On the existence of non-oscillatory phase functions for second order ordinary differential equations in the high-frequency regime

Transcription

1 We observe that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In addition, we describe numerical experiments which illustrate computational implications of this fact. For example, many special functions of interest such as the Bessel functions J ν and Y ν can be evaluated accurately using a number of operations which is Opq in the order ν. The present paper is devoted to the development of an analytical apparatus. Numerical aspects of this work will be reported at a later date. On the existence of non-oscillatory phase functions for second order ordinary differential equations in the high-frequency regime Z. Heitman ;, J. Bremer, V. Rokhlin ; a, Technical Report YALEU/DCS/TR-493 August 4, 04 This author s work was supported in part under ONR N , ONR N This author s work was supported in part by a fellowship from the Alfred P. Sloan foundation a This author s work was supported in part under ONR N , ONR N , AFOSR FA Department of Mathematics, University of California, Davis, CA 9566 ; Dept. of Mathematics, Yale University, New Haven CT 065 Approved for public release: distribution is unlimited. Keywords: Kummer s Equation, WKB Approximation, Phase Functions

2 . Introduction Given a differential equation y ptq ` λ qptqyptq 0 for all 0 ď t ď, () where λ is a real number and q : r0, s Ñ R is smooth and strictly positive, a sufficiently smooth α : r0, s Ñ R is a phase function for () if the pair of functions u, v defined by the formulas uptq cospαptqq α ptq { () and vptq sinpαptqq α ptq { (3) form a basis in the space of solutions of (). Phase functions have been extensively studied: they were first introduced in [9], play a key role in the theory of global transformations of ordinary differential equations [3, 0], and are an important element in the theory of special functions [6, 6,, ]. Despite this long history, an important property of phase functions appears to have been overlooked. Specifically, that when the function q is nonoscillatory, solutions of the equation () can be accurately represented using a nonoscillatory phase function. This is somewhat surprising since α is a phase function for () if and only if it satisfies the third order nonlinear ordinary differential equation pα ptqq λ qptq α 3 ptq α ptq ` 3 ˆα ptq for all 0 ď t ď. (4) 4 α ptq The equation (4) was introduced in [9], and and we will refer to it as Kummer s equation. The form of (4) and the appearance of λ in it suggests that its solutions will be oscillatory and most of them are. However, Bessel s equation y ptq ` ˆ λ {4 t yptq 0 for all 0 ă t ă 8 (5) furnishes a nontrivial example of an equation which admits a nonoscillatory phase function regardless of the value of λ. If we define u, v by the formulas c πt uptq J λptq (6) and c πt vptq Y λptq, (7) where J λ and Y λ denote the Bessel functions of the first and second kinds of order λ, and let α be defined by the relations (),(3), then α ptq πt Jλ ptq ` Y (8) λ ptq. It can be easily verified that (8) is a strictly negative, monotonically decreasing function of t

3 on p0, 8q. That this nonoscillatory phase function for Bessel s equation exists is the basis of several methods for the evaluation of Bessel functions of large orders and for the computation of their zeros [6, 8, 5]. The general situation is not quite so favorable: there need not exist a nonoscillatory function α such that () and (3) are exact solutions of (). However, assuming that q is nonoscillatory and λ is sufficiently large, there exists a nonoscillatory function α such that (), (3) approximate solutions of () with spectral accuracy (i.e., the approximation errors decay exponentially with λ). To see that this claim is plausible, we apply Newton s method for the solution of nonlinear equations to Kummer s equation (4). In doing so, it will be convenient to move the setting of our analysis from the interval r0, s to the real line so that we can use the Fourier transform to quantity the notion of nonoscillatory. Suppose that the extension of q to the real line is smooth and strictly positive, and that the Fourier transform of logpqq is smooth and decays rapidly. Letting pα ptqq λ expprptqq (9) in (4) yields the logarithm form of Kummer s equation: r ptq 4 pr ptqq ` 4λ pexpprptqq qptqq 0 for all t P R. (0) We use tr n u to denote the sequence of Newton iterates for the equation (0) obtained from the initial guess r 0 ptq logpqptqq. () The function r 0 corresponds to the first order WKB approximations for (). That is to say that if we insert the associated phase function ż t ż ˆ t α 0 ptq λ exp r a 0puq du λ qpuqdu () into (),(3), then and 0 ˆ uptq q {4 ptq cos λ ˆ vptq q {4 ptq sin λ ż t 0 ż t 0 0 a qpuq du (3) a qpuq du. (4) For each n ě 0, r n` is obtained from r n by solving the linearized equation h ptq r nptqh ptq ` 4λ exp pr n ptqq hptq f n ptq for all t P R, (5) where f n ptq r nptq ` 4 pr nptqq 4λ pexp pr n ptqq qptqq, (6) and letting r n` ptq r n ptq ` hptq. (7) 3

4 By introducing the change of variables xptq ż t 0 ˆrn puq exp du (8) into (5), we transform it into the inhomogeneous Helmholtz equation where h pxq ` 4λ hpxq g n pxq for all x P R, (9) g n pxq exp p r n pxqq f n pxq. (0) Suppose that pg n decays rapidly (when n 0, this is a consequence of our assumption that logpqq has a rapidly decaying Fourier transform) and let h be the solution of (9) whose Fourier transform is xh pξq pg npξq 4λ ξ. () Since x h pξq is singular when ξ λ, h will necessarily have a component which oscillates at frequency λ. However, according to (), the L 8 prq norm of that component is In fact, by rearranging () as pg n pλq 4λ. () xh pξq ˆ pgn pξq 4λ λ ξ ` pg npξq λ ` ξ and decomposing each of the terms on the right-hand side of (3) as pg n pξq λ ξ ˆ pgn pξq pg n p λq exp p pλ ξq q 4λ λ ξ we obtain where h 0 is defined by the formula ph 0 pξq ˆ pgn pξq pg n p λq exp p pλ ` ξq q 4λ λ ` ξ and h is defined by the formula ph pξq 4λ ` pg n p λq exp p pλ ξq q λ ξ (3), (4) h pxq h 0 pxq ` h pxq, (5) ` pg npξq pg n pλq exp p pλ ξq q, (6) λ ξ ˆ pg n p λq exp p pλ ` ξq q ` pg n pλq exp p pλ ξq q λ ` ξ λ ξ. (7) Since the factor in the denominator in (6) has been canceled and both pg n and the Gaussian function are smooth and rapidly decaying, h0 p is also smooth and rapidly decaying. Meanwhile, a straightforward calculation shows that the Fourier transform of is x i erf exppλixq (8) exp p pλ ξq q, (9) λ ξ 4

