular, λ n = i n λ n. We define the self-adjoint operator Q c : L 2 [ 1, 1] L 2 [ 1, 1] via the formula Q c [ϕ]x = 1 π 1 1 sinc x t x t ϕt dt. 6 Clearly, if we denote by F : L 2 R L 2 R the unitary Fourier transform, then Q c [ϕ]x = χ [ 1,1] x F 1 [ χ [ c,c] ξ F [ϕ] ξ ] x, 7 i.e. Q c represents low-passing followed by time-limiting. Q c relates to F c, defined via 4, by and the eigenvalues µ n of Q n satisfy the identity Q c = c 2π F c F c, 8 µ n = c 2π λ n 2, 9 for all integer n. Moreover, Q c has the same eigenfunctions ψ n as F c. In other words, µ n ψ n x = 1 π 1 1 sincx t x t ψ n t dt, 1 for all integer n and all 1 x 1. Also, Q c is closely related to the operator P c : L 2 R L 2 R, defined via the formula P c [ϕ] x = 1 π sinc x t x t ϕt dt, 11 which is a widely known orthogonal projection onto the space of functions of band limit c > on the real line R. The following theorem about the eigenvalues µ n of the operator Q c, defined via 6, can be traced back to [7]: 1 This convention agrees with that of [4], [5] and differs from that of [9]. 4

5 Theorem 2. Suppose that c > and < α < 1 are positive real numbers, and that the operator Q c : L 2 [ 1, 1] L 2 [ 1, 1] is defined via 6 above. Suppose also that the integer Nc, α is the number of the eigenvalues µ n of Q c that are greater than α. In other words, Nc, α = max {k = 1, 2,... : µ k 1 > }. 12 Then, Nc, α = 2 1 π c + π 2 log 1 α log c + O log c. 13 α According to 13, there are about 2c/π eigenvalues whose absolute value is close to one, order of log c eigenvalues that decay exponentially, and the rest of them are very close to zero. The eigenfunctions ψ n of Q c turn out to be the PSWFs, well known from classical mathematical physics [16]. The following theorem, proved in a more general form in [12], formalizes this statement. Theorem 3. For any c >, there exists a strictly increasing unbounded sequence of positive numbers χ < χ 1 <... such that, for each integer n, the differential equation 1 x 2 ψ x 2x ψ x + χ n c 2 x 2 ψx = 14 has a solution that is continuous on [ 1, 1]. Moreover, all such solutions are constant multiples of the eigenfunction ψ n of F c, defined via 4 above. For all real c > and all integer n, the following inequality holds: The following result provides an upper bound on ψ 2 n1. Theorem 4. For all c > and all natural n, n n + 1 < χ n < n n c ψ 2 n1 < n Elliptic Integrals In this subsection, we summarize several facts about elliptic integrals. These facts can be found, for example, in section 8.1 in [8], and in [21]. The incomplete elliptic integrals of the first and second kind are given, respectively, by the formulae Fy, k = Ey, k = y y dt 1 k 2 sin 2 t, 17 1 k 2 sin 2 t dt, 18 5

6 where y π/2 and k 1. By performing the substitution x = sint, we can write 17 and 18 as Fy, k = siny dx 1 x 2 1 k 2 x 2, 19 Ey, k = siny 1 k 2 x 2 1 x 2 dx. 2 The complete elliptic integrals of the first and second kind are given, respectively, by the formulae where k 1. π Fk = F 2, k = π Ek = E 2, k = π/2 π/2 2.3 Oscillation Properties of Second Order ODEs dt 1 k 2 sin 2 t, 21 1 k 2 sin 2 t dt, 22 In this subsection, we state several well known facts from the general theory of second order ordinary differential equations see e.g. [1]. The following two theorems appear in Section 3.6 of [1] in a slightly different form. Theorem 5 distance between roots. Suppose that ht is a solution of the ODE y t + Qtyt =. 23 Suppose also that x < y are two consecutive roots of ht, and that A 2 Qt B 2, 24 for all x t y. Then, π B < y x < π A. 25 Theorem 6. Suppose that a < b are real numbers, and that g : a, b R is a continuous monotone function. Suppose also that yt is a solution of the ODE in the interval a, b. Suppose furthermore that y t + gt yt =, 26 t 1 < t 2 < t 3 <

7 are consecutive roots of yt. If g is non-decreasing, then t 2 t 1 t 3 t 2 t 4 t If g is non-increasing, then t 2 t 1 t 3 t 2 t 4 t The following theorem is a special case of Theorem 6.2 from Section 3.6 in [1]: Theorem 7. Suppose that g 1, g 2 are continuous functions, and that, for all real t in the interval a, b, the inequality g 1 t < g 2 t holds. Suppose also that the function φ 1, φ 2 satisfy, for all a < t < b, φ 1t + g 1 t φ 1 t =, φ 2t + g 2 t φ 2 t =. 3 Then, φ 2 has a root between every two consecutive roots of φ 1. Corollary 1. Suppose that the functions φ 1, φ 2 are those of Theorem 7 above. Suppose also that φ 1 t = φ 2 t, φ 1t = φ 2t, 31 for some a < t < b. Then, φ 2 has at least as many roots in t, b as φ 1. Proof. By Theorem 7, we only need to show that if t 1 is the minimal root of φ 1 in t, b, then there exists a root of φ 2 in t, t 1. By contradiction, suppose that this is not the case. In addition, without loss of generality, suppose that φ 1 t, φ 2 t are positive in t, t 1. Then, due to 3, and hence < φ 1φ 2 φ 2φ 1 = g 2 g 1 φ 1 φ 2, 32 t1 t g 2 s g 1 s φ 1 sφ 2 sds = [ φ 1sφ 2 s φ 1 sφ 2s ] t 1 t = φ 1t 1 φ 2 t 1, 33 which is a contradiction. 7

