Improved estimates for nonoscillatory phase functions

Recently, it was observed that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In particular, under mild assumptions on the coefficients and wavenumber $\lambda$ of the equation, there exists a function whose Fourier transform decays as $\exp(-\mu |\xi|)$ and which represents solutions of the differential equation with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$. In this article, we establish an improved existence theorem for nonoscillatory phase functions. Among other things, we show that solutions of second order linear ordinary differential equations can be represented with accuracy on the order of $\lambda^{-1} \exp(-\mu \lambda)$ using functions in the space of rapidly decaying Schwartz functions whose Fourier transforms are both exponentially decaying and compactly supported. These new observations play an important role in the analysis of a method for the numerical solution of second order ordinary differential equations whose running time is independent of the parameter $\lambda$. This algorithm will be reported at a later date.


Introduction
Given a differential equation where λ is a real number and q : r0, 1s Ñ R is smooth and strictly positive, a sufficiently smooth α : ra, bs Ñ R is a phase function for (1) if the pair of functions u, v defined by the formulas uptq " cospαptqq and vptq " sinpαptqq form a basis in the space of solutions of (1). Phase functions have been extensively studied: they were first introduced in [1], play a key role in the theory of global transformations of ordinary differential equations [2,3], and are an important element in the theory of special functions [4,5,6,7].
It was observed by E.E. Kummer in [1] that α is a phase function for (1) if and only if it satisfies the third order nonlinear differential equatioǹ α 1 ptq˘2 " λ 2 qptq´1 2 α 3 ptq α 1 ptq`3 4ˆα 2 ptq on the interval ra, bs. The presence of quotients in (4) is often inconvenient, and we prefer the more tractable equation obtained from (4) by letting Of course, if r is a solution of (5) then the function α defined by the formula is a solution of (4). We will refer to (4) as Kummer's equation and (5) as the logarithm form of Kummer's equation. The form of these equations and the appearance of λ in them suggests that their solutions will be oscillatory -and most of them are. However, there are several well-known examples of second order ordinary differential equations which admit nonoscillatory phase functions. For example, the function is a phase function for Chebyshev's equation y 2 ptq`ˆ2`t 2`4 λ 2 p1´t 2 q 4p1´t 2 q 2˙y ptq " 0 for all´1 ď t ď 1.
Its existence is the basis of many numerical algorithms, including the fast Chebyshev transform (see, for instance, [8]). Bessel's equation t 2˙y ptq " 0 for all 0 ă t ă 8 (10) also admits a nonoscillatory phase function, although it cannot be expressed in terms of elementary functions (see, for instance, [9]).
Exact solutions of (4) which are nonoscillatory need not exist in the general case. However, as we show in this article, when the coefficient q appearing in (4) is nonoscillatory, there exists a nonoscillatory function α such that (2), (3) approximate solutions of (1) in the space L 8 pra, bsq with accuracy on the order of λ´1 expp´µλq. In order to make this statement rigorous, we will use the Fourier transform to quantify the notion of "nonoscillatory function." Accordingly, we assume that the coefficient q in (1) extends to a strictly positive function on the entire real line. Moreover, we define the function x via the formula xptq " ż t a a qpuq du, (11) and let ppxq be twice the Schwarzian derivative of the variable t with respect to the variable x (see Section 2.8). We suppose that there exist constants Γ and µ such that |p ppξq| ď Γ expp´µ|ξ|q for all ξ P R, that λ ą 2 µ , and that λ ą 2Γ. Then there exists a function δ in the space SpRq of rapidly decaying Schwartz functions (see Section 2.1) such that rptq " logpqptqq`δpxptqq (13) closely approximates a solution of (5), the support of the Fourier transform p δ of δ is contained in the interval p´?2λ, ? 2λq, andˇˇˇp δpξqˇˇď Γ λ exp p´µ |ξ|q for all |ξ| ă ? 2λ.
Moreover the function α defined via the formula is a phase function for a second order differential equation of the form where ν is an element of SpRq whose L 8 pRq norm is on the order of expp´µλq. While α is not a phase function for the original equation (1), the functions u, v obtained by inserting (15) into the formulas (2) and (3) approximate solutions of (1) with accuracy on the order of λ´1 exp p´µλq.
The bound (14) implies that when δ is approximated using various series expansions, the number of terms required to represent it to a specified precision is independent of λ. For instance, the minimum number of terms in the Legendre expansion of the restriction of δ to the interval ra, bs required to achieve a specified precision is independent of λ. Assuming that q is nonoscillatory, it follows that r and α can be represented likewise; that is, using finite series expansions whose number of terms is independent of λ (it is in this sense that they are nonoscillatory). In other words: Op1q terms are required to represent the function δ which enables us to approximate solutions of (1) with accuracy on the order of O`λ´1 expp´µλq˘. This is in contrast to superasymptotic and hyperasymptotic expansions (see, for instance, [10,11]) which approximate solutions of (1) to accuracy on the order of expp´ρλq, but which require Opλq terms in order to do so. Note that we avoid discussing the Fourier transforms of r and α because they need not decay at infinity and so there is no assurance that their Fourier transforms are functions (as opposed to tempered distributions).
The results presented in this article improve upon those of [12], which observed that solutions of (1) can be represented with accuracy on the order of expp´µλq using functions which are in L 2 pRq X C 8 pRq and whose Fourier transforms decay exponentially. Here we show that the function δ which represents the solutions of (1) is an element of the space SpRq of rapidly decaying Schwartz functions (see Section 2.1) and that its Fourier transform is both exponentially decaying and compactly supported (so that δ is entire). Moreover, we substantially reduce the constants appearing in the bounds on the decay of the Fourier transform of δ and in the error of the associated approximations of the solutions of (1). Among other things, these new observations play an important role in the analysis of an algorithm for constructing a nonoscillatory solution α of (4) whose running time is independent of λ. This algorithm allows for the numerical evaluation of solutions of second order linear ordinary differential equations of the form (1) using a number of operations which is independent of λ. It will be reported at a later date.
The remainder of this paper is organized as follows. Section 2 summarizes a number of well-known mathematical facts and establishes the notation which is used throughout this article. In Section 3, we reformulate Kummer's equation as a nonlinear integral equation. The statement of the main result of this paper, Theorem 12, is given in Section 4 and its proof is divided among Sections 5, 6 and 7.

