An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions
aa r X i v : . [ m a t h . C V ] A ug AN ASYMPTOTIC ANALYSIS OF THE FOURIER-LAPLACETRANSFORMS OF CERTAIN OSCILLATORY FUNCTIONS
FREDERIK BROUCKE, GREGORY DEBRUYNE, AND JASSON VINDAS
Abstract.
We study the family of Fourier-Laplace transforms F α,β ( z ) = F . p . Z ∞ t β exp(i t α − i zt ) d t, Im z < , for α > β ∈ C , where Hadamard finite part is used to regularize the integral whenRe β ≤ −
1. We prove that each F α,β has analytic continuation to the whole complex planeand determine its asymptotics along any line through the origin. We also apply our ideasto show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a quantified version of the Ingham-Karamata theorem, supplying newsimple and constructive proofs of optimality results for these complex Tauberian theorems. Introduction
In this article we study the Fourier-Laplace transforms of the family of oscillatory functions(1.1) f α,β ( t ) := t β exp(i t α ) , t > , where α > β ∈ C . When Re β > −
1, these functions are locally integrable and theirFourier-Laplace transforms are given by(1.2) F α,β ( z ) := Z ∞ t β exp(i t α − i zt ) d t for Im z < . One can extend the definition of F α,β to include any value β ∈ C if one considers theHadamard finite part of the integrals in (1.2) (cf. Subsection 2.1). Note that some instancesof the parameters α and β lead to well-studied classical functions, such as the Gaussian andthe Airy function essentially corresponding to β = 0 and α = 2 or 3, respectively.We shall show here that the Fourier-Laplace transforms F α,β admit analytic continuationas entire functions and our main aim is to determine their asymptotics throughout thecomplex plane. Although the study of these questions is of significant intrinsic interest, letus point out that they naturally arise in several applications.When α ≥ β < −
1, the functions (1.1) and their Fourier transforms,namely, the extensions of (1.2) to the real axis, naturally occur in the study of multifractal
Mathematics Subject Classification.
Primary 41A60; Secondary 30E15, 40E05, 42A38, 44A10, 46F12.
Key words and phrases.
Fourier transform; Laplace transform; oscillatory integrals; optimality of Taube-rian theorems; moment asymptotic expansion; stationary phase principle; steepest descent; analyticcontinuation.F. Broucke was supported by the Ghent University BOF-grant 01J04017.G. Debruyne acknowledges support by Postdoctoral Research Fellowships of the Research Foundation–Flanders (grant number 12X9719N) and the Belgian American Educational Foundation. The latter oneallowed him to do part of this research at the University of Illinois at Urbana-Champaign.J. Vindas was partly supported by Ghent University through the BOF-grant 01J04017 and by the ResearchFoundation–Flanders through the FWO-grant 1510119N. properties of various lacunary Fourier series. Indeed, the first order asymptotics of (1.2) onthe real line in combination with the Poisson summation formula play a crucial role [5] in thedetermination of the pointwise H¨older exponent at the rationals of the family of Fourier series R α,β ( x ) = P ∞ n =1 n β exp(2 π i n α x ), which are generalizations of Riemann’s classical function[6, Chapter 7]. The results of the present paper might certainly be used to further refine [5,Theorem 2.1] and exhibit full trigonometric chirp expansions for R α,β at each rational point.In [7], the asymptotic behavior of F α, − on the real axis has been determined and has shownto be useful in the construction of concrete instances of Beurling prime number systems forthe comparison of abstract prime number theorems.As a new application of the functions f α,β , we now explain how they (and some other closerelatives) can be used to establish some optimality results in Tauberian theory. QuantifiedTauberian theorems have many important applications in several diverse areas of mathemat-ics, ranging from number theory to operator theory. Accordingly, this kind of theorems hasbeen extensively studied over the past decades. We shall consider a variant of the followingmodel theorem, which is a quantified version of the celebrated Ingham-Karamata Tauberiantheorem [13, 14] (cf. [6, Chapter 3] and [15, Chapter III]). Theorem 1 ([3, Proposition 3.2]) . Let τ ∈ L ∞ and κ > . Suppose that the Laplacetransform L{ τ ; s } = R ∞ τ ( t )e − st d t admits an analytic continuation to Ω = { s : Re s ≥ − C/ (1 + | Im s | ) κ } , where it has at most polynomial growth. Then, (1.3) Z x τ ( t )d t = L{ τ ; 0 } + O (cid:18)(cid:18) log xx (cid:19) /κ (cid:19) . In fact, motivated by applications in partial differential equations, the above theorem hasrecently seen numerous generalizations [1, 2]. The most general one in terms of the regionof analytic continuation and the growth inside such a region is currently given in [19]. Anatural question is then whether the error term in (1.3) is sharp. Concerning this questionof optimality in Theorem 1, two rather different