8 2.4 Prüfer Transformations In this subsection, we describe the classical Prüfer transformation of a second order ODE see e.g. [1],[22]. Also, we describe a modification of Prüfer transformation, introduced in [3] and used in the rest of the paper. Suppose that we are given the second order ODE d ptu t + qtut =, 34 dt where t varies over some interval I in which p and q are continuously differentiable and have no roots. We define the function θ : I R via ptu t ut = γttanθt, 35 where γ : I R is an arbitrary positive continuously differentiable function. The function θt satisfies, for all t in I, θ t = γt pt sin2 θt qt γ t sin2θt γt cos2 θt. 36 γt 2 One can observe that if u t = for t I, then by 35 Similarly, if u t = for t I, then θ t = kπ, k is integer. 37 θ t = k + 1/2π, k is integer. 38 The choice γt = 1 in 35 gives rise to the classical Prüfer transformation see e.g. section 4.2 in [1]. In [3], the choice γt = qtpt is suggested and shown to be more convenient numerically in several applications. In this paper, this choice also leads to a more convenient analytical tool than the classical Prüfer transformation. Writing 14 in the form of 34 yields pt = 1 t 2, qt = χ n c 2 t 2, 39 for t < min { χn /c,1 }. The equation 35 admits the form ptψ nt ψ n t = ptqt tanθt, 4 which implies that pt ψ θt = atan nt + πmt, 41 qt ψ n t 8

9 where mt is an integer determined for all t by an arbitrary choice at some t = t the role of πmt in 41 is to enforce the continuity of θ at the roots of ψ n. The first order ODE 36 admits the form see [3], [22] θ t = ft + sin2θtvt, 42 where the functions f, v are defined, respectively, via the formulae qt ft = pt = χn c 2 t 2 1 t 2 43 and vt = 1 4 ptq t + qtp t ptqt = 1 2 t 1 t 2 + c 2 t χ n c 2 t Summary In this section, we summarize some of the properties of prolate spheroidal wave functions PSWFs, proved in Section 4. The PSWFs and the related notation were introduced in Section 2.1. Throughout this section, the band limit c > is assumed to be a fixed positive real number. Many properties of the PSWF ψ n depend on whether the eigenvalue χ n of the ODE 14 is greater than or less than c 2. The following simple relation between c, n and χ n is proved in Theorem 14 in Section Proposition 1. Suppose that n 2 is a non-negative integer. If n 2c/π 1, then χ n < c 2. If n 2c/π, then χ n > c 2. If 2c/π 1 < n < 2c/π, then either inequality is possible. In the following proposition, we describe the location of special points roots of ψ n, roots of ψ n, turning points of the ODE 14 that depends on whether χ n > c 2 or χ n < c 2. It is proved in Lemma 1 in Section and is illustrated on Figures 1, 2. Proposition 2. Suppose that n 2 is a positive integer. Suppose also that t 1 < < t n are the roots of ψ n in 1, 1, and x 1 < < x n 1 are the roots of ψ n in t 1, t n. Suppose furthermore that the real number x n is defined via the formula { maximal root of ψ n in 1, 1, if χ n < c 2, x n = 1, if χ n > c Then, χn c < x n < t 1 < x 1 < t 2 < < t n 1 < t n < x n < χn c. 46 9

10 In particular, if χ n < c 2, then t n < x n < χn and ψ n has n + 1 roots in the interval 1, 1; and, if χ n > c 2, then c t n < x n = 1 < and ψ n has n 1 roots in the interval 1, 1. < 1, 47 χn c, 48 The following two inequalities improve the inequality 15 in Section 2.1. Their proof can be found in Theorems 8,9 in Section This is one of the principal analytical results of this paper. The inequalities 49, 5 below are illustrated in Tables 1, 2, 3, 4. Proposition 3. Suppose that n 2 is a positive integer. Suppose also that t n and T are the maximal roots of ψ n and ψ n in the interval 1, 1, respectively. If χ n > c 2, then If χ n < c 2, then π tn χn c 2 t 2 1 t 2 dt < n < 2 π 1 χn c 2 t 2 1 t 2 dt π tn χn c 2 t 2 1 t 2 dt < n < 2 π T χn c 2 t 2 1 t 2 dt. 5 Note that 49 and 5 differ only in the range of integration on their right-hand sides. In the following proposition, we simplify the inequality 49 in Proposition 3. It is proven in Theorem 17 and Corollary 3 in Section Proposition 4. Suppose that n 2 is a positive integer, and that χ n > c 2. Then, n < 2 1 χn c 2 t 2 π 1 t 2 dt = 2 c χn E < n + 3, 51 π χn where the function E : [, 1] R is defined via 22 in Section 2.2. In the following proposition, we describe a relation between χ n and the maximal root t n of ψ n in 1, 1, by providing a lower and upper bounds on 1 t n in terms of χ n and c. It is proved in Theorem 16, 18 in Section Proposition 5. Suppose that n 2 is a positive integer, and that χ n > c 2. Suppose also that t n is the maximal root of ψ n in the interval 1, 1. Then, π 2 /8 χ n c 2 + χ n c πc/2 2 < 1 t n 4π 2 < χ n c χ n c πc 2. 1

11 In the following proposition, we provide yet another upper bound on χ n in terms of n. Its proof can be found in Theorem 12 in Section Proposition 6. Suppose that n 2 is a positive integer, and that χ n > c 2. Then, χ n < π 2 n We observe that, for sufficiently large n, the inequality 53 is even weaker than 15. On the other hand, 53 can be useful for n near 2c/π, as illustrated in Tables 5, 6. The following proposition summarizes Theorem 1 in Section and Theorems 13, 15 in Section It is illustrated in Tables 5, 6, 7, 8. Proposition 7. Suppose that n 2 is a positive integer, and that χ n > c 2. Suppose also that 1 < t 1 < t 2 < < t n < 1 are the roots of ψ n in the interval 1, 1. Suppose furthermore that the functions f, v are defined, respectively, via 43,44 in Section 2.4. Then: For all integer n + 1/2 i n 1, i.e. for all integer i such that t i < t n, π ft i+1 + vt i+1 /2 < t i+1 t i < π ft i. 54 For all integer n + 1/2 i n 1, i.e. for all integer i such that t i < t n, For all integer j = 1,...,n 1, t i+1 t i > t i+2 t i+1 > > t n t n t j+1 t j < π χn The following proposition summarizes Theorem 15 in Section Proposition 8. Suppose that n 2 is an integer, and that χ n < c 2 c 2. Suppose also that 1 < t 1 < t 2 < < t n < 1 are the roots of ψ n in the interval 1, 1. Then, t i+1 t i < t i+2 t i+1 < < t n t n 1, 57 for all integer n + 1/2 i n 1, i.e. for all integer i such that t i < t n. The following proposition summarizes Theorem 4 in Section 2.1 and Theorem 19 in Section 4.2. Proposition 9. Suppose that n is a non-negative integer, and that χ n > c 2. Then, 1 2 < ψ2 n1 < n