Function spaces
We denote by C pRq the set of continuous functions R Ñ C. If f P C pRq and, for each ǫ ą 0, there exists a compact set K such that |f pxq| ă ǫ for all x R K, then we say that f vanishes at infinity. We denote the set of continuous functions which vanish at infinity by C 0 pRq. By C 8 pRq we mean the set of infinitely differentiable functions R Ñ C, and C 8 c pRq is the set of compact supported functions in C 8 pRq.
We say that ϕ P C 8 pRq is a Schwartz function if ϕ and all of its derivatives decay faster than any polynomial. That is, if sup tPR |t i ϕ pjq ptq| ă 8 (17) for all pairs i, j of nonnegative integers. The set of all Schwartz functions is denoted by SpRq; clearly, it contains the set C 8 c pRq. We endow SpRq with the topology generated by the family of seminorms so that a sequence tϕ n u of functions in SpRq converges to ϕ in SpRq if and only if We denote the space of continuous linear functionals on SpRq, which are known as tempered distributions, by S 1 pRq. We endow S 1 pRq with the weak-* topology so that a sequence tω n u in S 1 pRq converges to ω P S 1 pRq if and only if lim nÑ8 ω n pϕq " ωpϕq (20) for all ϕ P SpRq. We refer the reader to [13] for a thorough discussion of the properties of Schwartz functions and tempered distributions.

The Fourier transform
We define the Fourier transform of a function f P SpRq via the formula p f pξq " The Fourier transform is an isomorphism SpRq Ñ SpRq (meaning that it is a continuous, invertible mapping SpRq Ñ SpRq whose inverse is also continuous). The formula extends the Fourier transform to an isomorphism S 1 pRq Ñ S 1 pRq. The definition (22) coincides with (21) when f P L 1 pRq. Moreover, when f P L 2 pRq, Owing to our choice of convention for the Fourier transform, and y f¨gpξq " 1 2π whenever f and g are elements of L 1 pRq. Moreover, whenever f and p f are elements of L 1 pRq. The observation that f is an entire function when p f is a compactly supported distribution is one consequence of the well-known Paley-Wiener theorem. See [14,15] for a thorough treatment of the Fourier transform.