5 so that (7) implies that h pxq ˆ pg n p λq x 4λ i erf exppλixq pg n pλq x i erf expp λixq. (30) Since g n is real-valued, pg n pλq pg n p λq. Inserting this into (30) yields h pxq pgpλq x 4λ erf sin pλxq, (3) which makes it clear that the L 8 prq norm of h is p4λq pg n pλq. In (5), the solution of (9) is decomposed as the sum of a nonoscillatory function h 0 and a highly oscillatory function h of small magnitude. However, the solution of (5) is actually given by the function h pxptqq h 0 pxptqq ` h pxptqq (3) obtained by reversing the change of variables (8). But since xptq is nonoscillatory and the Fourier transform of h 0 pxq decays rapidly, we expect that the composition h 0 pxptqq will also have a rapidly decaying Fourier transform. The L 8 prq norm of h pxptqq is, of course, the same as that of h pxq. So the solution of the linearized equation (5) can be written as the sum of a nonoscillatory function h 0 pxptqq and a highly oscillatory function h pxptqq of negligible magnitude. If, in each iteration of the Newton procedure, we approximate the solution of (5) by constructing h pxptqq and discarding the oscillatory term h ptpxqq of small magnitude, then it is plausible that we will arrive at an approximate solution rptq of the logarithm form of Kummer s equation which is nonoscillatory, assuming the Fourier transform of r 0 ptq logpqptqq decays rapidly enough and λ is sufficiently large. Most of the remainder of this paper is devoted to developing a rigorous argument to replace the preceding heuristic discussion. In Section, we summarize a number of well-known mathematical facts to be used throughout this article. In Section 3, we reformulate Kummer s equation as a nonlinear integral equation. Once that is accomplished, we are in a position to state the principal result of the paper and discuss its implications; this is done in Section 4. The proof of this principal result is contained in Sections 5, 6, 7 and 8. Section 9 contains an elementary proof of a relevant error estimate for second order differential equations of the form (). In Section 0, we present the results of numerical experiments concerning the evaluation of special functions. The details of our numerical algorithm will be reported at a later date. We conclude with a few brief remarks in Section.. Preliminaries.. Modified Bessel functions The modified Bessel function K ν ptq of the first kind of order ν is defined for t P R and ν P C by the formula K ν ptq ż 8 0 exp p t cosh ptqq coshpνtq dt. (33) 5

6 The following bound on the ratio of K ν` to K ν can be found in [4]. Theorem. Suppose that t ą 0 and ν ą 0 are real numbers. Then K ν` ptq K ν ptq ă ν `?ν ` t t ď ν t `. (34).. The binomial theorem A proof of the following can be found in [3], as well as many other sources. Theorem. Suppose that r is a real number, and that y is a real number such that y ă. Then 8ÿ p ` yq r Γpr ` q Γpk ` qγpr k ` q yk. (35) k 0.3. The Lambert W function The Lambert W function or product logarithm is the multiple-valued inverse of the function fpzq z exppzq. (36) We follow [5] in using W 0 to denote the branch of W which is real-valued and greater than or equal to on the interval r {e, 8q and W to refer to the branch which is real-valued and less than or equal to on r {e, 0q. We will make use of the following elementary facts concerning W 0 and W (all of which can be found in [5] and its references). Theorem 3. Suppose that y ě {e is a real number. Then x exppxq ď y if and only if x ď W 0 pyq. (37) Theorem 4. Suppose that 0 ă y ď {e is a real number. Then x expp xq ď y if and only if x ě W p yq. (38) Theorem 5. Suppose that 0 ď x ď is a real number. Then x ď W 0pxq ď x. (39) Theorem 6. Suppose that x ą exppq is a real number. Then ˆ logpxq ď W ď logpxq. (40) x 6

7 .4. Fréchet derivatives and the contraction mapping principle Given Banach spaces X, Y and a mapping f : X Ñ Y between them, we say that f is Fréchet differentiable at x P X if there exists a bounded linear operator X Ñ Y, denoted by fx, such that }fpx ` hq fpxq fx rhs} lim 0. (4) hñ0 }h} Theorem 7. Suppose that X and Y are a Banach spaces and that f : X Ñ Y is Fréchet differentiable at every point of X. Suppose also that D is a convex subset of X, and that there exists a real number M ą 0 such that for all x P D. Then for all x and y in D. }f x} ď M (4) }fpxq fpyq} ď M}x y} (43) Suppose that f : X Ñ X is a mapping of the Banach space X into itself. We say that f is contractive on a subset D of X if there exists a real number 0 ă α ă such that }fpxq fpyq} ď α}x y} (44) for all x, y P D. Moreover, we say that tx n u 8 n 0 is a sequence of fixed point iterates for f if x n` fpx n q for all n ě 0. Theorem 7 is often used to show that a mapping is contractive so that the following result can be applied. Theorem 8. (The Contraction Mapping Principle) Suppose that D is a closed subset of a Banach space X. Suppose also that f : X Ñ X is contractive on D and f pdq Ă D. Then the equation x fpxq (45) has a unique solution σ P D. Moreover, any sequence of fixed point iterates for the function f which contains an element in D converges to σ. A discussion of Fréchet derivatives and proofs of Theorems 7 and 8 can be found, for instance, in [8]..5. Gronwall s inequality The following well-known inequality can be found in, for instance, []. Theorem 9. Suppose that f and g are continuous functions on the interval ra, bs such that fptq ě 0 and gptq ě 0 for all a ď t ď b. (46) Suppose further that there exists a real number C ą 0 such that fptq ď C ` ż t s fpsqgpsq ds for all a ď t ď b. (47) 7

8 Then ˆż t fptq ď C exp gpsq ds a for all a ď t ď b. (48).6. Schwarzian derivatives The Schwarzian derivative of a smooth function f : R Ñ R is tf, tu f 3 ptq f ptq 3 ˆf ptq. (49) f ptq If the function xptq is a diffeomorphism of the real line (that is, a smooth, invertible mapping R Ñ R), then the Schwarzian derivative of xptq can be related to the Schwarzian derivative of its inverse tpxq; in particular, ˆdx tx, tu tt, xu. (50) dt The identity (50) can be found, for instance, in Section.3 of []. 3. Integral equation formulation In this section, we reformulate Kummer s equation pα ptqq λ qptq α 3 ptq α ptq ` 3 4 ˆα ptq α ptq as a nonlinear integral equation. As in the introduction, we assume that the function q has been extended to the real line and we seek a function α which satisfies (5) on the real line. By letting (5) pα ptqq λ expprptqq (5) in (5), we obtain the equation r ptq 4 pr ptqq ` 4λ pexpprptqq qptqq 0 for all t P R. (53) We next take r to be of the form rptq logpqptqq ` δptq, (54) which results in δ ptq q ptq qptq δ ptq 4 pδ ptqq ` 4λ qptq pexppδptqq q qptqpptq, for all t P R, (55) where p is defined by the formula pptq qptq 5 4 ˆq ptq q ptq. (56) qptq qptq Note that the function p appears in the standard error analysis of WKB approximations (see, for instance, []). Expanding the exponential in a power series and rearranging terms yields 8