12 The following proposition is illustrated on Figures 1, 2. It is proved in Theorem 2 in Section Proposition 1. Suppose that n is a non-negative integer, and that x, y are two arbitrary extremum points of ψ n in 1, 1. If x < y, then If, in addition, χ n > c 2, then ψ n x < ψ n y. 59 ψ n x < ψ n y < ψ n Analytical Apparatus The purpose of this section is to provide the analytical apparatus to be used in the rest of the paper, as well as to prove the results summarized in Section Oscillation Properties of PSWFs In this subsection, we prove several facts about the distance between consecutive roots of PSWFs 5 and find a more subtle relationship between n and χ n 14 than the one given by 15. Throughout this subsection c > is a positive real number and n is a non-negative integer. The principal results of this subsection are Theorems 8, 9, 11, and Special Points of ψ n We refer to the roots of ψ n, the roots of ψ n and the turning points of the ODE 14 as special points. Some of them play an important role in the subsequent analysis. These points are introduced in the following definition. Definition 1 Special points. Suppose that n 2 is a positive integer. We define t 1 < t 2 < < t n to be the roots of ψ n in 1, 1, x 1 < < x n 1 to be the roots of ψ n in t 1, t n, x n via the formula x n = { maximal root of ψ n in 1, 1, if χ n < c 2, 1, if χ n > c This definition will be used throughout all of Section 4. The relative location of some of the special points depends on whether χ n > c 2 or χ n < c 2. This is illustrated in Figures 1, 2 and is described by the following lemma. 12

13 Lemma 1 Special points. Suppose that n 2 is a positive integer. Suppose also that t 1 < < t n and x 1 < < x n are those of Definition 1. Then, χn c < x n < t 1 < x 1 < t 2 < < t n 1 < t n < x n < χn c. 62 In particular, if χ n < c 2, then t n < x n < χn c < 1, 63 and ψ n has n + 1 roots in the interval 1, 1; and, if χ n > c 2, then t n < x n = 1 < and ψ n has n 1 roots in the interval 1, 1. χn c, 64 Proof. Without loss of generality, we assume that ψ n 1 >. 65 Obviously, 65 implies that ψ nt n >. 66 Suppose first that χ n < c 2. Then, due to the ODE 14 in Section 2.1, We combine 14 and 66 to obtain ψ n1 = χ n c 2 2 ψ n 1 <. 67 ψ nt n = 2t n 1 t 2 ψ nt n >. 68 n In addition, we combine 61, 66, 67 to conclude that the maximal root x n of ψ n in 1, 1 satisfies Moreover, 65 implies that, for any root x of ψ n in t n, 1, We combine 65, 69, 7 with 14 to obtain t n < x n < ψ nx = χ n c 2 x 2 1 x 2 ψ n x <. 7 c 2 x 2 n χ n 1 x 2 n = ψ nx n <, 71 ψ n x n 13

14 which implies both 62 and 63. In addition, we combine 14, 63 and 7 to conclude that x n is the only root of ψ n between t n and 1. Thus, ψ n indeed has n+1 roots in 1, 1. Suppose now that χ n > c 2. We combine 14 and 65 to obtain If t n < x < 1 is a root of ψ n, then ψ n1 = χ n c 2 2 ψ n 1 >. 72 ψ nx = χ n c 2 x 2 1 x 2 ψ n x <, 73 therefore ψ n can have at most one root in t n, 1. We combine this observation with 61, 66, 72 and 73 to conclude that, in fact, ψ n has no roots in t n, 1, and hence both 62 and 64 hold. In particular, ψ n has n 1 roots in 1, A Sharper Inequality for χ n In this subsection, we use the modified Prüfer transformation see Section 2.4 to analyze the relationship between n, c and χ n. In particular, this analysis yields fairly tight lower and upper bounds on χ n in terms of c and n. These bounds are described in Theorems 8,9 below. These theorems are not only one of the principal results of this paper, but are subsequently used in the proofs of Theorems 1, 11, 12, 17, 18. We start with developing the required analytical machinery. In the following lemma, we describe several properties of the modified Prüfer transformation see Section 2.4, applied to the prolate differential equation 14. Lemma 2. Suppose that n 2 is a positive integer. Suppose also that the numbers t 1,...,t n and x 1,...,x n are those of Definition 1 in Section 4.1.1, and that the function θ : [ x n, x n ] R is defined via the formula i 1 2 π, if t = ti for some 1 i n, θt = atan 1 t 2 χ n c ψ 2 t 2 n t ψ nt + mt π, if ψ n t, where mt is the number of the roots of ψ n in the interval 1, t. Then, θ has the following properties: θ is continuously differentiable in the interval [ x n, x n ]. θ satisfies, for all x n < t < x n, the differential equation 74 θ t = ft + vt sin2θt, 75 where the functions f, v are defined, respectively, via 43, 44 in Section 2.4. for each integer j 2n, there is a unique solution to the equation θt = j π 2, 76 14

15 for the unknown t in [ x n, x n ]. More specifically, for each i = 1,...,n. In particular, θx n = n π. θ x n =, 77 θt i = i 1 π, 78 2 θx i = i π, 79 Proof. We combine 62 in Lemma 1 with 74 to conclude that θ is well defined for all x n t x n, where x n is given via 61 in Definition 1. Obviously, θ is continuous, and the identities 77, 78, 79 follow immediately from the combination of Lemma 1 and 74. In addition, θ satisfies the ODE 75 in x n, x n due to 36, 4, 42 in Section 2.4. Finally, to establish the uniqueness of the solution to the equation 76, we make the following observation. Due to 74, for any point t in x n, x n, the value θt is an integer multiple of π/2 if and only if t is either a root of ψ n or a root of ψ n. We conclude the proof by combining this observation with 61, 77 and 79. Remark 1. We observe that, due to 77, 78, 79, for all i = 1,...,n, sin2θt i = sin2θx i =, 8 where t 1,...,t n, x 1,...,x n are those of Definition 1 in Section 4.1.1, and θ is that of Lemma 2. This observation will play an important role in the analysis of the ODE 75 throughout the rest of this section. In the following lemma, we prove that θ of Lemma 2 is monotonically increasing. Lemma 3. Suppose that n 2 is a positive integer. Suppose also that the real number x n and the function θ : [ x n, x n ] R are those of Lemma 2 above. Then, θ is strictly increasing in [ x n, x n ], in other words, for all x n < t < x n. θ t >, 81 Proof. We first prove that d dt v t >, 82 f for x n < t < x n, where the functions f, v are defined, respectively, via 43, 44 in 15