Convolution exponentials
Formally, the Fourier transform of is the sum where ψ denotes the Fourier transform of f and δ is the delta distribution. The expression (28), which is referred to as the convolution exponential of ψ, is typically denoted by exp˚rψs.
We will not encounter the expression (28) in this article; however, we will consider the Fourier transforms of functions of the form exppf pxqq´1 and exppf pxqq´f pxq´1.
That is, exp1 rψs is obtained by truncating the leading term of exp˚rψs and exp2 rψs is obtained by truncated the first two leading terms of exp˚rψs. By repeatedly applying the inequality which can be found in [16] (for instance), we obtain and from which we see that the series (31) and (32) converge absolutely in L 1 pRq when ψ P R and therefore define L 1 pRq functions. Suppose that f is the inverse Fourier transform of ψ P L 1 pRq. For each nonnegative integer n, we define f n by f n pxq " We observe that tf n u converges in L 8 pRq to exppf pxqq´1. Since x f n pξq " ψpξq`ψ˚ψ pξq 2!p2πq`¨¨¨`ψ˚ψ˚¨¨¨˚ψ pξq n!p2πq n´1 (37) for each nonnegative integer n, it follows from (34) that ! x f n ) converges in L 1 pRq to exp1 rψs. We conclude that the Fourier transform of exppf pxqq´1 is exp1 rψs. A nearly identical argument shows that the Fourier transform of exppf pxqq´f pxq´1 is exp2 rψs.
Proof. Since f P SpRq, expp|f pxq|q is bounded and for any nonnegative integer k. Consequently, for any nonnegative integer k. We conclude that exppf pxqq´1 decays faster than any polynomial. We observe that the n th derivative of exppf pxqq´1 is of the form P´f pxq, f 1 pxq, f 2 pxq, . . . , f pnq pxq¯exppf pxqq, where P is a polynomial in n variables of the form Since f, f 1 , f 2 , . . . , f pnq are elements of SpRq, decays faster than any polynomial whenever k 1 , k 2 , . . . , k n are nonnegative integers not all of which are 0. We combine this observation with the fact that exppf pxqq is bounded in order to conclude that the function (40) decays faster than any polynomial; i.e., the n th derivative of exppf pxqq´1 decays faster than any polynomial. Therefore exppf pxqq´1 is in SpRq. Since exppf pxqq´f pxq´1 is obtained from exppf pxqq´1 by subtracting the Schwartz function f , it is also an element of SpRq.
We combine Theorem 1 with the observation that the Fourier transform is a continuous mapping SpRq Ñ SpRq in order to obtain the following theorem.

The constant coefficient Helmholtz equation
Under certain conditions on the function f , a solution of the inhomogeneous Helmholtz equation can be obtained via the Fourier transform. For instance, the following theorem is a special case of a more general one which can be found in [17].
Theorem 3. Suppose that f P L 1 pRq X CpRq, and that λ is a positive real number. Then the function g defined by the formula gpxq " 1 2λ is twice continuously differentiable, and We interpret the Fourier transform (46) of g as a tempered distribution defined via principal value integrals; that is to say that for all ϕ P SpRq, We also note that the requirement that f is continuous ensures that (45) holds for all x P R; without such an assumption on f , we are only guaranteed that (45) holds for almost all x P R.
When f P L 2 pRq and λ is real-valued, the integral (44) defining the function g is not necessarily absolutely convergent and the expression which is formally the Fourier transform of g, need not define a tempered distribution. If, however, the support of p f is contained in p´λ, λq, then (48) is a compactly supported element of L 1 pRq. In this case, we define g through the formula gpxq " 1 2π Since g is the inverse Fourier transform of a compactly supported function, it is entire. Moreover, the Fourier transform of from which we conclude that almost everywhere. Since both g and f are both entire, (52) in fact holds for all x P R. We record these observations as follows.
Theorem 4. Suppose that f P L 2 pRq, that λ is a positive real number, and that the support of p f is contained in the interval p´λ, λq. Then the function g defined via the formula gpxq " 1 2π is entire, and The following variant of Theorem 3 can be found in [18].

Theorem 5.
Suppose that f is continuous on the interval ra, bs, and that λ is a positive real number. Suppose also that y : ra, bs Ñ C is twice continuously differentiable, and that

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 The following bound on the ratio of K ν`1 to K ν can be found in [19].
Theorem 6. Suppose that t ą 0 and ν ą 0 are real numbers. Then

The binomial theorem
A proof of the following can be found in [20], as well as many other sources.

