Detailed analysis of prolate quadratures and interpolation formulas

Transcription

1 As demonstrated by Slepian et. al. in a sequence of classical papers see [33], [34], [7], [35], [36]), prolate spheroidal wave functions PSWFs) provide a natural and efficient tool for computing with bandlimited functions defined on an interval. As a result, PSWFs are becoming increasing popular in various areas in which such function occur - this includes physics e.g. wave phenomena, fluid dynamics), engineering e.g. signal processing, filter design), etc. To use PSWFs as a computational tool, one needs fast and accurate numerical algorithms for the evaluation of PSWFs and related quantities, as well as for the construction of quadratures, interpolation formulas, etc. For the last half a century, substantial progress has been made in design of such algorithms - this includes both classical results see e.g. [4]) as well as more recent developments see e.g. [38]). The complexity of many of the existing algorithms, however, is at least quadratic in the band limit c. For example, the evaluation of the nth eigenvalue of the prolate integral operator requires at least Oc 2 ) operations see e.g. [38]); also, the construction of accurate quadrature rules for the integration of bandlimited functions of band limit c requires Oc 3 ) operations see e.g. [6]). Therefore, while the existing algorithms are quite satisfactory for moderate values of c e.g. c 0 3 ), they tend to be relatively slow when c is large e.g. c 0 4 ). In this paper, we describe several numerical algorithms for the evaluation of PSWFs and related quantities, and design a class of PSWF-based quadratures for the integration of bandlimited functions. Also, we perform detailed analysis of the related properties of PSWFs. While the analysis is somewhat involved, the resulting numerical algorithms are quite simple and efficient in practice. For example, the evaluation of the nth eigenvalue of the prolate integral operator requires On+c) operations; also, the construction of accurate quadrature rules for the integration of bandlimited functions of band limit c requires Oc) operations. Our results are illustrated via several numerical experiments. Detailed analysis of prolate quadratures and interpolation formulas Andrei Osipov, Vladimir Rokhlin Research Report YALEU/DCS/TR-458 Yale University June 28, 202 This author s research was supported in part by the AFOSR grant #FA This author s research was supported in part by the ONR grants #N , #N , the AFOSR grant #FA , and the ONR / Telcordia grant #N C-076 / PO# This author has a significant financial interest in the Fast Mathematical Algorithms and Hardware corporation FMAHc) of Connecticut. Approved for public release: distribution is unlimited. Keywords: bandlimited functions, prolate spheroidal wave functions, quadratures, interpolation

2 Contents Outline 3. Quadratures for Bandlimited Functions Intuition Behind Quadrature Weights Overview of the Analysis Partial Fractions Expansion of /ψ n Quadrature Weights Mathematical and Numerical Preliminaries 0 2. Prolate Spheroidal Wave Functions Legendre Polynomials and PSWFs Elliptic Integrals Oscillation Properties of Second Order ODEs Growth Properties of Second Order ODEs Prüfer Transformations Numerical Tools Newton s Method The Taylor Series Method for the Solution of ODEs A Second Order Runge-Kutta Method Power and Inverse Power Methods Sturm Sequence Miscellaneous tools Summary 3 4 Analytical Apparatus Oscillation Properties of PSWFs Elimination of the First-Order Term of the Prolate ODE Growth Properties of PSWFs Transformation of the Prolate ODE into a 2 2 System The Behavior of ψ n in the Upper-Half Plane Partial Fractions Expansion of /ψ n Contribution of the Head of the Series 8) Contribution of the Tail of the Series 8) Bound on the Right-Hand Side of 8) PSWF-based Quadrature and its Properties Expansion of ϕ j into a Prolate Series Quadrature Error The Principal Result Quadrature Weights Numerical Algorithms Evaluation of χ n and ψ n x), ψ nx) for x Evaluation of λ n Evaluation of the Quadrature Nodes

3 5.4 Evaluation of the Quadrature Weights Evaluation of ψ n and its roots outside, ) Evaluation of ψ n x) for x > Evaluation of ψ nx) for x > Evaluation of the roots of ψ n in, ) Numerical Results Properties of PSWFs Illustration of Results from Section Illustration of Results from Section Illustration of Results from Section Illustration of Results from Section Performance of the Quadrature Quadrature Error and its Relation to λ n Quadrature Weights Outline. Quadratures for Bandlimited Functions The principal goal of this paper is a quadrature designed for the integration of bandlimited functions of a specified band limit c > 0. A function f : R R is bandlimited of band limit c > 0, if there exists a function σ L 2 [, ] such that fx) = σt) e icxt dt. ) In other words, the Fourier transform of a bandlimited function is compactly supported. While ) defines f for all real x, one is often interested in bandlimited functions, whose argument is confined to an interval, e.g. x. Such functions are encountered in physics wave phenomena, fluid dynamics), engineering signal processing), etc. see e.g. [33], [0], [29]). By quadrature we mean a set of nodes and weights < t n) < < t n) n < 2) W n),...,w n) n. 3) If f :, ) R is a bandlimited function, we use the quadrature to approximate the integral of f over the interval, ) by a finite sum; more specifically, ft) dt n j= W n) j f t n) j ). 4) 3

4 About half a century ago it was observed that the eigenfunctions of the integral operator F c : L 2 [, ] L 2 [, ], defined via the formula F c [ϕ] x) = ϕt)e icxt dt, 5) provide a natural tool for dealing with bandlimited functions, defined on the interval [, ]. Moreover, it was observed see [34], [7], [35]) that the eigenfunctions of F c are precisely the prolate spheroidal wave functions PSWFs) of band limit c, well known from the mathematical physics see, for example, [24], [0]). Therefore, when designing a quadrature for the integration of bandlimited functions of band limit c > 0, it is natural to require that this quadrature integrate several first PSWFs of band limit c with high accuracy. We formulate the principal objective of this paper in a more precise manner, as follows. Principal goal of this paper. Suppose that c > 0 is a real number. For every integer n > 0, we define a quadrature of order n for the integration of bandlimited functions of band limit c over, )) by specifying n nodes and n weights see 2), 3)). Suppose also that ε > 0. We require that, for sufficiently large n, the quadrature of order n integrate the first n PSWFs of band limit c up to the error ε. More specifically, we find the integer M = Mc, ε) such that, for every integer n M and all integer m = 0,,...,n, n ) ψ m t) dt W n) j ψ m t n) j ε, 6) where ψ m :, ) R is the mth PSWF of band limit c see Section 2.). j= Quadratures for the integration of bandlimited functions which satisfy 6) have already been discussed in the literature, for example: Quadrature. Suppose that n > 0 is an integer. The existence and uniqueness of n nodes and weights, such that 6) holds for ε = 0 and all m = 0,,...,2n, was first observed more than 00 years ago see, for example, [5], [6], [2], [22]) for all Chebyshev systems, of which PSWFs are a special case see [38]). Although numerical algorithms for the design of this optimal quadrature were recently constructed see [6], [20], [39]), they tend to be rather expensive require order n 3 operations with a large proportionality constant). Quadrature 2. Another quadrature was suggested in [38]. The PSWF ψ n has n roots t,...,t n in the interval, ) see Theorem in Section 2.); the idea is to use these roots as the quadrature nodes, solve the linear system of n equations n ψ m t j )W j = j= ψ m t) dt for the unknowns W,...,W n, and use the resulting weights and nodes to define a quadrature for the integration of functions of band limit 2c. This approach is justified by the generalization of the Euclid s division algorithm for PSWFs see [38]), and is less expensive computationally than the previous one its cost is dominated by the cost of solving the n m=0 7) 4