9 the equation δ ptq q ptq qptq δ ptq`4λ qptqδptq 4 pδ ptqq `4λ qptq Applying the change of variables transforms (57) into δ pxq ` 4λ δpxq 4 pδ pxqq ` 4λ xptq ż t 0 pδpxqq pδptqq ` pδptqq3 3! ` qptqpptq. (57) a qpuq du (58) ` pδpxqq3 3! ` ppxq for all x P R. (59) At first glance, the relationship between the function ppxq appearing in (59) and the coefficient qptq in the ordinary differential equation () is complex. However, the function pptq defined via (56) is related to the Schwarzian derivative (see Section.6) of the function xptq defined in (58) via the formula pptq tx, tu qptq It follows from (60) and Formula (50) in Section.6 that ˆ dt tx, tu. (60) dx ppxq tt, xu. (6) That is to say: p, when viewed as a function of x, is simply twice the Schwarzian derivative of t with respect to x. It is also notable that the part of (59) which is linear in δ is a constant coefficient Helmholtz equation. This suggests that we form an integral equation for (59) using a Green s function for the Helmholtz equation. To that end, we define the integral operator T for functions f P L prq via the formula T rf s pxq 4λ ż 8 8 sin pλ x y q fpyq dy (6) The following theorem summarizes the relevant properties of the operator T. Theorem 0. Suppose that λ ą 0 is a real number, and that the operator T is defined as in (6). Suppose also that f P L prq X C prq. Then:. T rf s pxq is an element of C prq;. T rf s pxq is a solution of the ordinary differential equation y pxq ` 4λ ypxq fpxq for all x P R; and 3. the Fourier transform of T rf s pxq is the principal value of pfpξq 4λ ξ pfpξq 4λ λ ξ ` pfpξq. λ ` ξ 9

10 Proof. We observe that T rf s pxq 4λ ż x 8 4λ sinpλxq ` 4λ cospλxq ż 8 sin pλ px yqq fpyq dy ` sin pλpy xqq fpyq dy 4λ x ż x cospλyqfpyq dy ż x 4λ cospλxq sinpλyqfpyq dy 8 ż 8 x sinpλyqfpyq dy 4λ sinpλxq 8 ż 8 x cospλyqfpyq dy for all x P R. We differentiate both sides of (63) with respect to x, apply the Lebesgue dominated convergence theorem to each integral (this is permissible since the sine and cosine functions are bounded and f P L prq) and combine terms in order to conclude that T rf s is differentiable everywhere and d dx T rf s pxq ż 8 8 (63) cos pλ x y q signpx yqfpyq dy (64) for all x P R. In the same fashion, we conclude that ˆ ż d 8 T rf s pxq fpxq λ sin pλ x y q fpyq dy (65) dx for all x P R. Since f is continuous by assumption and the second term appearing on the right-hand side in (65) is a continuous function of x by the Lebesgue dominated convergence theorem, we see from (65) that T rf s is twice continuously differentiable. By combining (65) and (6), we conclude that T rf s is a solution of the ordinary differential equation 8 We now define the function g through the formula y pxq ` 4λ ypxq fpxq for all x P R. (66) pgpξq 4λ ˆ λ ξ ` λ ` ξ. (67) It is well known that the Fourier transform of the principal value of {x is the function iπ signpxq; (68) see, for instance, [7] or [7]. It follows readily that the inverse Fourier transform of the principal value of (69) λ ξ is exp p λixq signpxq. (70) i From this and (67), we conclude that gpxq 4λ ˆ exp p λixq signpxq exp pλixq signpxq i i sin pλ x q. 4λ (7) 0

11 In particular, T rf s is the convolution of f with g. As a consequence, {T rf spξq pgpξq fpξq p pfpξq 4λ λ ξ ` pfpξq, (7) λ ` ξ which is the third and final conclusion of the theorem. In light of Theorem 0, it is clear that introducing the representation into (59) yields the nonlinear integral equation δpxq T rσ s pxq (73) σpxq S rt rσ ss pxq ` ppxq for all x P R, (74) where S is the operator defined for functions f P C prq by the formula S rf s pxq pf pxqq 4λ pfpxqq ` pfpxqq3 ` pfpxqq4 `. (75) 4! 3! 4! The following theorem is immediately apparent from the procedure used to transform Kummer s equation (5) into the nonlinear integral equation (74). Theorem. Suppose that λ ą 0 is a real number, that q : R Ñ R is an infinitely differentiable, strictly positive function, that xptq is defined by (58), and that ppxq is defined via (6). Suppose also that σ P L prq X C prq is a solution of the integral equation (74), that δ is defined via the formula δpxq T rσ s pxq 4λ 0 ż 8 8 sin pλ x y q σpyq dy, (76) and that the function α is defined by the formula ż t a ˆδpxpuqq αptq λ qpuq exp du. (77) Then:. δpxq is a twice continuously differentiable solution of (59);. δpxptqq is a twice continuously differentiable solution of of (57); 3. αptq is three times continuously differentiable solution of (5); and 4. αptq is a phase function for the ordinary differential equation y ptq ` λ qptqyptq 0 for all 0 ď t ď. (78) 4. Overview and statement of the principal result The composition operator S T appearing in (74) does not map any Lebesgue space L p prq or Hölder space C k,α prq to itself, which complicates the analysis of (74). Moreover, the integral defining T rσ s only exists if either σ P L prq or pσp λq 0. Even if both of these conditions are satisfied, it is not necessarily the case that S rt rσ ss ` p will satisfy either condition. This casts doubts on whether (74) is solvable for arbitrary p.

12 We avoid a detailed discussion of which spaces and in what sense (74) admits solutions and instead show that, under mild conditions on p and λ, there exists a function p b which approximates p and such that the equation σpxq S rt rσ ss pxq ` p b pxq for all x P R (79) admits a solution. Moreover, we prove that if p is nonoscillatory then the solution of (79) is also nonoscillatory and }p p b } 8 decays exponentially in λ. The next theorem, which is the principal result of this article, makes these statements precise. Its proof is given in Sections 5, 6, 7 and 8. Theorem. Suppose that q P C 8 prq is a strictly positive, and that xptq is defined by the formula ż t a xptq qpuq du. (80) Suppose also that ppxq is defined via the formula 0 ppxq tt, xu; (8) that is, ppxq is twice the Schwarzian derivative of the variable t with respect to the variable x defined via (80). Suppose furthermore that there exist real numbers λ ą 0, Γ ą 0 and a ą 0 such that " * λ ě 6 max a, Γ (8) and ppξq ď Γ exp p a ξ q for all ξ P R. (83) Then there exist functions p b P C 8 prq and σ b P L prq X C 8 prq such that σ b is a solution of (79), ˆ pσ b pξq ď Γ exp 5 6 a ξ for all ξ P R, (84) and }p p b } 8 ă 4Γ ˆ 5a exp 5 6 aλ. (85) According to Theorem, if σ is a solution of the integral equation (74), then the function α defined by (77) is a phase function for the differential equation (). We define α b in analogy with (77) using the solution σ b of the modified equation (79). That is, we let δ b be the function defined by the formula and then define α b via δ b pxq T rσ b s pxq 4λ α b ptq λ ż t 0 ż 8 8 sinpλ x y qσ b pyq dy, (86) a ˆδb pxpuqq qpuq exp du. (87) Obviously, α b is not a phase function for the equation (). However, if we define q b : R Ñ R by the formula 5 ˆq b ptq q b ptq p b ptq, (88) q b ptq 4 q b ptq q b ptq