Theorem 7.
Suppose that r is a real number, and that y is a real number such that |y| ă 1. Then

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 differ- Theorem 8. 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.
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 1 such that 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`1 " f px n q for all n ě 0.
Theorem 8 is often used to show that a mapping is contractive so that the following result can be applied.

Theorem 9. (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 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 8 and 9 can be found in, for instance, [21].

Schwarzian derivatives
The Schwarzian derivative of a smooth function f : 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 is related to the Schwarzian derivative of its inverse tpxq; in particular, The identity (67) can be found, for instance, in Section 1.13 of [6].

A bump function
It is well known that the function ϕ defined by the formula is an element of C 8 pRq such that ϕpξq " 0 for all ξ ă´1, ϕpξq " 1 for all ξ ą 1 and 0 ď ϕpξq ď 1 for all ξ P R (see, for instance, [16]). We suppose that λ is a positive real number and define the bump function p b via the formula where c " ? 2λ`λ 2 (70) and α " c´λ We observe that p b is an element of C 8 c pRq, the support of p b is contained in p´?2λ, ? 2λq, 0 ď p bpξq ď 1 for all ξ P R, and p bpξq " 1 for all |ξ| ď λ. Since p b is an element of C 8 c pRq, its inverse Fourier transform b is an entire function and an element of SpRq.

The Liouville-Green transform
The Liouville-Green transform is a well-known tool for analyzing the variable coefficient Helmholtz equation The following can be found in Chapter 2 of [22] (for instance).
Theorem 10. Suppose that q : ra, bs Ñ R is twice continuously differentiable and strictly positive, that f : ra, bs Ñ C is continuous, that the function x is defined by the formula and that the function p is defined by the formula Suppose also that y : ra, bs Ñ R is twice continuously differentiable, and that Then the inverse tpxq of the function xptq is continuously differentiable, and the function ϕ : r0, xpbqs Ñ R defined by the formula is the unique solution of the initial value problem Remark 1. We observe that due to Formula (67), the function ppxq appearing in (77) is twice the Schwarzian derivative of the inverse tpxq of the function xptq defined in (73). That is, ppxq " 2tt, xu.

Gronwall's inequality
The following well-known inequality can be found in, for instance, [23].
Theorem 11. Suppose that f and g are continuous functions on the interval ra, bs such that f ptq ě 0 and gptq ě 0 for all a ď t ď b.
Suppose further that there exists a real number C ą 0 such that Then

Integral equation formulation
In this section, we reformulate Kummer's equatioǹ as a nonlinear integral equation. We assume that the function q has been extended to the real line and we seek a function α which satisfies (81) on the real line.
By letting`α in (81), we obtain the equation We next take r to be of the form which results in where p is defined by the formula Expanding the exponential in a power series and rearranging terms yields the equation Applying the change of variables transforms (87) into At first glance, the relationship between the function ppxq appearing in (89) and the coefficient qptq in the ordinary differential equation (1) is complex. However, the function pptq defined via (86) is related to the Schwarzian derivative (see Section 2.8) of the function xptq defined in (88) via the formula It follows from (90) and Formula (67) in Section 2.8 that 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 (89) which is linear in δ is the constant coefficient Helmholtz equation. This suggests that we form an integral equation for (89) using a Green's function for the Helmholtz equation. To that end, we define the linear integral operator T for functions f P L 1 pRq X C pRq via the formula T rf s pxq " 1 4λ We extend the domain of T to include functions f P L 2 pRq whose Fourier transforms have support in the interval p´2λ, 2λq through the formula T rf s pxq " 1 2π Introducing the representation δpxq " T rσs pxq into (89) yields the nonlinear integral equation where S is the nonlinear differential operator defined by the formula S rf s pxq " According to Theorems 3 and 4, if σ is a solution of the integral equation (95) and either σ P L 1 pRq X C pRq or σ P L 2 pRq and the support of p σ is contained in p´2λ, 2λq, then the function δ defined via formula (94) is a solution of (89). Moreover, the function r defined via the formula is a solution of (83), and is a phase function for (1).