5 linear system 7)). The same quadrature can be used to integrate functions of band limit c, since 7) implies that 6) holds with ε = 0, for all m = 0,...,n. In this paper, we describe another quadrature whose nodes are the n roots of ψ n in, ). However, its weights differ from the solution of 7), and can be evaluated in On) operations see Section 4.4 and Section 5 below). Thus, the quadratures of this paper are much faster to evaluate than those described above. Moreover, 6) ensures that their accuracy is similar to that of Quadrature 2. Also, their nodes and weights can be used as starting points for the scheme that computes the optimal Quadrature. In order to define the weights, to make sure that 6) holds and to be able to compute them efficiently, we need to analyze the PSWFs in a somewhat detailed manner. This analysis will be preceded by a heuristic explanation, which provides some intuition as well as prevents one from the danger of not seeing the forest for the trees see Section.2 below). Section.3 contains a short overview of the analysis. Section 2 contains mathematical and numerical preliminaries, to be used in the rest of the paper. In Section 3, we summarize the principal analytical results of the paper. Section 4 contains the corresponding theorems and proofs. Section 5 contains the description and analysis of the numerical algorithms for the evaluation of the quadrature and some related quantities. In Section 6, we report the results of several numerical experiments..2 Intuition Behind Quadrature Weights We recall the following classical interpolation problem. Suppose that t,...,t n are n distinct points in the interval, ). We need to find the real numbers W,...,W n such that pt) dt = n W i pt i ), 8) for all the polynomials p of degree at most n. In other words, the quadrature with nodes t,...,t n and weights W,...,W n integrates all the polynomials of degree up to n exactly see 2), 3), 4)). To solve the problem, one constructs n polynomials l,...,l n of degree n with the property { 0 i j, l j t i ) = 9) i = j for every integer i, j =,...,n see, for example,[4]). It is easy to verify that, for every j =,...,n, the polynomial l j is defined via the formula l j t) = i= w n t) w nt j ) t t j ), 0) where w n is the polynomial of degree n whose roots are precisely t,...,t n. The weights W,...,W n are defined via the formula W j = l j t) dt = w nt j ) w n t) dt t t j, ) 5

6 for every integer j =,...,n. We observe that a single function w n is used to define all the n weights; also, w n is a polynomial of degree n, and hence does not belong to the space of the polynomials of degree up to n. In our case, the basis functions are the PSWFs and not the polynomials. Suppose that the roots t,...,t n of ψ n in the interval, ) are chosen to be the nodes of the quadrature. If we choose the weights W,...,W n such that the resulting quadrature integrates the first n PSWFs exactly, this will lead to the linear system 7), and hence to Quadrature 2 from Section.. Instead, we define the weights via using ψ n in the same way we used w n in ). More specifically, similar to 0), for every integer j =,...,n, we define the function ϕ j :, ) R via the formula ϕ j t) = We observe that, for every integer i, j =,...,n, { 0 i j, ϕ j t i ) = i = j, ψ n t) ψ nt j ) t t j ). 2) analogous to 9). Viewed as a function on the whole real line, each ϕ j is bandlimited with the same band limit c see, for example, Theorem 59 in Section 4.4. or Theorem 9.3 in [3]). On the other hand, ϕ j does not belong to the span of ψ 0, ψ,...,ψ n see Theorem 59 in Section 4.4.). We define the weights W,...,W n via the formula W j = 3) ϕ j t) dt, 4) for j =, 2,...,n. The weights W,...,W n, defined via 4), are different from the solution of the linear system 7). Nevertheless, the resulting quadrature is expected to satisfy 6) with ε of order λ n see Theorem 60 in Section 4.4.2), since the reciprocal of ψ n can be approximated well by a rational function with n poles. Making the latter statement precise is the principal purpose of Section 4 of this paper. While the analysis of the issue is somewhat detailed, the principal idea is simple enough to be presented in the next few sentences. If P is a polynomial with m simple roots z,...,z m in, ), then the function z Pz) is meromorphic in the complex plane; moreover, m Pz) = P z j ) z z j ), 5) j= for all complex z different from z,...,z m see Theorem 27 in Section 2.8). The righthand side of 5) is referred to as partial fractions expansion of P. Similarly, the function z ψ n z) is meromorphic; however, it has infinitely many poles, all of which are real and simple see Corollary 3 in Section 4..), and exactly n of which lie in, ) see Theorem in Section 2.). Suppose that the roots of ψ n in, ) are denoted by t < < t n. Motivated by 5), we analyze the partial fractions expansion of ψn. It turns out that n ψ n t) = ψ nt j ) t t j ) + O λ n ), 6) j= 6