13 then Theorem implies that α b is a phase function for the differential equation y ptq ` λ q b ptqyptq 0 for all 0 ď t ď. (89) Since }q q b } 8 is bounded in terms of }p p b } 8 and the solutions of (89) closely approximate those of () when }q q b } 8 is small, the function α b can be used to approximation solutions of () when }p p b } 8 is small. Theorem 3, which appears below, gives a relevant error estimate. Given ɛ ą 0, it specifies a bound on }p p b } 8 which ensures that solutions of (89) approximate those of () to relative precision ɛ. Its proof appears in Section 9. Defintion. We say that α is an ɛ-approximate phase function for the ordinary differential equation () if there exists a basis of solutions tru, rvu of () such that and where u, v are defined by and uptq ruptq ď ɛ sup ruptq for all 0 ď t ď (90) 0ďtď vptq rvptq ď ɛ sup rvptq for all 0 ď t ď, (9) 0ďtď uptq sinpαptqq α ptq { (9) vptq cospαptqq. (93) α { ptq Theorem 3. Suppose that q P C 8 prq is strictly positive, that the function p is defined by the formula (56), and that there exist real numbers 0 ă η ă η such that and η ď qptq ď η for all 0 ď t ď, (94) q ptq ˇ qptq ˇ ď η for all 0 ď t ď, (95) pptq ď η for all 0 ď t ď. (96) Suppose also that λ ą 0, ɛ ą 0 are real numbers such that η 3{4 0 ă ɛ ă λ exp, (97) 4 and that k is the real number defined by ˆη k 0 η ` 8η ` 0 η η `. (98) Suppose furthermore that p b : R Ñ R is an infinitely differentiable function such that }p p b } 8 ď η exp p kq exp η3{4 ɛ. (99) λ 4 3

14 Then there exists a function q b : r0, s Ñ R such that (88) holds and any phase function for (89) is an ɛ-approximate phase function for (). By combining Theorems, and 3 we obtain the following. Corollary. Suppose that q, xptq, p, η, η, λ, ɛ, k, Γ, a and η are as in the hypotheses of Theorem and 3. Suppose that, in addition, λ ě 6 ˆ k ` η3{4 Γ 5a 4 ` log. (00) η a ɛ Then there exists a function σ b P L prq X C 8 prq such that ˆ pσ b pξq ď Γ exp 5 6 a ξ for all ξ P R (0) and the function α b ptq defined by α b ptq λ ż t 0 a ˆδb pxpuqq qpuq exp du, (0) where δ b is the function defined through the formula δ b pxq ż sinpλ x y qσ b pyq dy, (03) 4λ is an ɛ-approximate phase function for the ordinary differential equation (). Remark. The hypothesis (00) can be replaced with the weaker condition λ ą 6 5a W 5 η a exp p kq exp η3{4 ɛ, (04) 88 Γ 4 where W denotes the branch of the Lambert W function which is real-valued and less than on the interval r {e, 0q (see Section.3). The proof of Theorem is divided amongst Sections 5, 6, 7 and 8. The principal difficulty lies in constructing a function p b which approximates p and for which (79) admits a solution. We accomplish this by introducing a modified integral equation σpxq S rt b rσ ss pxq ` ppxq, (05) where T b is a band-limited version of T. That is, T b rf s is defined via the formula prq bump function. This modified integral equation is introduced in Sec- where bpξq is a Cc 8 tion 5. {T b rf spξq { T rf spξqbpξq, (06) In Section 6, we show that under mild conditions on p and λ, (05) admits a solution σ. The argument proceeds by applying the Fourier transform to (05) and using the contraction mapping principle to show that the resulting equation admits a solution. In Section 7, we estimate the Fourier transform of the solution σ of (05) under the assumption that p is exponentially decaying. We show that pσ is also exponentially decaying, albeit at a slightly slower rate. 4

15 In Section 8, we define a band-limited version σ b of the solution σ of (05) via the formula pσ b pξq pσpξqbpξq. (07) By combining the observation that T b rσ s T rσ b s with (05), we obtain σpxq S rt rσ b ss pxq ` ppxq. (08) We then define p b by the formula p b pxq ppxq ` σ b pxq σpxq (09) and rearrange (08) as σ b pxq S rt rσ b ss pxq ` p b pxq. (0) The decay estimate on pσ is then used to bound }p p b } 8 }σ σ b } 8 ď }pσ pσ b }. 5. Band-limited integral equation In this section, we introduce a band-limited version of the operator T and use it to form an alternative to the integral equation (74). Let bpξq be any infinitely differentiable function such that. bpξq for all ξ ď λ,. 0 ď bpξq ď for all λ ď ξ ď? λ, and 3. bpξq 0 for all ξ ě? λ. We define T b rf s for functions f P L prq via the formula {T b rf spξq fpξq p bpξq () 4λ ξ We will refer to T b as the band-limited version of the operator T and and we call the nonlinear integral equation σpxq S rt b rσ ss pxq ` ppxq for all x P R () obtained by replacing T with T b in (74) the band-limited version of (74). Since T b is a Fourier multiplier, it is convenient to analyze () in the Fourier domain rather than the space domain. We now introduce notation which will allow us to write down the equation obtained by applying the Fourier transform to both sides of (). We let W b and Ă W b be the linear operators defined for f P L prq via the formulas and bpξq W b rf s pξq fpξq (3) 4λ ξ ĂW b rf s pξq fpξq bpξqiξ 4λ ξ, (4) where bpξq is the function used to define the operator T b. 5

16 For functions f P L prq, it is standard to denote the Fourier transform of the function exppfpxqq by exp rfs; that is, exp rfs pξq δpξq ` fpξq ` f fpξq ` f f fpξq `. (5)! 3! In (5), δ defers to the delta distribution and f f f denotes repeated convolution of the function f with itself. The Fourier transform of exppf pxqq never appears in this paper; however, we will encounter the Fourier transforms of the functions and exppf pxqq (6) exppfpxqq fpxq. (7) So, in analogy with the definition (5), we define exp rfs for f P L prq by the formula exp rfs pξq fpξq ` f fpξq ` f f fpξq `, (8)! 3! and we define exp rfs for f P L prq via the formula exp rfs pξq f fpξq ` f f fpξq `. (9)! 3! That is, exp rfs is obtained by truncating the leading term of exp rfs and exp rfs is obtained by truncated the first two leading terms of exp rfs. Finally, we define functions ψpξq and vpξq using the formulas and ψpξq pσpξq (0) vpξq ppξq. () Applying the Fourier transform to both sides of () results in the nonlinear equation where R rfs is defined for f P L prq by the formula ψpξq R rψs pξq, () R rfs pξq 4 Ă W b rf s ĂW b rf s pξq 4λ exp rw b rf s s pξq ` vpξq. (3) 6. Existence of solutions of the band-limited equation. In this section, we give conditions under which the sequence tψ n u 8 n 0 of fixed point iterates for () obtained by using the function v defined by () as an initial approximation converges. More explicitly, ψ 0 is defined by the formula and for each integer n ě 0, ψ n` is obtained from ψ n via ψ 0 pξq vpξq, (4) ψ n` pξq R rψ n s pξq. (5) 6