Existence of nonoscillatory phase functions
The nonlinear integral equation (95) is not solvable for arbitrary p. However, when the Fourier transform of the function p decays exponentially, there exists a function σ whose Fourier transform is compactly supported and a function ν of magnitude on the order of expp´ρλq, where ρ is a real constant, such that σpxq " S rT rσss pxq`ppxq`νpxq for all x P R.
The following theorem, which is the principal result of this article, makes these statements precise. Its proof is given in Sections 5, 6 and 7.
Theorem 12. Suppose that q P C 8 pRq is strictly positive, that xptq is defined by the formula and that the function p defined via the formula is an element of SpRq. Suppose furthermore that there exist positive real numbers λ, Γ and µ such that and |p ppξq| ď Γ exp p´µ |ξ|q for all ξ P R.
Then there exist functions σ and ν in SpRq such that σ is a solution of the nonlinear integral equation σpxq " S rT rσss pxq`ppxq`νpxq, for all x P R, p σpξq " 0 for all |ξ| ą ? 2λ, and  (102) and (107) that Suppose that σ and ν are the functions obtained by invoking Theorem 12, and that xptq is the function defined by the formula xptq " ż t a a qpuq du.
We define δ by the formula δpxq " T rσs pxq, r by the formula and α by the formula From the discussion in Section 3, we conclude that δpxq is a solution of the nonlinear differential equation that rptq is a solution of the nonlinear differential equation and that α is a solution of the nonlinear differential equatioǹ From (116), we see that α is a phase function for the second order linear ordinary differential equation Since the magnitude of ν is small, we expect that the difference between solutions of (117) and those of (1) will be small as well. Indeed, the following theorem follows easily by applying the Liouville-Green transform and invoking Gronwall's inequality.
Theorem 13. Suppose that q is continuous and strictly positive on the interval ra, bs, that f is a continuous function ra, bs Ñ C, and that λ is a positive real number. Suppose also that z : ra, bs Ñ C is a twice continuously differentiable function such that and that y : ra, bs Ñ C is a twice continuously differentiable function such that ypaq " zpaq, and y 1 paq " z 1 paq. Then there exists a positive real number C such that The constant C depends on q but not on λ or f .
Proof. We define the function ∆ by the formula and observe that ∆ is the unique solution of the boundary value problem # ∆ 2 ptq`λ 2 qptq∆ptq " f ptq for all a ď t ď b ∆paq " ∆ 1 paq " 0.
We define the function xptq via the formula and use tpxq to denote its inverse; moreover, we define the function pptq via the formula According to Theorem 10, the function ϕ defined via is the unique solution of the initial value problem We conclude from (126) and Theorem 5 that for all 0 ď x ď xpbq. We let and }f } 8 " sup aďuďb |f puq| .
We observe thatˇˇˇˇ1 for all 0 ď x ď xpbq. Now we let and observe thatˇˇˇˇ1 4λ for all 0 ď x ď xpbq. We combine (127), (130) and (132) in order to conclude that for all 0 ď x ď xpbq. By invoking Gronwall's inequality (which is Theorem 11 in Section 2.11) we conclude that for all 0 ď x ď xpbq. We conclude from (121), (125) and (134) that for all a ď t ď b, where C is defined via the formula By applying Theorem 13 to (117) and (1) we obtain the following.
The constant C depends on the coefficient q appearing in (1), but not on the parameter λ.
The rest of this article is devoted to the proof of Theorem 12. It is divided among Sections 5, 6 and 7. The principal difficulty lies in constructing a function ν such that (99) admits a solution. We accom-plish this by introducing a modified integral equation where T b is a "band-limited" version of T . That is, T b rf s is defined via the formula where p bpξq is the C 8 c pRq bump function given by Formula (69) of Section 2.9. In Section 5, we apply the Fourier transform to (141) and use the contraction mapping principle to show that under mild conditions on p and λ the resulting equation admits a solution. This gives rise to a solution σ b of (141). In Section 6, we show that if the Fourier transform of the function p is exponentially decaying, then the Fourier transform of σ b is as well. In Section 7 we use the solution σ b of (141) in order to construct functions σ and ν which satisfy (99). Moreover, we show that σ can be taken to be an element of the space SpRq of rapidly decaying Schwartz functions (see Section 2.1), that the Fourier transform of σ is compactly supported and exponentially decaying, and that ν is an element of SpRq whose L 8 pRq norm decays exponentially with λ.

Band-limited integral equation
In this section, we introduce a "band-limited" version of the operator T , use it to form an alternative to the integral equation (95), and apply the contraction mapping principle in order to show that this alternate equation admits a solution under mild conditions on the function p and the parameter λ.
We define the operator T b rf s for functions f P C 0 pRq such that p f P L 1 pRq via the formula We will refer to T b as the band-limited version of the operator T and and we call the nonlinear integral equation obtained by replacing T with T b in (95) the "band-limited" version of (95).