approaches are thus far known. The firstone arose in [3] and consisted in a delicate function-theoretic construction, where exactly theoptimality of Theorem 1 was proved. The technique was then refined to show optimality formore general versions of Theorem 1 in [1] and the most general optimality results achievedvia this technique can be found in [19]. The second approach only appeared very recently in[8] and crucially depends on a careful application of the open mapping theorem, see also [9]for the most general results obtained by this method. The question then remains whetherone can find “simple” functions showing optimality results. Indeed, the first approach givesa rather non-explicit complicated function, whereas the second functional analysis approachdoes not even construct a example, it merely shows the existence of one.We wish to indicate here that one can indeed find such “simple” functions, in this wayeffectively providing a new third approach for addressing optimality questions. Our focushere lies on the simplicity of the functions and any attempt to generality is beyond thescope of this article. Furthermore, we shall not directly answer the optimality questionfor Theorem 1, but for a slightly differently formulated version, although the interestedreader may verify via the same techniques developed in this paper that the function τ ( x ) = HE FOURIER-LAPLACE TRANSFORMS OF CERTAIN OSCILLATORY FUNCTIONS 3 exp(i x /κ / log /κ x ) if x ≥ e, τ ( x ) = 0 if x < e, satisfies the hypotheses of Theorem 1 yet Z x τ ( t ) d t = L{ τ ; 0 } + exp (cid:0) i x ( x/ log x ) /κ (cid:1) i(1 + 1 /κ ) (cid:18) log xx (cid:19) /κ + O (cid:18) log /κ − xx /κ (cid:19) . We shall show the optimality of Theorem 1 where the region of analytic continuation Ω isaltered to Ω = (cid:26) s : Re s ≥ − C log( | Im s | + 2)(1 + | Im s | ) κ (cid:27) . The error term in (1.3) then becomes O ( x − /κ ). The function τ ( x ) = f /κ, ( x ) = exp(i x /κ )provides then an extremal example for this theorem. Indeed, τ is clearly bounded, and thepolynomial bounds on the analytic extension of the Laplace transform in Ω follow from ourresults in Section 5. However, by integration by parts one can see that Z x τ ( t ) d t = L{ τ ; 0 } + exp (cid:0) i x /κ (cid:1) i(1 + 1 /κ ) x /κ + O (cid:0) x − − /κ (cid:1) . As a final application to Tauberian theory, we now discuss a recent conjecture by M¨ugeron the Wiener-Ikehara Tauberian theorem, another cornerstone in Tauberian theory, see[15, Chapter III] for a historical overview of this theorem and some of its applications.In [16], it was conjectured that any non-negative non-decreasing function S , whose Mellintransform R ∞ S ( x ) x − s − d x after subtraction of a pole term A/ ( s −
1) admits an analyticextension to a half-plane Re s > σ with 0 < σ <
1, should obey the asymptotic relation S ( x ) = Ax + O ( x σ + ε ), for each ε >
0. This conjecture was already disproved in [10].In fact, in that paper it was even shown that for each positive function δ ( x ) → S satisfying all requirements, yet ( S ( x ) − Ax ) / ( xδ ( x )) is unbounded. However,the proof depended on the open mapping theorem, and therefore this functional analysisapproach is unable to produce any counterexamples. Here, we can give an explicit functiondisproving the conjecture of M¨uger, namely, S ( x ) = Z x (1 + cos(log α u )) d u for α > . Indeed, a simple computation shows that its Mellin transform equals 1 / ( s − − /s +( F α, ( i (1 − s )) + F α, ( i (1 − s ))) / (2 s ) which after subtraction of the pole at 1 admits ananalytic extension to the half-plane Re s > F α, is an entire function. On the other hand, integrating by parts yields S ( x ) = x + x sin (cid:0) log α x (cid:1) α log α − x + O (cid:18) x log α − x (cid:19) . We also mention that it is possible to give a constructive proof of the more general negativeresult from [10] that holds for any function δ ( x ) →
0; see our forthcoming article [4].Let us now return to the main subject of our paper, the Fourier-Laplace transforms F α,β .In Section 3 we show they are entire functions; we will actually provide an explicit formulafor their analytic continuations. We point out that the entire extension of F α,β for β = − .Section 4 is devoted to an asymptotic analysis of F α,β . We shall obtain full asymptotic series The estimates [7, Eq. (3.12) and (3.13)] are unclear for σ >
0. However, upon replacing entire extensionby C ∞ -extension to Re s ≥ F. BROUCKE, G. DEBRUYNE, AND J. VINDAS on any line through the origin. The asymptotic behavior will display Stokes phenomenon,having qualitatively different asymptotic behavior on the two sectors { z : − π − π/α < arg z < } and { z : 0 < arg z < π − π/α } ; see the asymptotic formulas (4.1) and (4.5),respectively. On their boundary rays, the asymptotic behavior will essentially be a mixtureof the previous two cases. When z = x is real and positive for example, we have the followingasymptotic series. Proposition 1.