17 Theorem 4. Suppose that λ ą 0 is a real number, and that v P L prq such that }v} ď λ 8. (6) Theen the sequence tψ n u 8 n 0 defined by (4) and (5) converges in L prq norm to a function ψ which satisfies the equation () for almost all ξ P R. Proof. We observe that the Fréchet derivative (see Section.4) of R at f is the linear operator R f : L prq Ñ L prq given by the formula Rf W rhs pξq Ă b rf s ĂW b rhs pξq 4λ exp rw b rf ss W b rhs pξq. (7) From formulas (3) and (4) and the definition of bpξq we see that and }W b rf s} ď }f} λ (8) W Ă b rf s ď }f}? (9) λ for all f P L prq. From (7), (8) and (9) we conclude that R f rhs ď W Ă b rf s Wb Ă rhs ` 4λ }W b rf s} exp p}w b rf s} q }W b rhs ď }f} }h} ` 4λ }f} }h} ˆ}f} 4λ λ λ exp λ ˆ}f} ď 4λ ` }f} ˆ}f} exp }h} λ λ (30) for all f and h in L prq. Similarly, by combining (3), (8) and (9) we conclude that }R rf s} ď W 4 Ă b rf s ` 4λ }W b rf s} exp p}w b rf s} q ` }v} (3) ď }f} 8λ ` }f} ˆ}f} exp ` }v} λ λ whenever f P L prq. Now we set r λ {9 and let B denote the closed ball of radius r centered at 0 in L prq. Since ˆ 9 ă W 0, (3) 8 where W 0 denotes the branch of the Lambert W function which is real-valued and greater than or equal to on the interval r {e, 8q (see Section.3), we conclude that r r λ exp ă λ 4. (33) According to Theorem 5 of Section.3 and (3), it also the case that r λ ă 8. (34) 7

18 We combine (33), (34) with (30) to conclude that R f rhs r ď 4λ ` r r ˆ λ exp }h} λ ă 3 ` }h} ă 4 }h} (35) for all h P L prq and f P B. In other words, the L prq Ñ L prq operator norm of the linear operator Rf is bounded by { whenever f is in the ball B. Similarly, we insert (33), (34) and (6) into (3) to conclude that }R rf s} ď r 8λ ` r r λ exp ` r ˆ λ r ď r 8λ ` r r λ exp ` λ ˆ ď r 64 ` 4 ` for all f P B. ď r (36) Together with Theorem 7 of Section.4, formula (35) implies that the operator R is a contraction on the ball B while (36) says that it maps the ball B into itself. We now apply the contraction mapping theorem (Theorem 8 in Section.4) to conclude that any sequence of fixed point iterates for () which originates in B will converge to a solution of (). Since tψ n u is such a sequence, we are done. If ψ is a solution of () then the function σ defined by the formula σpxq ż 8 8 exppixξqψpξq dξ (37) is clearly a solution of the band-limited integral equation (). Note that because ψ P L prq, the integral in (37) is well-defined and σ is an element of the space C 0 prq of continuous functions which vanish at infinity. We record this observation as follows: Theorem 5. Suppose that λ ą 0 is a real number. Suppose also that p P L prq such that p P L prq and }p} ă λ 8. (38) Then there exists a function σ P C 0 prq which is a solution of the integral equation (). Remark. Since σ is not necessarily in L prq, the integral ż 8 need not exist. Nor is the existence of the improper integral 8 ż R lim RÑ8 R expp ixξqσpxq dx (39) expp ixξqσpxq dx (40) guaranteed. However, when viewed as a tempered distribution, the Fourier transform of σ exists and 8

19 is ψ; that is to say, for all functions f P SpRq. ż 8 8 ψpxqf pxq dx ż 8 8 σpxq p fpxq dx (4) In the next section we will prove that under additional assumptions on v, ψ lies in L prq. This implies that σ P L prq and ensures the convergence of the improper Riemann integrals (40). 7. Fourier estimate In this section, we derive a pointwise estimate on the solution ψ of Equation () under additional assumptions on the function v. Lemma. Suppose that a and C are real numbers such that Suppose also that f P L prq, and that Then 0 ď C ă a. (4) f pξq ď C expp a ξ q for all ξ P R. (43) ˆ ˆ exp rfs pξq ď C ` a ξ C C expp a ξ q exp exp π a πa π ξ where exp is the operator defined in (9). Proof. Let for all ξ P R, (44) gpξq C expp a ξ q (45) and for each integer m ą 0, denote by g m the m-fold convolution product of the function g. That is to say that g is defined via the formula g pξq gpξq (46) and for each integer m ą 0, g m` is defined in terms of g m by the formula g m` pξq g m gpξq. (47) We observe that for each integer m ą 0 and all ξ P R, g m pξq? m { ˆC ξ K m { pa ξ q ac, (48) π Γpmq where K ν denotes the modified Bessel function of the second kind of order ν (see Section.). By repeatedly applying Theorem of Section., we conclude that for all integers m ą 0 and all real t, K m { ptq ď K { ptq K { ptq m ź j ˆ t `j ` t m Γ ` `t m Γ `. `t (49) 9

20 We insert the identity c π K { ptq expp tq (50) t into (48) in order to conclude that for all integers m ą 0 and all real numbers t ą 0,? ˆ { m π t K m { ptq ď expp tq Γ ` `t m Γ `. (5) `t By combining (5) and (48) we conclude that ˆ m C Γ `a ξ ` m g m pξq ď C expp a ξ q (5) πa ΓpmqΓ `a ξ for all integers m ą 0 and all ξ 0. Moreover, the limit as ξ Ñ 0 of each side of (5) is finite and the two limits are equal, so (5) in fact holds for all ξ P R. We sum (5) over m, 3,... in order to conclude that 8ÿ ˆ m C Γ `a ξ ` m exp rgs pξq ď C expp a ξ q πa m Γpm ` qγpmqγ `a ξ C expp a ξ q for all ξ P R. Now we observe that Γpm ` q ď Inserting (54) into (53) yields 8ÿ m m ˆ exp rgs pξq ď C expp a ξ q ˆ m C Γ `a ξ πa 8ÿ m ` m Γpm ` qγpm ` qγ `a ξ (53) for all m 0,,,.... (54) ˆ C m Γ `a ξ πa Γpm ` qγ ` m (55) `a ξ for all ξ P R. Now we apply the binomial theorem (Theorem of Section.), which is justified since C ă a, to conclude that ˆ exp rgs pξq ď C expp a ξ q C `a ξ πa for all ξ P R. We observe that and C expp a ξ q exp ` a ξ log C πa (56) exppxq ď x exppxq for all x ě 0, (57) ˆ ď log ď x for all 0 ď x ď x π. (58) 0