Remark 3. The function
is an element of C 8 c pRq since the support of p b is bounded away from the points˘2λ at which the denominator vanishes. Consequently, (143) is a compactly supported tempered distribution and T b rf s is an entire function whenever f is a tempered distribution. For our purposes, it suffices to know that T b rf s is defined for p f P L 1 pRq.
It is convenient to analyze (144) in the Fourier domain rather than the space domain. We denote by W b and Ă W b the linear operators defined for f P L 1 pRq via the formulas and where p bpξq is the function used to define the operator T b . We define functions ψpξq and wpξq using the formulas and wpξq " p ppξq.
Finally, we define R rf s for functions f P L 1 pRq via where exp2 is the operator defined by Formula (32) of Section 2.3. Applying Fourier transform to both sides of (144) results in the nonlinear equation The following theorem gives conditions under which the sequence tψ n u 8 n"0 of fixed point iterates for (151) obtained by using the function w defined by (149) as an initial approximation converges. More explicitly, ψ 0 is defined by the formula and for each integer n ě 0, ψ n`1 is obtained from ψ n via ψ n`1 pξq " R rψ n s pξq.
Theorem 15. Suppose that λ ą 0 is a real number, and that w is an element of L 1 pRq such that Then the sequence tψ n u defined by (152) and (153) converges in L 1 pRq norm to a function ψ P L 1 pRq such that ψpξq " R rψs pξq for all ξ P R.
Proof. We observe that the Fréchet derivative (see Section 2.7) of R at f is the linear operator R 1 f : L 1 pRq Ñ L 1 pRq given by the formula From formulas (146) and (147) and the definition of p bpξq we see that and for all f P L 1 pRq. We combine (156) with (34) in order to conclude that › › R 1 f rhs for all f and h in L 1 pRq. By inserting (157) and (158) into (159) we see that for all f and h in L 1 pRq. Similarly, by combining (150), (35), (157) and (158) we conclude that whenever f P L 1 pRq. We now let r " πλ 2 and denote by Ω the closed ball of radius r centered at 0 in L 1 pRq. Suppose that f P L 1 pRq such that and that We insert (162) and (163) into (161) in order to obtain from which we conclude that R maps Ω into itself. Next, we insert (162) into (160) in order to obtain which shows that R is a contraction on Ω. We now invoke the contraction mapping theorem (Theorem 9 in Section 2.7) in order to conclude that any sequence of fixed point iterates for (151) which originates in Ω will converge in L 1 pRq to a solution of (151). Since tψ n u is such a sequence, it converges in L 1 pRq to a function ψ such that ψpξq " R rψs pξq for almost all ξ P R.
We change the values of ψ on a set of measure zero (which does not affect R rψs) in order to ensure that (166) holds for all ξ P R.

Fourier estimate
In this section, we derive a pointwise estimate on the solution ψ of Equation (151) under additional assumptions on the function w.

Lemma 1.
Suppose that µ and C are real numbers such that Suppose also that f P L 1 pRq, and that |f pξq| ď C expp´µ|ξ|q for all ξ P R.
Proof. We let and for each integer m ą 0, we define g m`1 in terms of g m through the formula We observe that for each integer m ą 0 and all ξ P R, where K ν denotes the modified Bessel function of the second kind of order ν (see Section 2.5). By repeatedly applying Theorem 6 of Section 2.5, we conclude that for all integers m ą 0 and all real t, We insert the identity into (173) in order to conclude that for all integers m ą 0 and all real numbers t ą 0, By combining (175) and (172) we conclude that for all integers m ą 0 and all ξ ‰ 0. Moreover, the limit as ξ Ñ 0 of each side of (176) is finite and the two limits are equal, so (176) in fact holds for all ξ P R. We sum (176) over m " 2, 3, . . . in order to conclude that exp2 rgs pξq ď C expp´µ|ξ|q for all ξ P R. Now we observe that 1 Γpm`2q ďˆ1 2˙m for all m " 0, 1, 2, . . . .
By combining (181) and (182) with (180) we conclude that for all ξ P R. Note that in (183), we used the assumption that C ă a in order to apply the inequality (182). Owing to (168), |exp2 rf s pξq| ď exp2 rgs pξq for all ξ P R.
By combining this observation with (183), we obtain (169), which completes the proof.
The weaker bound (169) is sufficient for our immediate purposes, but formula (187) might serve as a basis for improved estimates on solutions of Kummer's equation.
The following lemma is a special case of Formula (172).