16 Section 2.4. We differentiate v/f with respect to t to obtain v p q + q p p p q + q p = = f 4pq q 4q 3/2 p 1/2 = q 3 p 1 [ q1/2 p 1/2 q + 1 p 2 q3/2 p 1/2 p q + q p p q + 2p q + q p q 3/2 p 1/2] = q 5/2 p 3/2 4 = q 5/2 p 3/2 4 [ 3 2 q p + 1 p 2 p q q + q p pq p q + 2p q + q p ] [ 3 2 p2 q q2 p ] 2 q 2 pp p 2 qq >, 83 since, due to 39, pt >, p t = 2 <, qt >, q t = 2c 2 <. 84 We now proceed to prove 81 for < t < x n. Suppose that, by contradiction, there exists < x < x n such that Combined with 75 in Lemma 2 above, 85 implies that and, in particular, that θ x < vx fx + vx sin2θx sin2θx = <, 86 fx fx sin2θx <. 87 Due to Lemma 2 above, there exists an integer n + 1/2 i n such that i 1 π < θx < i π Moreover, due to 77, 78, 79, 85, 87, 88, there exists a point y such that and also t i < x < y < x i x n, 89 θx = θy, θ y >. 9 for otherwise 79 would be impossible. We combine 75 and 9 to obtain 1 + vy fy + vy sin2θy sin2θx = = θ y >, 91 fy fy fy in contradiction to 82, 86 and 87. This concludes the proof of 81 for < t < x n. For x n < t <, the identity 81 follows now from the symmetry considerations. 16

17 The right-hand side of the ODE 75 of Lemma 2 contains a monotone term and an oscillatory term. In the following lemma, we study the integrals of the oscillatory term between various special points, introduced in Definition 1 in Section Lemma 4. Suppose that n 2 is an integer. Suppose also that the real numbers t 1 < < t n and x 1 < < x n are those of Definition 1 in Section 4.1.1, and the function θ : [ x n, x n ] R is that of Lemma 2 above. Suppose furthermore that the function v is defined via 44 in Section 2.4. Then, ti+1 x i xi+1 vt sin2θt dt >, 92 t i+1 xi+1 vt sin2θt dt <, 93 x i vt sin2θt dt <, 94 for all integer n 1/2 i n 1, i.e. for all integer i such that x i < x n. Note that the integral in 94 is the sum of the integrals in 92 and 93. Proof. Suppose that i is a positive integer such that n 1/2 i n 1. Suppose also that the function s : [, n π] [ x n, x n ] is the inverse of θ. In other words, for all η n π, θsη = η. 95 Using 75, 78, 79 in Lemma 2, we expand the left-hand side of 92 to obtain ti+1 x i vt sin2θt dt = θti+1 θx i i+1/2 π i π π/2 vsη sin2η s η dη = vsη sin2η dη fsη + vsη sin2η = vsη + i π sin2η dη fsη + i π + vsη + i π sin2η, 96 from which 92 readily follows due to 44 in Section 2.4 and 81 in Lemma 3. By the same token, we expand the left-hand side of 93 to obtain xi+1 t i+1 vt sin2θt dt = i+1 π i+1/2 π π/2 vsη sin2η dη fsη + vsη sin2η = vsη + i + 1/2 π sin2η dη fsη + i + 1/2 π vsη + i + 1/2 π sin2η, 97 17

18 which, combined with 44 in Section 2.4 and 81 in Lemma 3, implies 93. Finally, for all < η < π/2, sin2η f/vsη + i + 1/2 π sin2η > sin2η f/vsη + i π + sin2η, 98 since the function f/v is decreasing due to 82 in the proof of Lemma 3. The inequality 94 now follows from the combination of 96, 97 and 98. We are now ready to prove one of the principal results of this paper. It is illustrated in Tables 1, 2, 3, 4. Theorem 8. Suppose that n 2 is a positive integer. If χ n > c 2, then If χ n < c 2, then n < 2 π 1 χn c 2 t 2 1 t 2 dt. 99 n < 2 π T χn c 2 t 2 1 t 2 dt, 1 where T is the maximal root of ψ n in 1, 1. Note that 99 and 1 differ only in the range of integration on their right-hand sides. Proof. Suppose that the real numbers 1 x n < t 1 < x 1 < t 2 < < t n 1 < x n 1 < t n < x n 1 11 are those of Definition 1 in Section 4.1.1, and the function θ : [ x n, x n ] R is that of Lemma 2 above. Suppose also that the functions f, v are defined, respectively, via 43, 44 in Section 2.4. If n is even, then we combine 75, 78, 79 in Lemma 2 with 94 in Lemma 4 to obtain n 2 π = < xn x n/2 θ t dt = xn xn ft dt + n 1 i=n/2 xi+1 x i vt sin2θt dt ft dt. 12 If n is odd, then we combine 75, 78, 79 in Lemma 2 with 93, 94 in Lemma 4 to obtain n xn xn 2 π = θ t dt = ft dt + t n+1/2 < xn+1/2 t n+1/2 vt sin2θt dt + xn n 1 i=n+1/2 xi+1 x i vt sin2θt dt ft dt. 13 We combine 12 and 13 with 61 in Lemma 1 to conclude both 99 and 1. 18

19 To prove Theorem 9, we need to develop a number of technical tools. In the following two lemmas, we describe several properties of the equation ft = vt in the unknown t, where f, v are defined, respectively, via 43, 44 in Section 2.4. Lemma 5. Suppose that n is a non-negative integer. Suppose also that the functions f, v are defined, respectively, via 43, 44 in Section 2.4. Suppose furthermore that the real number x n is that of Definition 1 in Section Then, there exists a unique point ˆt in the interval, x n such that fˆt = vˆt. 14 Proof. We observe that, due to 43,44 in Section 2.4, for all < t < x n. Moreover, vt ft > 15 v f =, lim vt =. 16 t x n. t<x n ft We combine 82 in the proof of Lemma 3 with 15 and 16 to conclude both existence and uniqueness of the solution to the equation ft = vt in the unknown < t < x n. Lemma 6. Suppose that n 2 is a positive integer. Suppose also that the real number x n and the function θ : [ x n, x n ] R are those of Lemma 2 above. Suppose furthermore that the point < ˆt < x n is that of Lemma 5 above. Then, n 1 4 π < θˆt < n π. 17 Proof. Suppose that the point < x < x n is defined via the formula x = θ 1 n 1 π, 18 4 where θ 1 denotes the inverse of θ. By contradiction, suppose that 17 does not hold. In other words, < ˆt < x. 19 It follows from the combination of Lemma 5, 82 in the proof of Lemma 3, and 19, that fx < vx. On the other hand, due to 75 in Lemma 2 and 18, θ x = fx + vx sin2θx = fx + vx sin 2nπ π = fx vx <, 11 2 in contradiction to 81 in Lemma 3. 19