21 By combining (57) and (58) with (56) we conclude that exp rgs pξq ď C expp a ξ q ` a ξ log exp C πa ˆ ď C ` a ξ C expp a ξ q exp exp π a πa for all ξ P R. Owing to (43), ` a ξ ˆ C π ξ log C πa (59) exp rfs pξq ď exp rgs pξq for all ξ P R. (60) By combining this observation with (59), we obtain (44), which completes the proof. Remark 3. Kummer s confluent hypergeometric function Mpa, b, zq is defined by the series where paq n is the Pochhammer symbol Mpa, b, zq ` az b ` paq z pbq! ` paq 3z 3 pbq 3 3! `, (6) paq n Γpa ` nq Γpaq apa ` qpa ` q... pa ` n q. (6) By comparing the definition of Mpa, b, zq with (53), we conclude that ˆ ˆ ` a ξ exp rfs pξq ď C expp a ξ q M,, C πa provided for all ξ P R (63) f pξq ď C expp a ξ q for all ξ P R. (64) The weaker bound (44) is sufficient for our immediate purposes, but formula (63) might serve as a basis for improved estimates on solutions of Kummer s equation. The following lemma is a special case of Formula (48). Lemma. Suppose that C ě 0 and a ą 0 are real numbers, and that f P L prq such that Then fpξq ď C exp p a ξ q for all ξ P R. (65) ˆ ` a ξ f fpξq ď C expp a ξ q a We will also make use of the following elementary observation. for all ξ P R. (66) Lemma 3. Suppose that a ą 0 is a real number. Then expp a ξ q ξ ď a exppq for all ξ P R. (67) We combine Lemmas and with (8) and (9) in order to obtain the following key estimate.

22 Theorem 6. Suppose that Γ ą 0, λ ą 0, a ą 0 and C ě 0 are real numbers such that Suppose also that f P L prq such that and that v P L prq such that 0 ď C ă aλ. (68) fpξq ď C expp a ξ q for all ξ ď? λ, (69) Suppose further that R is the operator defined via (3). Then ˆC ˆ ` a ξ ˆ R rf s pξq ď expp a ξ q λ a 8 ` ˆ π exp for all ξ P R. Proof. We define the operator R via the formula and R by the formula where W b and Ă W b are defined as in Section 5. Then for all ξ P R. We observe that ˇ ˇĂW b rf s pξqˇ ď vpξq ď Γ expp a ξ q for all ξ P R. (70) C 4πλ a ˆ C exp 4πλ ξ ` Γ (7) R rfs pξq 4 Ă W b rf s ĂW b rf s pξq (7) R rfs pξq 4λ exp rw b rf ss pξq, (73) R rf s pξq R rfs pξq ` R rfs pξq ` vpξq (74) By combining Lemma with (75) we obtain ˆ R rfs pξq ď C ` a ξ 8λ expp a ξ q a Now we observe that C? λ expp a ξ q for all ξ P R. (75) for all ξ P R. (76) W b rf s pξq ď C expp a ξ q for all ξ P R. (77) λ Combining Lemma with (77) yields ˆ ˆ ˆ R rfs pξq ď C ` a ξ C C πλ expp a ξ q exp exp a 4πλ a 4πλ ξ (78) for all ξ P R. Note that (68) ensures that the hypothesis (4) in Lemma is satisfied. We combine (76) with (78) and (70) in order to obtain (7), and by so doing we complete the proof. Remark 4. Note that Theorem 6 only requires that fpξq satisfy a bound on the interval r?λ,? λs and not on the entire real line. In the next theorem, we use Theorem 6 to bound the solution of () under an assumption on the decay of v.

23 Theorem 7. Suppose that λ ą 0, a ą 0 and Γ ě 0 are real numbers such that " λ ě 6 max Γ, *. (79) a Suppose also that v P L prq such that vpξq ď Γ expp a ξ q for all ξ P R. (80) Then there exists a solution of ψpξq of equation () such that ˆ ˆ ψpξq ď Γ exp a ξ for all ξ P R. (8) λ Proof. Due to (79) and (80), }v} ď λ 8. (8) It follows from Theorem 4 and (8) that a solution ψpξq of () is obtained as the limit of the sequence of fixed point iterates tψ n pξqu defined by the formula and the recurrence ψ 0 pξq vpξq (83) ψ n` pξq R rψ n s pξq. (84) We now derive pointwise estimates on the iterates ψ n pξq in order to establish (8). Let tβ k u 8 k 0 be the sequence of real numbers be generated by the recurrence relation with the initial value β k` β k λ ` Γ (85) β 0 Γ. (86) It can be established by induction that (79) implies that this sequence is bounded above by Γ and monotonically increasing, and hence β k converges to a real number β such that 0 ď β ď Γ. Now suppose that n ě 0 is an integer, and that ψ n pξq ď β n expp a ξ q for all ξ ď? λ. (87) When n 0, this is simply the assumption (80). The function ψ n` pξq is obtained from ψ n pξq via the formula We combine Theorem 6 with (88) and (87) to conclude that ˆβ ψ n` pξq ď expp a ξ q n ˆ ` a ξ ˆ λ a 8 ` ˆ π exp βn 4πλ a ψ n` pξq R rψ s pξq. (88) exp ˆ βn 4πλ ξ ` Γ (89) for all ξ P R. The application of Theorem 6 is justified: the hypothesis (68) is satisfied since β n ď Γ ď λ a (90) 3

24 for all integers n ě 0. We restrict ξ to the interval r?λ,? λs in (89) and use the fact that aλ ă 6, (9) which is a consequence of (79), in order to conclude that? ˆβ ψ n` pξq ď expp a ξ q n ˆ ` a λ ˆ λ a 8 ` ˆ ˆ π exp βn βn? exp λ 4πλ a 4πλ ˆβ ď expp a ξ q n ˆ λ 6 `? ˆ 8 ` ˆ ˆ π exp βn βn exp 4πλ? ` Γ πλ for all ξ ď? λ. Now we combine (9) with the inequality β n λ ď Γ λ ă 3 and the observation that ˆ 6 `? ˆ 8 ` ˆ ˆ π exp exp 7π 6? ď π in order to conclude that ˆβ ψ n` pξq ď expp a ξ q n λ ď expp a ξ q ˆβ n λ ` Γ β n` expp a ξ q ˆ 6 `? ˆ 8 ` ˆ ˆ π exp exp 7π 6? ` Γ π for all ξ ď? λ. We conclude by induction that (87) holds for all integers n ě 0. ` Γ (9) (93) (94) (95) The sequence tψ n pξqu converges to ψpξq in L prq norm (and hence a subsequence of ψ n pξq converges to ψpξq pointwise almost everywhere) and (87) holds for all integers n ě 0. Moreover, for all integers n ě 0, β n` ď Γ. From these observations we conclude that ψpξq ď Γ expp a ξ q (96) for almost all ξ ď? λ. We also observe that ψpξq is a fixed point of the operator R, so that for all ξ P R. Clearly, ψpξq R rψ s pξq (97) R rfs pξq R rgs pξq (98) for all ξ P R if fpξq gpξq for almost all ξ ď? λ, so (96) in fact holds for all ξ ď? λ. We now apply Theorem 6 to the function ψpξq (which is justified since Γ ă λ a) to conclude that ˆ4Γ ˆ ` a ξ ˆ ψpξq ď expp a ξ q λ a 8 ` ˆ ˆ Γ Γ π exp exp 4πλ a 4πλ ξ ` Γ (99) for all ξ P R. Note the distinction between (87) and (99) is that the former only holds for all ξ in the interval r?λ,? λs, while the later holds for all ξ on the real line. It follows from (79) that λa ă 6 and Γ λ ă 6. (00) 4