Lemma 2.
Suppose that C ě 0 and µ ą 0 are real numbers, and that f P L 1 pRq such that |f pξq| ď C exp p´µ|ξ|q for all ξ P R.
We will also make use of the following elementary observation.

Lemma 3.
Suppose that µ ą 0 is a real number. Then We combine Lemmas 1 and 2 with (157) and (158) in order to obtain the following key estimate.
Suppose further that R is the operator defined via (150). Then Proof. We define the operator R 1 via the formula and R 2 by the formula where W b and Ă W b are defined as in Section 5. Then R rf s pξq " R 1 rf s pξq`R 2 rf s pξq`wpξq for all ξ P R. We observe thatˇˇˇĂ By combining Lemma 2 with (199) we obtain Now we observe that In the following theorem, we use Theorem 16 to bound the solution of (151) under an assumption on the decay of w.
Proof. From (203) and (204) we obtain It follows from Theorem 15 and (207) that a solution ψpξq of (151) is obtained as the limit of the sequence of fixed point iterates tψ n pξqu defined by the formula ψ 0 pξq " wpξq (208) and the recurrence ψ n`1 pξq " R rψ n s pξq.
We now derive pointwise estimates on the iterates ψ n pξq in order to establish (205) and (206).
We denote by tβ k u be the sequence of real numbers generated by the recurrence relation with the initial value From mathematical induction and (203), we conclude that tβ k u is a monotonically increasing sequence which converges to We also observe that Now suppose that n ě 0 is an integer, and that |ψ n pξq| ď β n expp´µ|ξ|q for all |ξ| ď ? 2λ.
When n " 0, this is simply the assumption (204). The function ψ n`1 pξq is obtained from ψ n pξq via the formula ψ n`1 pξq " R r ψ s pξq.
The sequence tψ n u converges to ψ in L 1 pRq norm and so a subsequence of ψ n converges to ψ pointwise almost everywhere. From (213) and (214) which is the conclusion (205).
Suppose that p is an element of L 1 pRq, and that there exist positive real numbers µ and Γ such that |p ppξq| ď Γ exp p´µ |ξ|q for all ξ P R.
Suppose further that ψ P L 1 pRq is the solution of (151) obtained by invoking Theorem 15. Then the function σ b defined by the formula is a solution of the integral equation (144). Moreover, according to Theorem 17, if λ ą 2µ´1 and λ ą 2Γ, then the Fourier transform of σ b (which is, of course, ψ) decays faster than any polynomial. In this event, σ b is infinitely differentiable, p σ b P L 2 pRq, and σ b P L 2 pRq. We record these observations in the following theorem.
Theorem 18. Suppose that there exist real numbers λ ą 0, Γ ą 0 and µ ą 0 such that Suppose also that p is an element of L 1 pRq such that |p ppξq| ď Γ exp p´µ|ξ|q for all ξ P R.
Then there exists a solution σ b P L 2 pRq X C 8 pRq of the integral equation (144) such that and