7 for real < t < see Section 4.3 and Theorem 27 in Section 2.8). In other words, 6) means that the reciprocal of ψ n differs from a certain rational function with n poles by a function, whose magnitude in the interval, ) is of order λ n. A rigorous version of 6) is established and proven in Section 4.3. The relation between 6), 2), 4) and 6) is studied in Section 4.4. The results of these two sections rely on the machinery, developed in Sections 4., Overview of the Analysis.3. Partial Fractions Expansion of /ψ n To establish 6), we proceed as follows. Suppose that x < x 2 <... are the roots of ψ n in, ) see Corollary 3 in Section 4..). Suppose also that M >, and R > is a point between x M and x M+. In other words, Then, for all real < t <, < x < x 2 < < x M < R < x M+ <.... 7) n ψ n t) ψ nt j ) t t j ) = M k= j= ψ nx k ) t x k ) + ψ n x k ) t + x k ) ) + 2πi Γ R dz ψ n z) z t), 8) where Γ R is the boundary of the square [ R, R] [ ir, ir], traversed in the counterclockwise direction see Theorem 27 in Section 2.8). Suppose now that x > is a root of ψ n. We observe that ψ n is a holomorphic function defined in the entire complex plane. We use the integral equation 37) in Section 2. and Theorem 25 in Section 2.8 to show that ψ n x + it) 2 + ψ nx + it) 2 x + it) 2 c 2 x + it) 2 χ n e ct ψ n ) 2, t 9) ct λ n see Theorem 36 in Section 4.2.2). On the other hand, we use the differential equation 48) in Section 2. and Theorem 22 in Section 2.5 to show that ψ n x + it) 2 + ψ nx + it) 2 x + it) 2 c 2 x + it) 2 χ n e /4 e ct ψ nx) x 2 ) 3/4 ct x 2 χ n /c 2 )) /4 20) see Theorems 37, 38, 39, 40, 42 in Section 4.2.2). We combine 9) and 20) to establish the inequality ψ nx) e/4 λ n x 2 ) 3 4 x 2 χ n /c 2 )) 4 2) 7

8 see Theorem 43 in Section 4.2.2). Then, we use 2) to show that, for every integer M >, M t x k ) ψ nx k ) < 5 λ n log2 x M ) + χ n ) /4) 22) k= see Theorems 44, 45 in Section 4.3. for a more precise statement). We observe that 22) provides an upper bound on the first summand in right-hand side of 8). While this bound is of order λ n for x M < O λ n ), it diverges if we let M go to infinity see, however, 24) below). To overcome this obstacle, we use the integral equation 44) in Section 2. to analyze the behavior of ψ n x) and ψ nx) for x > λ n 2 see Section 4.3.2). In particular, if x > is a root of ψ n and if x > λ n 2, then ψ nx) = 2ψ n ) λ n x [ + O x λ n )] 23) see Theorem 5 in Section for a more precise statement). More detailed analysis reveals that, if y > x > λ n 2 are two consecutive roots of ψ n and < t < is a real number, then ψ nx) x t) + y ψ ny) y t) 20 c x ds s 2 24) see Theorem 52 in Section 4.3.2). In Theorem 53 of Section 4.3.3, we establish, for all real < t <, the inequality of the form ) ) ψ nx k ) x k t) const λ n log + χ n ) /4, 25) λ n k= where 22), 24) are used to bound the head and the tail of the infinite sum, respectively. Eventually, we analyze the behavior of ψ n of the complex argument to demonstrate that, for all real < t <, dz lim sup k 2πi ψ n z) z t) < 2 2 λ n, 26) Γ Rk where {R k } is a certain sequence that tends to infinity, and the contours Γ Rk are as in 8) see Theorems 54, 55 in Section for more details). We substitute 25) and 26) into 8) to obtain, for all real < t <, an inequality of the form n ψ n t) ψ nt j ) t t j ) const λ n j= ) ) log + χ n ) /4 λ n see Theorems 56, 58 in Section 4.3.3). In the next subsection, we overview the implications of 27) to the analysis of the quadrature, discussed in Section.2. 27) 8

9 .3.2 Quadrature Weights Roughly speaking, 27) asserts that, for all real < t <, n ψ n t) ψ nt j ) t t j ) = O λ n ). 28) j= In other words, the left-hand side of 28) is uniformly bounded in, ), and its magnitude is of order λ n. If we multiply both sides of 28) by ψ n t) and use 2), we obtain = ϕ t) + + ϕ n t) + ψ n t) O λ n ) 29) In other words, ϕ,...,ϕ n constitute a partition of unity in the interval, ), up to an error of order λ n. We integrate both sides of 29) over, ) and use Theorem in Section 2. and 4) in Section.2 to obtain 2 = W + + W n + O λ n ) 30) see Section for more details). Suppose now that m n is an integer. We multiply both sides of 29) by ψ m to obtain ψ m t) = n ψ m t) ϕ j t) + ψ m t) ψ n t) O λ n ). 3) j= On the other hand, for every integer j =,...,n, we use integration by parts to evaluate ϕ j t) ψ m t) dt = λ m 2 ψ m t j ) λ m 2 λ n 2 [ W j + icλ n ψ nt j ) 0 ] ψ n x) e icxt j dx 32) see Theorem 59 in Section 4.4.). We combine 27), 3) and 32) with some additional analysis to conclude that, for all integer 0 m < n, n ψ m t) dt ψ m t j ) W j const λ n log ) λ n + χ n 33) j= see Theorems 60, 62 in Section 4.4.2). According to 33), the quadrature error 6) in Section. is roughly of order λ n. It remains to establish for what values of n this error is smaller than the predefined accuracy parameter ε > 0. In Section 4.4.3, we combine Theorems 6, 7, with 33) to achieve that goal. Namely, we show that, if n > 2c π + const logc) logc) + log ), 34) ε 9

10 then ψ m t) dt n ψ m t j ) W j ε, 35) j= for all integer 0 m < n see Theorem 65). Numerical experiments seem to indicate that the situation is even better in practice: namely, to achieve the desired accuracy it suffices to pick the minimal n such that λ n < ε, which occurs for n = 2c/π + Olog c) log ε)) see Section 6, in particular, Conjecture 2 and Experiment 4 in Section 6.2.). 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. Prolate Spheroidal Wave Functions In this subsection, we summarize several facts about the PSWFs. Unless stated otherwise, all these facts can be found in [38], [30], [8], [34], [7], [25], [26]. Given a real number c > 0, we define the operator F c : L 2 [, ] L 2 [, ] via the formula F c [ϕ] x) = ϕt)e icxt dt. 36) Obviously, F c is compact. We denote its eigenvalues by λ 0, λ,...,λ n,... and assume that they are ordered such that λ n λ n+ for all natural n 0. We denote by ψ n the eigenfunction corresponding to λ n. In other words, the following identity holds for all integer n 0 and all real x : λ n ψ n x) = ψ n t)e icxt dt. 37) We adopt the convention that ψ n L 2 [,] =. The following theorem describes the eigenvalues and eigenfunctions of F c. Theorem. Suppose that c > 0 is a real number, and that the operator F c is defined via 36) above. Then, the eigenfunctions ψ 0, ψ,... of F c are purely real, are orthonormal and are complete in L 2 [, ]. The even-numbered functions are even, the odd-numbered ones are odd. Each function ψ n has exactly n simple roots in, ). 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. This convention agrees with that of [38], [30] and differs from that of [34]. 0