20 In the following three lemmas, we study some of the properties of the ratio f/v, where f, v are defined, respectively, via 43, 44 in Section 2.4. Lemma 7. Suppose that n is a non-negative integer, and that the functions f, v are defined, respectively, via 43, 44 in Section 2.4. Then, for all real < t < 1, d f t = h t a ft, 111 dt v where the real number a > is defined via the formula a = χ n c 2, 112 and, for all < t < 1, the function h t :, R is defined via the formula h t a = 4t6 + 2a 6 t a t 2 + 2a a + 1 t a 2t Moreover, for all real < t < min { a,1}, d f t 3 ft. 114 dt v 2 Proof. The identity 111 is obtained from 43, 44 via straightforward algebraic manipulations. To establish 114, it suffices to show that, for a fixed < t < 1, inf a We start with observing that, for all < t < 1, { ht a : t 2 < a < } lim h ta = 6, a t 2, a>t 2 lim h ta = 2 a t Then, we differentiate h t a, given via 113, with respect to a to obtain dh t da a = 2 1 t 2 t a 2t 2 3 6t 4 + a 9 t 2 + a It follows from 116, 117, that if t 2 < â t < is a local extremum of h t a, then â t = 6t4 + 9t 2 1 t > t 2, 118 which is possible if and only if 1 > t 2 > 1/7. Then we substitute â t, given via 118, into 113 to obtain It is trivial to verify that inf t { ht, â t : ht, â t = t4 + 14t 2 1 8t } 1 7 < t < 1 = lim ht, â t = 3 t 1, t> Now 115 follows from the combination of 116, 118, 119 and 12. 2

21 Lemma 8. Suppose that n 2 is a positive integer, and that t n is the maximal zero of ψ n is the interval 1, 1. Suppose also that the real number Z is defined via the formula Then, for all < t t n, Z = π vt < ft Z, 122 where the functions f, v are defined, respectively, via 43,44 in Section 2.4. Proof. Due to 82 in the proof of Lemma 3, the function f/v decreases monotonically in the interval, t n, and therefore, to prove 122, it suffices to show that ft n vt n > 1 Z = 1 + 3π Suppose that the point ˆt is that of Lemma 5. Suppose also that the real number x n and the function θ : [ x n, x n ] R are those of Lemma 2. Suppose furthermore that the function s : [, n π] [ x n, x n ] is the inverse of θ. In other words, for all η n π, t n θsη = η. 124 We combine Lemma 2, Lemma 3, Lemma 5, Lemma 6 and Lemma 7 to obtain ft n vt n 1 = f ˆt f t n = v v ˆt d f θˆt d dt f v sη dη t dt = dt v fsη + vsη sin2η > which implies 123. n 1/4π n 1/2π n 1/4π n 1/2π θt n d dt f v sη dη fsη + vsη sin2η > d dt f v sη dη fsη > π = 3π 8, 125 Lemma 9. Suppose that n 2 and n+1/2 i n 1 are positive integers. Suppose also that the real number x n and the function θ : [ x n, x n ] R are those of Lemma 2. Suppose furthermore that < δ < π/4 is a real number, and that the real number Z δ is defined via the formula [ Z δ = ] π δ 1, Z sin2δ where Z is defined via 121 in Lemma 8 above. Then, vt < ft Z δ, 127 for all < t si + 1/2 π δ, where the functions f, v are defined, respectively, via 43, 44 in Section 2.4, and the function s : [, n π] [ x n, x n ] is the inverse of θ. 21

22 Proof. Suppose that the point t δ is defined via the formula t δ = si + 1/2 π δ. 128 Due to 82 in the proof of Lemma 3, the function f/v decreases monotonically in the interval, t δ, and therefore to prove 127 it suffices to show that ft δ vt δ > 1 = π Z δ δ Z sin2δ We observe that, due to Lemma 3, sin2θt sin2δ, 13 for all t δ t si+1/2π. We combine 128, 13 with Lemma 2, Lemma 3, Lemma 6, Lemma 7 and Lemma 8 to obtain ft δ fsi + 1/2π vt δ vsi + 1/2π = si+1/2π t δ i 1/2π i 1/2π δ d dt d dt f i 1/2π t dt = v f v sη fsη i 1/2π δ dη 1 + v/fsη sin2δ > d dt f v sη dη fsη + vsη sin2η > 3 2 δ Z sin2δ. 131 We combine 131 with 122 in Lemma 8 to obtain 129, which, in turn, implies 127. In the following two lemmas, we estimate the rate of decay of the ratio f/v and its relationship with θ of the ODE 75 in Lemma 2. Lemma 1. Suppose that n 2 and n + 1/2 i n 1 are positive integers. Suppose also that the real number x n and the function θ : [ x n, x n ] R are those of Lemma 2. Suppose furthermore that < δ < π/4 is a real number. Then, f v siπ δ f v siπ δ + π/2 > 2 sin2δ, 132 where the functions f, v are defined, respectively, via 43, 44 in Section 2.4, and the function s : [, n π] [ x n, x n ] is the inverse of θ. Proof. We observe that, due to Lemma 2 and Lemma 3, sin2θt >,

23 for all siπ < t < siπ δ + π/2. We combine 133 with Lemma 2, Lemma 3, Lemma 6, Lemma 7 and Lemma 9 to obtain fsiπ fsiπ δ + π/2 vsiπ vsiπ δ + π/2 = siπ δ+π/2 siπ iπ δ+π/2 iπ d dt d dt f v f iπ δ+π/2 t dt = v sη fsη iπ d dt f v sη dη fsη + vsη sin2η > dη 1 + v/fsη > 3 π 2 2 δ 1, Z δ where Z δ is defined via 126 in Lemma 9. We also observe that, due to Lemma 2 and Lemma 3, sin2θt <, 135 for all siπ δ < t < siπ. We combine 135 with Lemma 2, Lemma 3, Lemma 6 and Lemma 7 to obtain fsiπ δ vsiπ δ fsiπ vsiπ = siπ siπ δ iπ iπ δ d dt f v d dt f v iπ t dt = iπ δ sη dη fsη d dt f v sη dη fsη + vsη sin2η > > 3 δ Next, suppose that the function h : [, π/4] R is defined via the formula hδ = 3 π 2 2 δ δ 2 sin2δ, Z δ 2 where Z δ is defined via 126 in Lemma 9. One can easily verify that min δ {hδ : δ π/4} > 1 25, 138 and, in particular, that hδ > for all δ π/4. We combine 134, 136, 137 and 138 to obtain 132. Lemma 11. Suppose that n 2 and n + 1/2 i n 1 are positive integers. Suppose also that the real number x n and the function θ : [ x n, x n ] R are those of Lemma 2. Suppose furthermore that < δ < π/4 is a real number. Then, f v siπ + δ π/2 f v siπ + δ > 2 sin2δ, 139 where the functions f, v are defined, respectively, via 43, 44 in Section 2.4, and the function s : [, n π] [ x n, x n ] is the inverse of θ. 23