Perturbed integral equation
Suppose that σ b is the function obtained by invoking Theorem 18 so that We rearrange (240) as σ b pxq " S rT rσ b ss pxq`ppxq`pS rT b rσ b ss pxq´S rT rσ b ss pxqq (241) and define ν b via the formula ν b pxq " S rT b rσ b ss pxq´S rT rσ b ss pxq (242) so that σ b is a solution of the perturbed integral equation From the discussion in Section 4, we see that the phase function arising from σ b approximates solutions of (1) with accuracy on the order of λ´1}ν b } 8 . However, it is not immediately apparent that T rσ b s is defined: the integral 1 2λ need not converge absolutely, and without an estimate on the derivative of p σ b , it is not clear that which is formally the Fourier transform of T rσ b s, defines a tempered distribution (although, in fact, it does). It is possible obtain a bound on the derivative of p σ b by modifying the argument of Section 6. That bound can be used show that T rσ b s is defined and to estimate the magnitude of ν b . We prefer the following, simpler approach to constructing an appropriate perturbation ν of the function p.
We define the function σ via the formula where p bpξq is the function used to define the operator T b . We observe that, unlike T rσ b s, there is no difficulty in defining T rσs since σ P L 2 pRq and the support of p σ is contained in p´?2λ, ? 2λq. Moreover, T b rσ b s " T rσs so that σ b pxq " S r T rσs s pxq`ppxq for all x P R.
Rearranging (247), we obtain σpxq " S rT rσs s pxq`ppxq`νpxq for all x P R, where ν is defined the formula Using (239), (236) and (249), we conclude that under the hypotheses of Theorem 18, By combining Theorem 18 with (250) we arrive at the following theorem.
Theorem 19. Suppose that q P C 8 pRq is strictly positive, and that xptq is defined by the formula Suppose also that ppxq is defined via the formula ppxq " 2tt, xu; that is, ppxq is twice the Schwarzian derivative of the variable t with respect to the variable x defined via (251). Suppose furthermore that there exist positive real numbers λ, Γ and µ such that and |p ppξq| ď Γ exp p´µ |ξ|q for all ξ P R.
Then there exist functions ν and σ in L 2 pRq X C 8 pRq such that σ is a solution of the nonlinear integral equation σpxq " S rT rσss pxq`ppxq`νpxq, |p σpξq| ďˆ1`2 Γ λ˙Γ exp p´µ|ξ|q for all |ξ| ă ? 2λ, p σpξq " 0 for all |ξ| ě ? 2λ, and The function σ b is obtained as the inverse Fourier transform of the limit ψ of a sequence of fixed point iterates tψ n u for the equation (151). As a consequence of Theorem 2, the functions ψ n are elements of the space SpRq of Schwartz functions if p p is an element of SpRq. Thus σ b is the limit in L 1 pRq of a sequence of Schwartz functions. The following proof of the principal result of this paper, Theorem 12, proceeds by approximating σ b using the inverse Fourier transform of an appropriately chosen ψ n .
Proof of Theorem 12. We denote by tψ n u the sequence of fixed point iterates for (151) generated by the function p p. That is, ψ 0 is defined by the formula and, for each integer n " 0, 1, 2, . . ., ψ n`1 is defined via the formula ψ n`1 pξq " R rψ n s pξq, where R is as in (150). According to Theorem 15, the sequence tψ n u converges in L 1 pRq to a function ψ such that ψpξq " R rψs pξq for all ξ P R.
Moreover, as a consequence of Theorem 2 and the assumption that p p P SpRq, each of the ψ n is an element of SpRq. We denote the inverse Fourier transform of ψ byσ b . According to Theorem 18,σ b is a solution of the nonlinear integral equatioñ σ b pxq " S rT b rσ b ss pxq`ppxq.
We now defineσ via the formula p σpξq " ψpξq p bpξq, where p b is given in (69), so that T b rσ b s pξq " T rσs pξq.
The function ψ n is in SpRq and p b is an element of C 8 c pRq, so the σ n are contained in SpRq. We combine (263) and (268) in order to obtain }σ n´σ } 8 ď 1 2π from which we conclude that the derivative of T rσ n s converges to the derivative of T rσs in L 8 pRq.
We combine these observations with the definition (96) in order to conclude that }S rT rσ n ss´S rT rσss} 8 Ñ 0 as n Ñ 8.
Together (275) and (267) imply when n is sufficiently large. Note that the inequality (267) is strict.
We have already established that ψ n and σ n are elements of SpRq for all nonnegative integers n. Now we observe that { T rσ n spξq " x σ n pξq 4λ 2´ξ2 " x ψ n pξq p bpξq 4λ 2´ξ2 .
Since the support of p b is contained in p´?2λ, ? 2λq and ψ n is an element of SpRq, we conclude from (277) that the Fourier transform of T rσ n s -and hence T rσ n s -is an element of SpRq for each nonnegative integer n. Next, we combine this observation with Theorem 1 in order to conclude that S rT rψ n ss P SpRq for all nonnegative integers n. We rearrange (273) as ν n pxq " σ n pxq´S rT rσ n ss pxq´ppxq (278) and observe that all of the functions appearing on the right-hand side of (278) are elements of SpRq.
We conclude that ν n P SpRq for all nonnegative integers n.
It follows from Formulas (213) and (214), which appear in the proof of Theorem 18, that |ψ n pξq| ďˆ1`2 Γ λ˙Γ exp p´µ|ξ|q for all |ξ| ď ? 2λ and all nonnegative integers n. We conclude from (273) (276) and (279), and our observation that σ n and ν n are elements of SpRq for all nonnegative integers n that we obtain Theorem 12 by letting σ " σ n and ν " ν n for a sufficiently large n.