11 We define the self-adjoint operator Q c : L 2 [, ] L 2 [, ] via the formula Q c [ϕ]x) = π sinc x t)) x t ϕt) dt. 38) Clearly, if we denote by F : L 2 R) L 2 R) the unitary Fourier transform, then Q c [ϕ]x) = χ [,] x) F [ χ [ c,c] ξ) F [ϕ] ξ) ] x), 39) where χ [ a,a] : R R is the characteristic function of the interval [ a, a], defined via the formula { a x a, χ [ a,a] x) = 40) 0 otherwise, for all real x. In other words, Q c represents low-passing followed by time-limiting. Q c relates to F c, defined via 36), by and the eigenvalues µ n of Q n satisfy the identity for all integer n 0. Obviously, Q c = c 2π F c F c, 4) µ n = c 2π λ n 2, 42) µ n <, 43) for all integer n 0, due to 39). Moreover, Q c has the same eigenfunctions ψ n as F c. In other words, µ n ψ n x) = π sinc x t)) x t ψ n t) dt, 44) for all integer n 0 and all x. 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) = π sinc x t)) x t ϕt) dt, 45) which is a widely known orthogonal projection onto the space of functions of band limit c > 0 on the real line R. The following theorem about the eigenvalues µ n of the operator Q c, defined via 38), can be traced back to [8]: Theorem 2. Suppose that c > 0 and 0 < α < are positive real numbers, and that the operator Q c : L 2 [, ] L 2 [, ] is defined via 38) 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 =, 2,... : µ k > α}. 46)

12 Then, Nc, α) = 2c π + π 2 log α ) log c + O log c). 47) α According to 47), 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 [24]. The following theorem, proved in a more general form in [35], formalizes this statement. Theorem 3. For any c > 0, there exists a strictly increasing unbounded sequence of positive numbers χ 0 < χ <... such that, for each integer n 0, the differential equation x 2 ) ψ x) 2x ψ x) + χ n c 2 x 2) ψx) = 0 48) has a solution that is continuous on [, ]. Moreover, all such solutions are constant multiples of the eigenfunction ψ n of F c, defined via 36) above. Remark. For all real c > 0 and all integer n 0, 37) defines an analytic continuation of ψ n onto the entire complex plane. All the roots of ψ n are simple and real. In addition, the ODE 48) is satisfied for all complex x. Many properties of the PSWF ψ n depend on whether the eigenvalue χ n of the ODE 48) is greater than or less than c 2. In the following theorem from [25], [26], we describe a simple relationship between c, n and χ n. Theorem 4. Suppose that n 2 is a non-negative integer. If n 2c/π), then χ n < c 2. If n 2c/π), then χ n > c 2. If 2c/π) < n < 2c/π), then either inequality is possible. In the following theorem, upper and lower bounds on χ n in terms of c and n are provided. Theorem 5. Suppose that c > 0 is a real number, and n 0 is an integer. Then, n n + ) < χ n < n n + ) + c 2. 49) It turns out that, for the purposes of this paper, the inequality 49) is insufficiently sharp. More accurate bounds on χ n are described in the following three theorems see [25], [26], [27], [28]). 2

13 Theorem 6. Suppose that n 2 is a positive integer, and that χ n > c 2. Then, n < 2 χn c 2 t 2 π 0 t 2 dt = ) 2 c χn E < n + 3, 50) π χn where the function E : [0, ] R is defined via 09) in Section 2.3. Theorem 7. Suppose that n is a positive integer, and that n > 2c π + 2 ) 4eπc π 2 δ log, 5) δ for some 0 < δ < 5π 4 c. 52) Then, χ n > c π δ c. 53) Theorem 8. Suppose that n is a positive integer, and that 2c π n 2c π + 2 ) 4eπc π 2 δ log 3, 54) δ for some 3 < δ < 5π 4 c. 55) Then, χ n < c π δ c. 56) The following theorem is a direct consequence of Theorem 6. Theorem 9. Suppose that n > 0 is a positive integer, and that Then, n > 2c π +. 57) χ n > c ) 3

14 Proof. It follows from 50) of Theorem 6 that n < 2c + χ n c 2 π c 2 t 2 dt 0 < 2c π + 2 π χ n c 2 We combine 59) with 57) to obtain 58). 0 dt = 2c t 2 π + χ n c 2. 59) In the following theorem from [27], [28], we provide an upper bound on λ n in terms of n and c. Theorem 0. Suppose that c > 0 is a real number, and that Suppose also that δ > 0 is a real number, and that c > ) 3 < δ < πc 6. 6) Suppose, in addition, that n is a positive integer, and that n > 2c π + 2 ) 4eπc π 2 δ log. 62) δ Suppose furthermore that the real number ξn, c) is defined via the formula [ ξn, c) = 7056 c exp δ δ )]. 63) 2πc Then, λ n < ξn, c). 64) In the following theorem from [27], [28], we provide another upper bound on λ n. Theorem. Suppose that n > 0 is a positive integer, and that Suppose also that the real number x n is defined via the formula n > 2c π ) Then, x n = χ n c 2. 66) λ n < 95 c x n ) 3 4 xn ) 4 x n 2) 3 [ exp π 4 xn ) ] c. 67) xn 4

15 The following theorem is a combination of certain results from [30] and [25], [26]. Theorem 2. Suppose that c > 0 is a real number, and that χ n > c 2. Then, 2 < ψ2 n) < n ) The following theorem appears in [25], [26]. Theorem 3. Suppose that n 0 is a non-negative integer, and that x, y are two arbitrary extremum points of ψ n in, ). If x < y, then If, in addition, χ n > c 2, then ψ n x) < ψ n y). 69) ψ n x) < ψ n y) < ψ n ). 70) The following theorem appears in [32]. Theorem 4. For all real c > 0 and all natural n, max max ψ mt) 2 n. 7) m n+ t In the following theorem, we provide a recurrence relation between the derivatives of ψ n of arbitrary order see Lemma 9. in [38]). Theorem 5. Suppose that c > 0 is a real number, and that n 0 is an integer. Then, t 2 ) ψ n t) 4tψ nt) + χ n c 2 t 2 2 ) ψ nt) 2c 2 tψ n t) = 0 72) for all real t. Moreover, for all integer k 2 and all real t, t 2 ) ψ k+2) n t) 2 k + )tψ n k+) t) + χ n k k + ) c 2 t 2) ψ n k) t) c 2 ktψ n k ) t) c 2 k k )ψ n k 2) t) = 0. 73) We refer to the roots of ψ n, the roots of ψ n and the turning points of the ODE 48) as special points. In the following theorem from [25], [26], we describe the location of some of the special points. 5