24 Proof. We observe that, due to Lemma 2 and Lemma 3, sin2θt >, 14 for all siπ < t < siπ + δ. We combine 14 with Lemma 2, Lemma 3, Lemma 6, Lemma 7 and Lemma 9 to obtain fsiπ fsiπ + δ vsiπ vsiπ + δ = siπ+δ siπ iπ+δ iπ d dt f v d dt f v iπ+δ t dt = iπ sη fsη d dt f v sη dη fsη + vsη sin2η > dη 1 + v/fsη > 3 2 δ, Z δ where Z δ is defined via 126 in Lemma 9. We also observe that, due to Lemma 2 and Lemma 3, sin2θt <, 142 for all siπ + δ π/2 < t < siπ. We combine 142 with Lemma 2, Lemma 3, Lemma 6 and Lemma 7 to obtain fsiπ + δ π/2 vsiπ + δ π/2 fsiπ vsiπ = siπ siπ+δ π/2 iπ iπ+δ π/2 d dt f v d dt f v iπ t dt = iπ+δ π/2 sη dη fsη Obviously, for all < δ < π/4, 3 2 δ + 3 π 1 + Z δ 2 2 δ > 3 π 2 2 δ d dt f v sη dη fsη + vsη sin2η > > 3 π 2 2 δ δ Z δ 2 We combine 141,143, 144 with 137, 138 in the proof of Lemma 1 to obtain 139. In the following lemma, we analyze the integral of the oscillatory part of the right-hand side of the ODE 75 between consecutive roots of ψ n. This lemma can be viewed as an extention of Lemma 4, and is used in the proof of Theorem 9 below. Lemma 12. Suppose that n 2 is an integer, 1 < t 1 < t 2 < < t n < 1 are the roots of ψ n in the interval 1, 1, and x 1 < < x n 1 are the roots of ψ n in the interval t 1, t n. Suppose also, that the real number x n and the function θ : [ x n, x n ] R are those of 24

25 Lemma 2 above. Suppose furthermore that the function v is defined via 44 in Section 2.4. Then, ti+1 t i vt sin2θt dt >, 145 for all integer n + 1/2 i n 1, i.e. for all integer i such that t i < t n. Proof. Suppose that i is a positive integer such that n 1/2 i n 1. Suppose also that the function s : [, n π] [ x n, x n ] is the inverse of θ. In other words, for all η n π, Due to 97 in the proof of Lemma 4 above, xi vt sin2θt dt = t i π/2 θsη = η. 146 vsiπ + η π/2 sin2η dη fsiπ + η π/2 vsiπ + η π/2 sin2η. 147 We proceed to compare the integrand in 147 to the integrand in 96 in the proof of Lemma 4. First, for all < η < π/4, 1 f/vsiπ + η π/2 sin2η < 1 f/vsiπ + η + sin2η, 148 due to 139 in Lemma 11. Moreover, for all π/4 < η < π/2, we substitute δ = π/2 η to obtain 1 f/vsiπ + η π/2 sin2η = 1 f/vsiπ δ sin2δ < 1 f/vsiπ δ + π/2 + sin2δ = 1 f/vsiπ + η + sin2η, 149 due to 132 in Lemma 1. We combine 96 in the proof of Lemma 4 with 147, 148, 149 to obtain 145. The following theorem is a counterpart of Theorem 8 above. It is illustrated in Tables 1, 2, 3, 4. Theorem 9. Suppose that n 2 is a positive integer. Suppose also that t n is the maximal root of ψ n in 1, 1. Then, tn χn c 2 t 2 π 1 t 2 dt < n

26 Proof. Suppose that the real numbers 1 x n < t 1 < x 1 < t 2 < < t n 1 < x n 1 < t n < x n and the function θ : [ x n, x n ] R are those of Lemma 2 above. Suppose also that the functions f, v are defined, respectively, via 43,44 in Section 2.4. If n is odd, then we combine 75, 78, in Lemma 2 with 145 in Lemma 12 to obtain n 1 2 π = > tn t n+1/2 θ t dt = tn tn ft dt + n 1 i=n+1/2 ti+1 t i vt sin2θt dt ft dt. 152 If n is even, then we combine 75, 78, 79 in Lemma 2 with 92 in Lemma 4 and 145 in Lemma 12 to obtain n 1 2 π = > tn x n/2 θ t dt = tn/2+1 tn ft dt + x n/2 vt sin2θt dt + tn n 1 i=n/2+1 ti+1 t i vt sin2θt dt ft dt. 153 We combine 152 and 153 to conclude 15. Corollary 2. Suppose that n 2 is a positive integer, and that χ n > c 2. Suppose also that t n is the maximal root of ψ n in the interval 1, 1. Then, c χn E asint n, < n < 2 c χn E, 154 π χn π χn where Ey, k and Ek are defined, respectively, via 18 and 22 in Section 2.2. Proof. It follows immediately from 2, 99 in Theorem 8 and 15 in Theorem 9. The following theorem, illustrated in Tables 7, 8, provides upper and lower bounds on the distance between consecutive roots of ψ n inside 1, 1. Theorem 1. Suppose that n 2 is a positive integer, and that χ n > c 2. Suppose also that 1 < t 1 < t 2 < < t n < 1 are the roots of ψ n in the interval 1, 1. Suppose furthermore that the functions f, v are defined, respectively, via 43,44 in Section 2.4. Then, π ft i+1 + vt i+1 /2 < t i+1 t i < π ft i, 155 for all integer n + 1/2 i n 1, i.e. for all integer i such that t i < t n. 26