25 We insert these bounds into (99) in order to conclude that ˆ ˆ ` a ξ ˆ ψpξq ď Γ expp a ξ q 3λ a 8 ` ˆ ˆ π exp exp 7π πλ ξ ` for all ξ P R. Now we observe that which, when combined with (0), yields ˆˆ ψpξq ď Γ expp a ξ q 9 ` ξ 3λ (0) ˆ π exp ă 7π 6, (0) ˆ 8 ` 6 exp By rearranging the right-hand side of (03) as ˆ Γ expp a ξ q 7 ` ˆ 56 exp πλ ξ ˆ ˆ Γ exp a ˆ ξ exp λ λ ξ ˆ 7 ` ˆ 56 exp πλ ξ ` ξ λ ` ξ 9λ exp and applying Lemma 3, we arrive at the inequality ˆ ˆ ψpξq ď Γ exp a ˆ ξ λ 7 ` 56 ` from which (8) follows immediately. ˆ πλ ξ ` for all ξ P R. (03) ` ξ λ ` ξ ˆ 9λ exp πλ ξ ` exppq ` ˆ πλ ξ ` 4 33 exppq ` for all ξ P R, Suppose that ψ P L prq is a solution of (). Then the function σ defined by the formula σpxq ż 8 8 (04) exppixξqψpξq dξ (05) is a solution of the integral equation (). However, the Fourier transform of (05) might only be defined in the sense of tempered distributions and not as a Lebesgue or improper Riemann integral. If, however, we assume the function p appearing in () is an element of L prq and impose the hypotheses of Theorem 8 on the Fourier transform of p, then ψ P L prq, from which we conclude that σ is also an element of L prq. In this event, there is no difficulty in defining the Fourier transform of σ. We record these observations in the following theorem. Theorem 8. Suppose that there exist real numbers λ ą 0, Γ ą 0 and a ą 0 such that " λ ą 6 max Γ, *. (06) a Suppose also that p P L prq such that and ppξq ď Γ exp p a ξ q for all ξ P R. (07) Then there exists a solution σpxq of the integral equation () such that ˆ ˆ pσpξq ď Γ exp a ξ for all ξ P R (08) λ 5

26 Remark 5. The bound (08) implies that pσ decays faster than any polynomial, from which we conclude that σ is infinitely differentiable. 8. Approximate solution of the original equation We would like to insert the solution σ of () into the original equation (74). However, we have no guarantee that σ is in L prq, nor do we expect that pσp λq 0. As a consequence, the integrals defining T rσ s might not exist. To remedy this problem, we define a band-limited version σ b of σ by the formula pσ b pξq pσpξqbpξq, (09) where bpξq is the function used to define the operator T b. We observe that there is no difficulty in applying T to σ b since Moreover, T b rσ s T rσ b s, so that Rearranging (), we obtain where p b pxq is defined the formula pσ b p λq 0. (0) σpxq S rt rσ b ss pxq ` ppxq for all x P R. () σ b pxq S rt rσ b ss pxq ` p b pxq for all x P R, () p b pxq ppxq ` σ b pxq σpxq. (3) Using (08) and (3), we conclude that under the hypotheses of Theorem 8, }p p b } 8 ď } pσ b pσ} ż ď Γ exp ď ξ ěλ 4Γ a exp λ ď 4Γ 5a exp ˆ ˆ a λ ˆ ˆ a λ ˆ 5 6 aλ. λ ξ dξ (4) Together Theorem 8 and (4) imply Theorem. 9. Backwards error estimate In this section, we prove Theorem 3. Although both p and p b are defined on the real line, we are only concerned with solutions of () on the interval r0, s, and so we only require estimates there. Accordingly, throughout this section we use } } 8 to denote the L 8 pr0, sq norm. 6

27 We omit the proof of the following lemma, which is somewhat long and technical but entirely elementary (it can be established with the techniques found in ordinary differential equation textbooks; see, for instance, Chapter of [4]). Lemma 4. Suppose that q : R Ñ R is infinitely differentiable and strictly positive, that pptq is defined by (56), and that there exist real numbers η ą 0 and η ą 0 such that and Let η ď qptq ď η for all 0 ď t ď, (5) pptq, q ptq ď η for all 0 ď t ď. (6) ˆη k 0 η and suppose also that ɛ ą 0 is a real number such that ` 8η ` 0 η η ` (7) ɛ ă η. (8) Suppose furthermore that p b : r0, s Ñ R is an infinitely differential function such that }p p b } 8 ď ɛ expp kq. (9) Then there exists an infinitely differentiable function q b : r0, s Ñ R such that 5 ˆq b ptq q b ptq p b ptq for all 0 ď t ď (0) q b ptq 4 q b ptq q b ptq and }q q b } 8 ď ɛ. () We derive a bound on the change in solutions of the ordinary differential equation () when the coefficient qptq is perturbed. Lemma 5. Suppose that λ ą 0, ɛ ą 0, η ą 0 and η ą 0 are real numbers. Suppose also that q : r0, s Ñ R is a continuously differentiable function such that that pptq is defined by the formula (56), and that η ď qptq ď η for all 0 ď t ď, () pptq ď η for all 0 ď t ď. (3) Suppose furthermore that q b : r0, s Ñ R is a continuously differentiable function such that qptq q b ptq ď η λ exp η3{4 ɛ for all 0 ď t ď. (4) 4 If zptq is a solution of the ordinary differential equation z ptq ` λ qptqzptq 0 for all 0 ď t ď (5) and z 0 ptq is the unique solution of the ordinary differential equation z 0ptq ` λ q b ptqz 0 ptq 0 for all 0 ď t ď (6) 7