16 Theorem 6 Special points). Suppose that n 2 is a positive integer. Suppose also that t < t 2 <... are the roots of ψ n in, ), and that s < s 2 <... are the roots of ψ n in, ). If χ n < c 2, then < χn c < s < t < s 2 < < t n < s n < t n < s n+ < χn c < 74) In particular, ψ n has n roots in, ), and ψ n has n + roots in, ). On the other hand, if χ n > c 2, then χn c < < t < s < t 2 < < t n < s n < t n < < In particular, ψ n has n roots in, ), and ψ n has n roots in, ). χn c. 75) In the following theorem, proven in [25], [26], we describe a relation between the magnitude of ψ n and ψ n in the interval, ). Theorem 7. Suppose that n 0 is a non-negative integer, and that the functions p, q : R R are defined via 40) in Section 2.6. Suppose also that the functions Q, Q : 0, min { χn /c, } ) R are defined, respectively, via the formulae and Qt) = ψnt) 2 + pt) qt) ψ nt) ) 2 = ψ 2 t 2 ) ψ n t) + nt)) 2 χ n c 2 t 2 76) Qt) = pt) qt) Qt) = t 2) χn c 2 t 2) ψ 2 nt) + t 2) ψ nt) ) 2 ). 77) Then, Q is increasing in the interval 0, min { χn /c, }), and Q is decreasing in the interval 0, min { χn /c, }). 2.2 Legendre Polynomials and PSWFs In this subsection, we list several well known facts about Legendre polynomials and the relationship between Legendre polynomials and PSWFs. All of these facts can be found, for example, in [2], [38], []. The Legendre polynomials P 0, P, P 2,... are defined via the formulae and the recurrence relation P 0 t) =, P t) = t, 78) k + ) P k+ t) = 2k + )tp k t) kp k t), 79) for all k =, 2,.... The even-indexed Legendre polynomials are even functions, and the odd-indexed Legendre polynomials are odd functions. The Legendre polynomials {P k } k=0 6

17 constitute a complete orthogonal system in L 2 [, ]. The normalized Legendre polynomials are defined via the formula P k t) = P k t) k + /2, 80) for all k = 0,, 2,.... The L 2 [, ]-norm of each normalized Legendre polynomial equals to one, i.e. Pk t) ) 2 dt =. 8) Therefore, the normalized Legendre polynomials constitute an orthonormal basis for L 2 [, ]. In particular, for every real c > 0 and every integer n 0, the prolate spheroidal wave function ψ n, corresponding to the band limit c, can be expanded into the series ψ n x) = k=0 β n) k P k x) = for all x, where β n) 0, β n),... are defined via the formula and α n) 0, αn) β n) k =,... are defined via the formula α n) k k=0 α n) k P k x), 82) ψ n x) P k x) dx, 83) = β n) k k + /2, 84) for all k = 0,, 2,.... Due to the combination of Theorem in Section 2. with 8), 82), 83), β n) 0 ) 2 + β n) ) 2 + β n) 2 The sequence β n) 0, β n),... satisfies the recurrence relation ) 2 + =. 85) A 0,0 β n) 0 + A 0,2 β n) 2 = χ n β n) 0, A, β n) + A,3 β n) 3 = χ n β n), A k,k 2 β n) k 2 + A k,k β n) k + A k,k+2 β n) k+2 = χ n β n) k, 86) for all k = 2, 3,..., where A k,k, A k+2,k, A k,k+2 are defined via the formulae 2kk + ) A k,k = kk + ) + 2k + 3)2k ) c2, k + 2)k + ) A k,k+2 = A k+2,k = 2k + 3) 2k + )2k + 5) c2, 87) 7

18 for all k = 0,, 2,.... In other words, the infinite vector identity ) β n) 0, β n),... satisfies the A χ n I) β n) 0, β n) T,...) = 0, 88) where I is the infinite identity matrix, and the non-zero entries of the infinite symmetric matrix A are given via 87). The matrix A naturally splits into two infinite symmetric tridiagonal matrices, A even and A odd, the former consisting of the elements of A with even-indexed rows and columns, and the latter consisting of the elements of A with odd-indexed rows and columns. Moreover, for every pair of integers n, k 0, β n) k = 0, if k + n is odd, 89) due to the combination of Theorem in Section 2. and 83). In the following theorem that appears in [38] in a slightly different form), we summarize the implications of these observations to the identity 88), that lead to numerical algorithms for the evaluation of PSWFs. Theorem 8. Suppose that c > 0 is a real number, and that the infinite tridiagonal symmetric matrices A even and A odd are defined, respectively, via A 0,0 A 0,2 A even A 2,0 A 2,2 A 2,4 = A 4,2 A 4,4 A 4,6 90) and A, A,3 A odd A 3, A 3,3 A 3,5 = A 5,3 A 5,5 A 5,7, 9) where the entries A k,j are defined via 87). Suppose also that the unit length infinite vector β n) l 2 is defined via the formula β n) β n) 0, β n) T 2,...) n is even, = β n), β n) T 92) 3,...) n is odd, where β n) 0, β n),... are defined via 83). If n is even, then If n is odd, then A even β n) = χ n β n). 93) A odd β n) = χ n β n). 94) 8