27 Proof. Suppose that the function θ : [ 1, 1] R is that of Lemma 2. We observe that f is increasing in, 1 due to 43 in Section 2.4, and combine this observation with 75, 78, in Lemma 2 and 145 in Lemma 12 to obtain π = > ti+1 t i θ t dt = ti+1 ti+1 t i ft dt + ti+1 t i vt sin2θt dt t i ft dt > t i+1 t i ft i, 156 which implies the right-hand side of 155. As in Lemma 2, suppose that x i is the zero of ψ n in the interval t i, t i+1. We combine 75, 78, 78 in Lemma 2 and 92, 93 in Lemma 4 to obtain xi t i ft dt > xi t i θ t dt = π = ti+1 x i θ t dt > ti+1 x i ft dt. 157 Since f is increasing in t i, t i+1 due to 43 in Section 2.4, the inequality 157 implies that x i t i > t i+1 x i. 158 Moreover, we observe that v is also increasing in, 1. We combine this observation with 158, 75, 78, in Lemma 2 and 92, 93 in Lemma 4 to obtain ti+1 ti+1 ti+1 π = θ t dt < ft dt + vt sin2θt dt t i t i x i < t i+1 t i ft i+1 + t i+1 x i vt i+1 < t i+1 t i ft i+1 + t i+1 t i 2 which implies the left-hand side of 155. vt i+1, 159 The following theorem is a direct consequence of Theorem 8 above. Theorem 11. Suppose that n 2 is a positive integer. If n 2c/π, then χ n > c 2. Proof. Suppose that χ n < c 2, and T is the maximal root of ψ n in, 1, as in Theorem 8 above. Then, n < 2 T χn c 2 t 2 π 1 t 2 dt = 2c T χn /c 2 t 2 π 1 t 2 dt < 2c π T < 2c π, 16 due to 1 in Theorem Elimination of the First-Order Term of the Prolate ODE In this subsection, we analyze the oscillation properties of ψ n via transforming the ODE 14 into a second-order linear ODE without the first-order term. The following lemma is the principal technical tool of this subsection. 27

28 Lemma 13. Suppose that n is a non-negative integer. Suppose also that that the functions Ψ n, Q n : 1, 1 R are defined, respectively, via the formulae Ψ n t = ψ n t 1 t and Q n t = χ n c 2 t 2 1 t t 2 2, 162 for 1 < t < 1. Then, Ψ nt + Q n t Ψ n t =, 163 for all 1 < t < 1. Proof. We differentiate Ψ n with respect to t to obtain Ψ nt = ψ nt 1 t 2 ψ n t t 1 t Then, using 164, we differentiate Ψ n with respect to t to obtain Ψ nt = ψ nt 1 t 2 ψ nt 2t 1 t t 2 1 ψ nt 2 + t 2 / 1 t 2 1 t 2 = ψ nt 1 t 2 ψ nt 2t ψ nt 1 t t 2 = = 1 1 t t 2 [ 1 t 2 ψ nt 2t ψ nt ψ ] nt 1 t [ 2 ψ n t χ n c 2 t 2 ψ ] nt 1 t 2 χn c 2 t 2 = Ψ n t 1 t t We observe that 163 follows from 165. In the next theorem, we provide an upper bound on χ n in terms of n. The results of the corresponding numerical experiments are reported in Tables 5, 6. Theorem 12. Suppose that n 2 is a positive integer, and that χ n > c 2. Then, χ n < π 2 n Proof. Suppose that the functions Ψ n, Q n : 1, 1 R are those of Lemma 13 above. We observe that, since χ n > c 2, Q n t > χ n + 1,

29 for 1 < t < 1. Suppose now that t n is the maximal root of ψ n in 1, 1. We combine 167 with 163 in Lemma 13 above and Theorem 7, Corollary 1 in Section 2.3 to obtain the inequality t n 1 π χn Then, we combine 168 with Theorem 9 above to obtain n > tn χn c 2 t 2 π 1 t 2 dt > t n χn π χn 1 > 2 χn 1, 169 π π χn + 1 π which implies 166. The following theorem is a consequence of the proof of Theorem 12. Theorem 13. Suppose that n 2 is a positive integer, and that χ n > c 2. Suppose also that t 1 < < t n are the roots of ψ n in 1, 1. Then, for all j = 1, 2,...,n 1. t j+1 t j < π χn + 1, 17 Proof. The inequality 17 follows from the combination of 167 in the proof of Theorem 12, 163 in Lemma 13 and Theorem 7, Corollary 1 in Section 2.3. The following theorem extends Theorem 11 in Section Theorem 14. Suppose that n 2 is a positive integer. If n 2c/π 1, then χ n < c 2. If n 2c/π, then χ n > c 2. If 2c/π 1 < n < 2c/π, then either inequality is possible. Proof. Suppose that χ n > c 2, and that the functions Ψ n, Q n : 1, 1 R are those of Lemma 13 above. Suppose also that t 1 < < t n are the roots of ψ n in 1, 1. We observe that, due to 162 in Lemma 13, Q n t = c 2 + χ n c 2 1 t t 2 2 > c We combine 171 with 163 in Lemma 13 above and Theorem 5 in Section 2.3 to conclude that t j+1 t j < π c,

30 for all j = 1,...,n 1, and, moreover, 1 t n < π c. 173 We combine 172 with 173 to obtain the inequality 2 1 π < 2t n = t n t 1 < n 1 π c c, 174 which implies that n > 2 c π We conclude the proof by combining Theorem 11 in Section with 175. The following theorem is yet another application of Lemma 13 above. Theorem 15. Suppose that n 2 is a positive integer. Suppose also that 1 < t 1 < t 2 < < t n < 1 are the roots of ψ n in the interval 1, 1. Suppose furthermore that i is an integer such that t i < t n, i.e. n + 1/2 i n 1. If χ n > c 2, then If χ n < c 2 c 2, then t i+1 t i > t i+2 t i+1 > > t n t n t i+1 t i < t i+2 t i+1 < < t n t n Proof. Suppose that the functions Ψ n, Q n : 1, 1 R are those of Lemma 13 above. If χ n > c 2, then, due to 162 in Lemma 13, Q n t = c 2 + χ n c 2 1 t t is obviously a monotonically increasing function. We combine this observation with 163 of Lemma 13 and 28 of Theorem 6 in Section 2.3 to conclude 176. Suppose now that χ n < c 2 c Suppose also that the function P n : 1, R is defined via the formula P n y = Q n 1 1 = y 2 + χ n c 2 y + c 2, 18 y for 1 < y <. Obviously, 1 Q n t = P n 1 t