28 such that z 0 p0q zp0q and z 0p0q z p0q, then }z z 0 } 8 ď ɛ}z} 8. (7) Proof. We start by observing that the function ψptq z 0 ptq zptq is the unique solution of the initial value problem # ψ ptq ` λ qptqψptq λ fptq for all 0 ď t ď ψp0q ψ (8) p0q 0, where fptq pqptq q 0 ptqq pψptq ` zptqq. (9) We now apply the well-known Liouville-Green transformation to (8) in two steps. First, we introduce the function φptq pqptqq {4 ψptq, (30) which is the solution of the initial value problem $ & φ ptq q ptq qptq φ ptq ` λ qptqφptq pqptqq {4 λ fptq qptqpptqφptq for all 0 ď t ď 4 % φp0q φ p0q 0. Next we introduce the change of variables xptq ż t 0 (3) a qpuq du, (3) which transforms (3) into $ & φ pxq ` λ φpxq pqpxqq 3{4 λ fpxq 4 ppxqφpxq for all 0 ď x ď x % φp0q φ p0q 0, where 0 x ż 0 (33) a qpuq du. (34) We now use the Green s function for the initial value problem (33) obtained from the Liouville- Green transform to conclude that for all 0 ď x ď x, ż x ˆ sinpλpx yqq φpxq pqpyqq 3{4 λ fpyq λ 4 ppyqφpyq dy. (35) By inserting (9) into (35) and we obtain the inequality ˆ ż λ x φpxq ď }q q b } 8 ` η φpyq dy ` λ }q q b } 8 }z} 8 for all 0 ď x ď x. (36) η 4 0 We apply Gronwall s inequality (Theorem 9 in Section.5) to (36) in order to conclude that φpxq ď λ ˆˆ λ }q q b } 8 }z} 8 exp }q q b } 8 ` η x for all 0 ď x ď x. (37) η 4 η 3{4 η 3{4 8

29 Combing (30), (37) and the observation that ż a { x qpuq du ď η (38) yields the inequality for all 0 ď t ď. Now if and only if φptq ď λ η }q q b } 8 exp λ η }q q b } 8 exp 0 ˆ λ }q q b } 8 η { η 3{4 exp }z} 8 (39) η 4 ˆ λ }q q b } 8 η { exp η λ η { }q q b } 8 ă W 0 ɛη { exp η η3{4 4 η 3{4 ă ɛ (40) 4, (4) where W 0 is the branch of the Lambert W function which is real-valued and greater than on the interval r {e, 8q (see Section.3). According to Theorem 5, λ η { }q q b } 8 ă ɛη { exp η3{4, (4) η 4 implies (4). We algebraically simplify (4) in order to conclude that }q q b } 8 ă η λ exp η3{4 ɛ (43) 4 implies (39). By combining Lemmas 4 and 5 we obtain Theorem Numerical experiments In this section, we describe numerical experiments which, inter alia, illustrate one of the important consequences of the existence of nonoscillatory phase functions. Namely, that a large class of special functions can be evaluated to high accuracy using a number of operations which does not grow with order. Although the proof of Theorem suggests a numerical procedure for the construction of nonoscillatory phase functions, we utilize a different procedure here. It has the advantage that the coefficient q in the ordinary differential equation () need not be extended outside of the interval on which the nonoscillatory phase function is constructed. A paper describing this work is in preparation. The code we used for these calculations was written in Fortran and compiled with the Intel Fortran Compiler version..3. All calculations were carried out in double precision arithmetic on a desktop computer equipped with an Intel Xeon X5690 CPU running at 3.47 GHz. 9

30 0.. A nonoscillatory solution of the logarithm form of Kummer s equation. In this experiment, we illustrate Theorem in Section 4. We first construct a nonoscillatory solution r of the logarithm form of Kummer s equation r ptq 4 pr ptqq ` 4λ pexpprptqq qptqq 0 (44) on the interval r, s, where λ =,000 and q is the function r, s Ñ R defined by the formula qptq ˆ 3 ` ` 0t ` t3 cosp5tq. (45) Then we compute the 500 leading Chebyshev coefficients of q and r. We display the results of this experiment in Figures and. Figure contains plots of the functions q and r, while Figure contains a plot of the base-0 logarithms of the absolute values of the leading Chebyshev coefficients of q and r. We observe that, consistent with Theorem, the Chebyshev coefficients of both r and q decay exponentially, although those of r decay at a slightly slower rate. 0.. Evaluation of Legendre polynomials. In this experiment, we compare the cost of evaluating Legendre polynomials of large order using the standard recurrence relation with the cost of doing so with a nonoscillatory phase function. For any integer n ě 0, the Legendre polynomial P n pxq of order n is a solution of the second order differential equation p t qy ptq ty ptq ` npn ` qyptq 0. (46) Equation (46) can be put into the standard form ˆ ` n ψ nt n pt q ptq ` ψptq 0 (47) p t q by introducing the transformation ψptq? t yptq. (48) Legendre polynomials satisfy the well-known three term recurrence relation pn ` qp n` ptq pn ` qtp n ptq np n ptq. (49) See, for instance, [] for a discussion of the these and other properties of Legendre polynomials. For each of 9 values of n, we proceed as follows. We sample 000 random points t, t,..., t 000 (50) from the uniform distribution on the interval p, q. Then we evaluate the Legendre polynomial of order n using the recurrence relation (49) at each of the points t, t,..., t 000. Next, we construct a nonoscillatory phase function for the ordinary differential equation (47) and 30

31 use it evaluate the Legendre polynomial of order n at each of the points t, t,..., t 000. Finally, for each integer j,..., 000, we compute the error in the approximation of P n pt j q obtained from the nonoscillatory phase function by comparing it to the value obtained using the recurrence relation (we regard the recurrence relation as giving the more accurate approximation). The results of this experiment are shown in Table. There, each row correponds to value of n. That value is listed n, as is the time required to compute each phase function for that value of n, the average time required to evaluate the Legendre polynomial of order n using the recurrence relation, the average cost of evaluating the Legendre polynomial of order n with the nonoscillatory phase function, and the largest of the absolute errors in the approximations of the quantities obtained via the phase function method. P n pt q, P n pt q,..., P n pt 000 q This experiment reveals that, as expected, the cost of evaluating P n ptq using the recurrence relation (49) grows as Opnq while the cost of doing so with nonoscillatory phase function is independet of order. However, it also exposes a limitation of phase functions. The values of P n ptq are obtained in part by evaluating sine and cosine of a phase function whose magnitude is on the order of n. This imposes limitations on the accuracy of the method due to the well-known difficulties in evaluating periodic functions of large arguments. Figure 3 contains a plot of the nonoscillatory phase function for the equation (47) when n,000, Evaluation of Bessel functions. In this experiment, we compare the cost of evaluating Bessel functions of integer order via the standard recurrence relation with that of doing so using a nonoscillatory phase function. We will denote by J ν the Bessel function of the first kind of order ν. It is a solution of the second order differential equation t y ptq ` ty ptq ` pt ν qyptq 0, (5) which can be brought into the standard form ˆ ψ ptq ` λ {4 ψptq 0 (5) t via the transformation ψptq? t yptq. (53) An inspection of (5) reveals that J ν is nonoscillatory on the interval ˆ 0, 4ν? (54) 3