19 Remark 2. While the matrices A even and A odd are infinite, and their entries do not decay with increasing row or column number, the coordinates of each eigenvector β n) decay superexponentially fast see e.g. [38] for estimates of this decay). In particular, suppose that we need to evaluate the first n + eigenvalues χ 0,...,χ n and the corresponding eigenvectors β 0),...,β n) numerically. Then, we can replace the matrices A even, A odd in 93), 94), respectively, with their N N upper left square submatrices, where N is of order n, and solve the resulting symmetric tridiagonal eigenproblem by any standard technique see, for example, [37], [7]; see also [38] for more details about this numerical algorithm). The cost of this algorithm is On 2 ) operations. The Legendre functions of the second kind Q 0, Q, Q 2,... are defined via the formulae and the recurrence relation for all k =, 2,.... In particular, Q 0 t) = 2 log + t t, Q t) = t 2 log + t, 95) t k + ) Q k+ t) = 2k + )tq k t) kq k t), 96) Q 2 t) = 3t2 4 Q 3 t) = 5t3 3t 4 log + t t 3 2 t, log + t t 5 2 t ) We observe that the recurrence relation 96) is the same as the recurrence relation 79), satisfied by the Legendre polynomials. It follows from 79), 96), that both the Legendre polynomials P 0, P,... and the Legendre functions of the second kind Q 0, Q,... satisfy another recurrence relation, namely for all k = 2, 3,..., where t 2 P k t) = A k 2 P k 2 t) + B k P k t) + C k+2 P k+2 t), t 2 Q k t) = A k 2 Q k 2 t) + B k Q k t) + C k+2 Q k+2 t), 98) k + )k + 2) A k = 2k + 3)2k + 5), 99) 2kk + ) B k = 2k + 3)2k ), 00) kk ) C k = 2k 3)2k ). 0) In addition, for every integer k = 0,, 2,..., the kth Legendre polynomial P k and the kth Legendre function of the second kind Q k are two independent solutions of the second order Legendre differential equation t 2 ) y t) 2t y t) + kk + ) yt) = 0. 02) 9

20 Also, for every integer k = 0,,... and all complex z such that arg z ) < π, Q k z) = 2 see, for example, Section 8.82 of [2]). P k t) z t dt 03) Remark 3. For any real number < x < and integer n 0, we can use the three-term recurrences 79), 96) to evaluate numerically P 0 x),..., P n x) and Q 0 x),..., Q n x) with high precision, in On) operations see, for example, [7] for more details). 2.3 Elliptic Integrals In this subsection, we summarize several facts about elliptic integrals. These facts can be found, for example, in section 8. in [2], and in []. The incomplete elliptic integrals of the first and second kind are defined, respectively, by the formulae Fy, k) = Ey, k) = y 0 y 0 dt k 2 sin 2 t, 04) k 2 sin 2 t dt, 05) where 0 y π/2 and 0 k. By performing the substitution x = sint, we can write 04) and 05) as Fy, k) = siny) 0 dx x 2 ) k 2 x 2 ), 06) Ey, k) = siny) 0 k 2 x 2 x 2 dx. 07) The complete elliptic integrals of the first and second kind are defined, respectively, by the formulae for all 0 k. Moreover, ) E k 2 = + π ) Fk) = F 2, k = π ) Ek) = E 2, k = π/2 0 π/2 0 dt k 2 sin 2 t, 08) k 2 sin 2 t dt, 09) ) logk) + log2) k 2 + O k 4 logk) ). 0)

21 2.4 Oscillation Properties of Second Order ODEs In this subsection, we state several well known facts from the general theory of second order ordinary differential equations see e.g. [23]). The following two theorems appear in Section 3.6 of [23] in a slightly different form. Theorem 9 distance between roots). Suppose that ht) is a solution of the ODE y t) + Qt) yt) = 0. ) Suppose also that x < y are two consecutive roots of ht), and that A 2 Qt) B 2, 2) for all x t y. Then, π B < y x < π A. 3) Theorem 20. 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 are consecutive roots of yt). If g is non-decreasing, then If g is non-increasing, then y t) + gt) yt) = 0, 4) t < t 2 < t 3 <... 5) t 2 t t 3 t 2 t 4 t ) t 2 t t 3 t 2 t 4 t ) The following theorem is a special case of Theorem 6.2 from Section 3.6 in [23]: Theorem 2. Suppose that g, g 2 are continuous functions, and that, for all real t in the interval a, b), the inequality g t) < g 2 t) holds. Suppose also that the function φ, φ 2 satisfy, for all a < t < b, φ t) + g t) φ t) = 0, φ 2t) + g 2 t) φ 2 t) = 0. 8) Then, φ 2 has a root between every two consecutive roots of φ. 2

22 Corollary. Suppose that the functions φ, φ 2 are those of Theorem 2 above. Suppose also that φ t 0 ) = φ 2 t 0 ), φ t 0 ) = φ 2t 0 ), 9) for some a < t 0 < b. Then, φ 2 has at least as many roots in t 0, b) as φ. Proof. By Theorem 2, we only need to show that if t is the minimal root of φ in t 0, b), then there exists a root of φ 2 in t 0, t ). By contradiction, suppose that this is not the case. In addition, without loss of generality, suppose that φ t), φ 2 t) are positive in t 0, t ). Then, due to 8), and hence 0 < φ φ 2 φ 2φ = g 2 g ) φ φ 2, 20) t t 0 g 2 s) g s)) φ s)φ 2 s)ds = [ φ s)φ 2 s) φ s)φ 2s) ] t t 0 = φ t )φ 2 t ) 0, 2) which is a contradiction. 2.5 Growth Properties of Second Order ODEs The following theorem appears in [9] in a more general form. We provide a proof for the sake of completeness. Theorem 22. Suppose that a < b are real numbers, and that the functions w, u, β, γ : a, b) C are continuously differentiable. Suppose also that, for all real a < t < b, and that w ) t) u = t) 0 βt) γt) 0 ) wt) ut) ), 22) βt) 0, γt) 0, 23) for all a < t < b. Suppose furthermore that the functions R, Q : a, b) R are defined, respectively, via the formulae Rt) = βt) γt) 24) and Qt) = w t) 2 + Rt) u t) 2. 25) 22