31 Suppose also that y is defined via the formula y = 1 1 χ n /c 2 = c 2 c χ n We combine 179, 18 and 182 to conclude that, for 1 < y < y, P ny = 2y c 2 χ n < 2y c 2 χ n = 2c2 c 2 χ n 2 Moreover, due to 18, 182, 183, c 2 χ n <. 183 c 2 2 P n y > P n y = χ n c 2 >, 184 for all 1 < y < y. We combine 18, 181, 182, 183 and 184 to conclude that Q n is monotonically decreasing and strictly positive in the interval, χ n /c. We combine this observation with 29 of Theorem 6 in Section 2.3, 63 of Lemma 1, and 163 of Lemma 13 to conclude 177. Remark 2. Numerical experiments confirm that there exist real c > and integer n > such that c 2 c 2 < χ n < c 2 and neither of 176, 177 is true. In the following theorem, we provide an upper bound on 1 t n, where t n is the maximal root of ψ n in the interval 1, 1. Theorem 16. Suppose that n 2 is a positive integer, and that χ n > c 2. Suppose also that t n is the maximal root of ψ n in the interval 1, 1. Then, Moreover, c 2 1 t n 2 + χ n c t n 1 t n < π t n < 4π 2 χ n c χ n c πc 2. Proof. Suppose that the functions Ψ n, Q n : 1, 1 R are those of Lemma 13 above. Since χ n > c 2, the function Q n is monotonically increasing, i.e. for all t n t < 1. We consider the solution ϕ n of the ODE with the initial conditions Q n t n Qt, 187 ϕ nt + Q n t n ϕ n t =, 188 ϕt n = Ψ n t n =, ϕ t n = Ψ nt n

32 The function ϕ n has a root y n given via the formula y n = t n + π Qn t n. 19 Suppose, by contradiction, that y n 1. Then, due to the combination of 163 of Lemma 13, Theorem 7, Corollary 1 in Section 2.3, and 187 above, Ψ n has a root in the interval t n, y n, in contradiction to 161. Therefore, t n + π > Qn t n We rewrite 191 as and plug 162 into 192 to obtain the inequality 1 t n 2 Q n t n < π 2, 192 c 2 1 t n 2 + χ n c t n t n 1 + t n 2 < π2, 193 which immediately yields 185. Since 1 t n is positive, 193 implies that 1 t n is bounded from above by the maximal root x max of the quadratic equation given via the formula c 2 x 2 + χ n c 2 x π 2 =, x max = 1 4c 2 χn c π 2 c 2 χ n c 2 = 16π2 c 2 1 4c 2 χ n c χ n c π 2 c2, which implies 186. The following theorem uses Theorem 16 to simplify the inequalities 99 in Theorem 8 and 15 in Theorem 9 in Section Theorem 17. Suppose that n 2 is a positive integer, and that χ n > c 2. Suppose also that t n is the maximal root of ψ n in the interval 1, 1. Then, 2 1 χn c 2 t 2 π 1 t 2 dt < Moreover, n < 2 π t n 1 χn c 2 t 2 1 t 2 dt < n

33 Proof. We observe that, for t n t < 1, χ n c 2 t t < χ n c 2 t 2 n 1 + t n = χ n c t n + c2 c t t 2 n 1 + t n = c 2 1 t n + χ n c t n. 198 We combine 198 with 185 in Theorem 16 to obtain the inequality χ n c 2 t t < π2 1 t n, 199 valid for t n t < 1. We conclude from 199 that 1 χn c 2 t 2 π 1 dt π t n 1 t 2 dt < = 2 1 t n = 2π, 2 1 tn t n 1 t 1 tn which implies 196. The inequality 197 follows from the combination of 196, 99 in Theorem 8 and 15 in Theorem 9 in Section Corollary 3. Suppose that n 2 is a positive integer, and that χ n > c 2. Then, n < 2 c χn E < n + 3, 21 π χn where Ek is defined via 22 in Section 2.2. Proof. The inequality 21 follows immediately from the combination of 22 in Section 2.2 and 197 in Theorem 17 above. The following theorem extends Theorem 16 above by providing a lower bound on 1 t n, where t n is the maximal root of ψ n in the interval 1, 1. Theorem 18. Suppose that n 2 is a positive integer, and that χ n > c 2. Suppose also that t n is the maximal root of ψ n in the interval 1, 1. Then, π 2 /8 χ n c 2 + χ n c πc/2 2 < 1 t n. 22 Proof. We combine the inequalities 99 in Theorem 8 and 15 in Theorem 9 in Section to conclude that 1 < 2 1 χn c 2 t 2 π 1 t 2 dt. 23 We combine 23 with 198 in the proof of Theorem 17 above to obtain 1 < 2 χ n c c π 1 + t 2 dt 1 t n n 1 t t n t n < 4 π c 2 1 t n 2 + χ n c 2 1 t n

34 We rewrite 24 as c 2 1 t n 2 + χ n c 2 1 t n π2 > Since 1 t n is positive, 25 implies that 1 t n it is bounded from below by the maximal root x max of the quadratic equation given via the formula c 2 x 2 + χ n c 2 2 x π2 =, x max = 1 2c 2 χn c π 2 c 2 /4 χ n c 2 = π2 c 2 8c 2 1 χ n c 2 + χ n c π 2 c 2 /4, 27 which implies Growth Properties of PSWFs In this subsection, we establish several bounds on ψ n and ψ n. Throughout this subsection c > is a fixed positive real number. The principal results of this subsection are Theorems 19, 2, 21. The following lemma is a technical tool to be used in the rest of this subsection. Lemma 14. Suppose that n is a non-negative integer, and that the functions p, q : R R are defined via 39 in Section 2.4. Suppose also that the functions Q, Q :, min { χn /c,1 } R are defined, respectively, via the formulae Qt = ψnt 2 + pt qt ψ nt 2 = ψ 2 1 t 2 ψ n t + nt 2 χ n c 2 t 2 28 and Qt = pt qt Qt = 1 t 2 χn c 2 t 2 ψ 2 nt + 1 t 2 ψ nt Then, Q is increasing in the interval, min { χn /c,1 }, and Q is decreasing in the interval, min { χn /c,1 }. Proof. We differentiate Q, defined via 28, with respect to t to obtain 2c Q t = 2 ψ n t ψ nt 2 t 1 t 2 + χ n c 2 t 2 2 2t χ n c 2 t 2 ψ nt t 2 χ n c 2 t 2 ψ nt ψ nt