There are constants c n,α,β , d n,α,β ∈ C such that, as x → ∞ , (1.4) F α,β ( x ) + X m,nβ + nα + m +1=0 i n − m x m n ! m ! (log x + π i / ∼ ∞ X n =0 c n,α,β x β + nα +1 + exp (cid:0) − i α − / ( α − (1 − /α ) x αα − (cid:1) x β +1 − α/ α − ∞ X n =0 d n,α,β x nα/ ( α − . The coefficients c n,α,β and d n,α,β are given by (4.8) and (4.9) , respectively. It should be noted that the main leading terms of the asymptotic expansion (1.4) ofProposition 1 were essentially obtained in [5, 7] in some cases by employing Littlewood-Paleydecompositions of the unity. Our approach in this article is based on different technology.We exploit here the Estrada-Kanwal moment asymptotic expansion [11, 12] in combinationwith contour integration to deduce asymptotic series expansions. This technique turns outto provide a unified way to deal with the distinct cases of asymptotic behavior that we shallencounter in Section 4; in addition, it directly yields desired uniformity of the asymptoticexpansions on closed subsectors. Finally, the article concludes with some polynomial boundsin Section 5 for F α,β on hourglass-shaped neighborhoods of the real line.2. Preliminaries
We collect in this section several useful notions that play a role in the article. We al-ways write z = x + i y for complex variables. We will often employ Vinogradov’s notation f ( t ) ≪ g ( t ) to denote the asymptotic bound f ( t ) = O ( g ( t )). We shall make extensiveuse of Schwartz distribution calculus in our manipulations; see the monographs [12, 18] forbackground material on asymptotic analysis in the distributional setting. Our notation fordistributions is as in [12]. In particular, we denote the dual pairing between a distribution f and a test function ψ as h f, ψ i , or as h f ( t ) , ψ ( t ) i with the use of a dummy variable ofevaluation.2.1. Distributional regularization.
Let us first clarify our interpretation of f α,β ( t ) = t β exp(i t α ) as Schwartz distributions. We take them as 0 on ( −∞ , β > −
1, then f α,β ∈ L , and since they are of polynomial growth, they can be viewed as elements ofthe space of tempered distributions S ′ . When Re β ≤ −
1, they do not define distributionsautomatically and we have to consider regularizations. This can be done in many ways (seee.g. [12, Section 2.4]), but it is desirable that the property tf α,β ( t ) = f α,β +1 ( t ) remains true.(Then F ′ α,β = − i F α,β +1 .) This is the case when we regularize them by taking Hadamard HE FOURIER-LAPLACE TRANSFORMS OF CERTAIN OSCILLATORY FUNCTIONS 5 finite part [12, page 67]. Suppose ψ ∈ S . Define h f α,β , ψ i := F . p . Z ∞ t β e i t α ψ ( t ) d t = Z ∞ t β e i t α ψ ( t ) d t + Z t β (cid:18) e i t α ψ ( t ) − X m,nm + nα +Re β ≤− i n ψ ( m ) (0) n ! m ! t nα + m (cid:19) d t + X ′ m,nm + nα +Re β ≤− i n ψ ( m ) (0) n ! m ! 1 β + nα + m + 1 . Here the notation P ′ means that the possible terms with β + nα + m + 1 = 0 are excluded.This choice for the regularizations also has the property that for fixed α the map β f α,β is meromorphic; it has poles at β = − nα − m − n, m ∈ N with residues X m,nm + nα + β +1=0 i n n ! ( − m δ ( m ) m ! , where δ ( m ) denotes the m -th derivative of the Dirac delta distribution.2.2. The moment asymptotic expansion.