23 Then, for all real a < t 0, t < b, ) Rt) 4 t exp Rt 0 ) t 0 Qt) Qt 0 ) ) Rt) 4 exp Rt 0 ) t t 0 R ) s) 2 + 4Rs) R ) s) 2 + 4Rs) ) 2 βs) γs) + R βs)γs)) 2 ) 2 βs) γs) + R βs)γs)) 2 ds ds. 26) Proof. We note that, for a each fixed t, the formula 25) can be written in the matrix notation as Qt) = wt) ūt) ) ) ) 0 wt). 27) 0 Rt) ut) We differentiate Qt) with respect to t to obtain, by using 22), Q t) = w t) wt) + wt) w t) + Rt)ūt)u t) + Rt)ū t)ut) + R t)ut)ūt) = βt)ut) wt) + βt)ūt)wt) + Rt)γt)wt)ūt) + Rt) γt) wt)ut) + R t)ut)ūt) = wt) ūt) ) ) ) 0 βt) + Rt) γt) wt) βt) + Rt)γt) R. 28) t) ut) Then, we define the functions x, y : a, b) R via the formulae We substitute 30) into 27), 28) to obtain xt) = wt), 29) yt) = ut) Rt). 30) Q t) Qt) = ) xt) ȳt) 0 βt)+rt)γt) Rt) βt)+rt) γt) Rt) R t) Rt) ) xt) yt) xt) 2 + yt) 2. 3) To find the eigenvalues of the matrix in 3), we solve, for each a < t < b, the quadratic equation ) ) λ 2 R t) Rt) λ βt) + Rt) γt) βt) + Rt)γt) = 0, 32) Rt) Rt) 23

24 in the unknown λ. Suppose that λ t) < λ 2 t) are the roots of 32) for a fixed a < t < b. We use 24) to obtain [ λ t) = R t) R ) 2Rt) t) βt) γt) + R βt)γt)))], 2Rt) [ λ 2 t) = R t) R ) 2Rt) + t) βt) γt) + R βt)γt)))]. 33) 2Rt) Due to 3), for all a < t < b, λ t) Q t) Qt) λ 2t). 34) We substitute 33) into 34), integrate it from t 0 to t and exponentiate the result to obtain 26). 2.6 Prüfer Transformations In this subsection, we describe the classical Prüfer transformation of a second order ODE see e.g. [23],[9]). Also, we describe a modification of Prüfer transformation, introduced in [] and used in the rest of the paper. Suppose that we are given the second order ODE d pt)u t) ) + qt)ut) = 0, 35) 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 pt)u t) ut) = γt)tanθt), 36) 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). 37) γt) 2 One can observe that if u t) = 0 for t I, then by 36) Similarly, if u t) = 0 for t I, then θ t) = kπ, k is integer. 38) θ t) = k + /2) π, k is integer. 39) The choice γt) = in 36) gives rise to the classical Prüfer transformation see e.g. section 4.2 in [23]). 24

25 In [], the choice γt) = qt)pt) 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 48) in the form of 35) yields pt) = t 2, qt) = c 2 t 2 χ n, 40) for all real t > max { χn /c, }. The equation 36) admits the form which implies that pt)ψ nt) ψ n t) = pt)qt) tanθt), 4) ) pt) ψ θt) = atan nt) + πmt), 42) qt) ψ n t) where mt) is an integer determined for all t by an arbitrary choice at some t = t 0 the role of πmt) in 42) is to enforce the continuity of θ at the roots of ψ n ). The first order ODE 37) admits the form see [], [9]) θ t) = ft) sin2θt))vt), 43) where the functions f, v are defined, respectively, via the formulae qt) c ft) = pt) = 2 t 2 χ n t 2 and vt) = 4 pt)q t) + qt)p t) pt)qt) = 2 t t 2 + c 2 t c 2 t 2 χ n 44) ). 45) Remark 4. In this paper, the variable t in 4), 42), 43) will be confined to the open ray max {, χ n /c}, ). 46) Nevertheless, a similar analysis is possible for t in the interval min {, χ n /c}, min {, χ n /c}). 47) The following theorem from [25], [26], summarizes such analysis for the case χ n > c 2. Theorem 23. Suppose that n 2 is a positive integer, and that χ n > c 2. Suppose also that t,...,t n are the roots of ψ n in, ), and s,...,s n are the roots of ψ n in, ) see Theorem 6 in Section 2.). Suppose furthermore that the function θ : [, ] R is defined via the formula i 2) π, if t = ti for some i n, θt) = atan t 2 χ n c ψ nt) 2 t 2 ψ nt) ) + mt) π, if ψ n t) 0, 48) where mt) is the number of the roots of ψ n in the interval, t). Then, θ has the following properties: 25

26 θ is continuously differentiable in the interval [, ]. θ satisfies, for all < t <, the differential equation θ t) = ft) vt) sin2θt)), 49) where the functions f, v are defined, respectively, via 44), 45) in Section 2.6. for each integer 0 k 2n, there is a unique solution to the equation for the unknown t in [, ]. More specifically, for each i =,...,n and each j =,...,n. For all real < t <, In other words, θ is monotonically increasing. θt) = k π 2, 50) θ ) = 0, 5) θt i ) = i ) π, 52) 2 θs j ) = j π, 53) θ) = n π, 54) θ t) > 0. 55) 2.7 Numerical Tools In this subsection, we summarize several numerical techniques to be used in this paper Newton s Method Newton s method solves the equation fx) = 0 iteratively given an initial approximation x 0 of the root x. The nth iteration is defined by x n = x n fx n ) f x n ). 56) The convergence is quadratic provided that x is a simple root and x 0 is close enough to x. More details can be found e.g. in [7]. 26

27 2.7.2 The Taylor Series Method for the Solution of ODEs The Taylor series method for the solution of a linear second order differential equation is based on the Taylor formula ux + h) = k j=0 u j) x) h j + Oh k+ ). 57) j! This method evaluates ux + h) and u x + h) by using 57) and depends on the ability to compute u j) x) for j = 0,...,k. When the latter satisfy a simple recurrence relation like 73) and hence can be computed in Ok) operations, this method is particularly useful. The reader is referred to [] for further details A Second Order Runge-Kutta Method We use the following second order Runge-Kutta Method, which can be found, for example, in [7]. It solves the initial value problem on the interval t 0 t t 0 + L by computing with i = 0,...,n and yt 0 ) = y 0, y t) = ft, y) 58) t i+ = t i + h, k i+ = hf t i+, y i + k i ), y i+ = y i + k i + k i+ )/2 59) h = L n, k 0 = ft 0, y 0 ). 60) Exactly n + evaluations of f are required for this algorithm, which results in the total cost being On). The global truncation error is Oh 2 ) Power and Inverse Power Methods The methods described in this subsection are widely known and can be found, for example, in [7]. Suppose that A is an n n real symmetric matrix, whose eigenvalues satisfy σ > σ 2 σ 3 σ n. 6) The Power Method approximates σ and the corresponding unit eigenvector in the following way. Set v 0 to be a random vector in R n such that v 0 = v0 Tv 0 =. Set j = and η 0 = 0. Compute ˆv j = Av j. 27

28 Set η j = v T j ˆv j. Set v j = ˆv j / ˆv j. If η j η j is sufficiently small, stop. Otherwise, set j = j + and repeat the iteration. The output value η j approximates σ, and v j approximates a unit eigenvector corresponding to σ. The cost of each iteration is dominated by the cost of evaluating Av j. The rate of convergence of the algorithm is linear and equals to σ 2 / σ, that is, the error after j iterations is of order σ 2 / σ ) j. Remark 5. A modification of the algorithm used in this paper defines η j by i = argmax { v j k) : k =,...,n}, η j = ˆv ji) v j i). 62) The Inverse Power Method finds the eigenvalue σ k of A and a corresponding unit eigenvector provided that an approximation σ of σ k is known such that σ σ k < max { σ σ j : j k}. 63) Conceptually, the Inverse Power Method is an application of the Power Method on the matrix B = A σi). In practice, B need not be evaluated explicitly and it suffices to be able to solve the linear system of equations for the unknown ˆv j on each iteration of the algorithm. A σi) ˆv j = v j 64) Remark 6. If the matrix A is tridiagonal, the system 64) can be solved in On) operations, for example, by means of Gaussian elimination or QR decomposition see e.g [37], [7]) Sturm Sequence The following theorem can be found, for example, in [37] see also [2]). It provides the basis for an algorithm of evaluating the kth smallest eigenvalue of a symmetric tridiagonal matrix. Theorem 24 Sturm sequence). Suppose that a b b 2 a 2 b C = b n a n b n 0 0 b n a n 65) 28

29 is a symmetric tridiagonal matrix such that none of b 2,...,b n is zero. Then, its n eigenvalues satisfy σ C) < < σ n C). 66) Suppose also that C k is the k k leading principal submatrix of C, for every integer k =,...,n. We define the polynomials p, p 0,...,p n via the formulae and p x) = 0, p 0 x) = 67) p k x) = detc k xi k ), 68) for k = 2,...,n. In other words, p k is the characteristic polynomials of C k. Then, p k x) = a k x)p k x) b 2 k p k 2x), 69) for every integer k =, 2,...,n. Suppose furthermore, that, for any real number σ, the integer Aσ) is defined to be the number of agreements of sign of consecutive elements of the sequence p 0 σ), p σ),..., p n σ), 70) where the sign of p k σ) is taken to be opposite to the sign of p k σ) if p k σ) is zero. Then, the number of eigenvalues of C that are strictly larger than σ is precisely Aσ). Corollary 2 Sturm bisection). The eigenvalue σ k C) of 65) can be found by means of bisection, each iteration of which costs On) operations. Proof. We initialize the bisection by choosing x 0 < σ k C) < y 0. Then we set j = 0 and iterate as follows. Set z j = x j + y j )/2. If y j x j is small enough, stop and return z j. Compute A j = Az j ) using 69) and 70). If A j k, set x j+ = z j and y j+ = y j. If A j < k, set x j+ = x j and y j+ = z j. Increase j by one and go to the first step. In the end σ k C) z j is at most y j x j. The cost of the algorithm is due to 69) and the definition of Aσ). 29

30 2.8 Miscellaneous tools In this subsection, we list some widely know theorems of real analysis. The following theorem can be found in section 6.4 of [3] in a more general form. In this theorem, we use the following widely used notation. Suppose that g, h : 0, ) C are complex-valued functions. The expression means that gt) ht), t, 7) ht) lim =. 72) t gt) Theorem 25 Watson s Lemma). Suppose that b > 0, and that the function f : [0, b] R is twice continuously differentiable. Then, b in the sense of 7). In other words, 0 lim t fs) e st ds f0), t, 73) t t f0) b 0 fs) e st ds =. 74) The following theorem appears, for example, in [8] in a more general form. Theorem 26. Suppose that x 0 is a real number, and u : R 2 R is a function of two real variables t, x), defined in the shifted upper half-plane H x0 = {t, x) : < t <, x 0 x < }. 75) Suppose also, that u is bounded in H x0 and is harmonic in the interior of Hx0. Suppose furthermore, that ut, x 0 ) dt <. 76) Then, for all real t and x > x 0, the value ut, x) is given by the formula ut, x) = π and, moreover, for all x > x 0, us, x 0 ) ut, x 0 ) dt = x x 0 t s) 2 ds, 77) + x x 0 ) 2 ut, x) dt. 78) 30

31 The following theorem is a special case of the well known Cauchy s integral formula see, for example, [3]). Theorem 27. Suppose that D C is an open bounded simply connected subset of the complex plane, and that the boundary Γ of D is piecewise continuously differentiable. Suppose also that the function g : C C is holomorphic in a neighborhood of D, and that none of the roots of g lies on Γ. Suppose furthermore that z, z 2,...,z m D are the roots of g in D, all of which are simple, and that z D is a complex number such that gz) 0. In other words, z D \ {z, z 2,...,z m }. 79) Then, where Γ m gz) = j= g z j ) z z j ) + 2πi Γ dζ gζ) ζ z), 80) denotes the contour integral over Γ in the counterclockwise direction. 3 Summary In this section, we summarize some of the properties of prolate spheroidal wave functions PSWFs), proved in the rest of the paper, mainly in Section 4. The PSWFs and the related notation were introduced in Section 2.. Throughout this section, the band limit c > 0 is assumed to be a positive real number. In the following proposition, we describe the location of special points roots of ψ n, roots of ψ n, turning points of the ODE 48)), in the case χ n > c 2. This proposition is proven in Theorem 29 and Corollary 3 in Section 4.. see also Theorem 6 in Section 2.). It is illustrated in Figures, 2 see Experiment in Section 6..). Proposition. Suppose that n 0 is a positive integer, and that χ n > c 2. Suppose also that x < x 2 <... are the roots of ψ n in, ), and y < y 2 <... are the roots of ψ n in, ). Then, < χn c < y < x < y 2 < x 2 <.... 8) Also, ψ n has infinitely many roots in, ); all of these roots are simple. The following proposition summarizes the statements of Theorems 3, 32 in Section 4.. It is illustrated in Tables, 2, 3. Proposition 2. Suppose that n 0 is an integer, and that χ n > c 2. Suppose also that x < x 2 <... are the roots of ψ n in, ). For each integer k =, 2,..., π c + c 2 x 2 k ) 2 x k+ x k π x 2 k c x 2 k χ n/c 2 ). 82) 3