To obtain asymptotics for the functions F α,β ,we will systematically employ various techniques based on the so-called moment asymptoticexpansion [12, Chapter 3] (cf. [11]). Let A be a test function space of smooth functions wheredilation and derivatives act continuously and consider the corresponding distribution space A ′ (its topological dual). A distribution f ∈ A ′ is said to satisfy the moment asymptoticexpansion if(2.1) f ( λt ) ∼ ∞ X n =0 ( − n µ n δ ( n ) ( t ) n ! λ n +1 , as λ → ∞ , where the µ n are the moments of f given by µ n := h f ( t ) , t n i . The relation (2.1) is to beinterpreted in the weak sense, meaning that the asymptotic expansion holds after evaluationat each test function ψ ∈ A . Such distributions are also called distributionally small at ∞ .We shall actually make use of a slight generalization of (2.1). Let ( α n ) n be a sequenceof complex numbers such that (Re α n ) n is increasing and tends to infinity. Suppose a testfunction ψ on (0 , ∞ ) has the following expansion(2.2) ψ ( t ) ∼ ∞ X n =0 c n t α n , as t → + . Then, we consider the generalized moment asymptotic expansion(2.3) h f ( λt ) , ψ ( t ) i ∼ ∞ X n =0 c n µ α n λ α n +1 , as λ → ∞ , where µ α n are the generalized moments of f given by µ α n := h f ( t ) , t α n i . The (generalized)moment asymptotic expansion and its validity have been thoroughly investigated by Estradaand Kanwal and we refer to [12] for a complete account on the subject. See also [17, 20] forsome recent contributions.In this article we shall exploit the fact that (2.3) holds for all elements of the ensuing twodistribution spaces on (0 , ∞ ). F. BROUCKE, G. DEBRUYNE, AND J. VINDAS • The test function space P{ t α n } consists of all functions ψ ∈ C ∞ (0 , ∞ ) having as-ymptotic expansions (2.2) and such that ∀ k ∈ N , ∀ c > ψ ( k ) ( t ) = O k,c (e ct ) , as t → ∞ . This space becomes a Fr´echet space via the ensuing family of norms k ψ k n := sup
We now show that the Fourier-Laplace transform of f α,β , given by F α,β ( z ) := h f α,β ( t ) , e − i zt i = F . p . Z ∞ t β exp (i t α − i zt ) d t for y = Im z < , has holomorphic extension to the whole complex plane. For it, we shift the contour ofintegration to the ray arg ζ = π/ (2 α ) where i ζ α is real and negative . By Cauchy’s theorem, F α,β ( z ) = ˆ f α,β ( z ) = F . p . lim ε → Z Γ ,ε ∪ Γ ,ε ζ β exp(i ζ α − i zζ ) d ζ , where Γ ,ε is the arc of the circle of radius ε and center at the origin between the points ε and ε e i π/ (2 α ) , and Γ ,ε is the half line { e i π/ (2 α ) t : t ∈ [ ε, ∞ ) } . Indeed, defining Γ R as the circle arcof radius R from e i π/ (2 α ) R to R , it is easy to see that, for y < R Γ R ζ β exp(i ζ α − i zζ ) d ζ → We use the principal branch of the logarithm.
HE FOURIER-LAPLACE TRANSFORMS OF CERTAIN OSCILLATORY FUNCTIONS 7 as R → ∞ . After a small computation, one gets the following expression, F α,β ( z ) = e i( β +1) π/ (2 α ) F . p . Z ∞ t β exp( − t α − ie i π/ (2 α ) zt ) d t + X m,nm + nα + β +1=0 i n ( − i z ) m n ! m ! π α i= e i( β +1) π/ (2 α ) "Z ∞ t β exp (cid:0) − t α − ie i π/ (2 α ) zt (cid:1) d t + Z t β (cid:18) exp (cid:0) − t α − ie i π/ (2 α ) zt (cid:1) − X m,nm + nα +Re β ≤− ( − n ( − i z ) m e i mπ/ (2 α ) n ! m ! t nα + m (cid:19) d t + X ′ m,nm + nα +Re β ≤− ( − n ( − i z ) m e i mπ/ (2 α ) n ! m ! 1 β + nα + m + 1 + X m,nm + nα + β +1=0 i n ( − i z ) m n ! m ! π α i . (3.1)The right hand side however is well defined for any z ∈ C since α >
1, so this expressionyields the desired entire continuation.4.
Asymptotic expansion on rays
Write z = R e i θ . We will derive in this section an asymptotic series expansion for F α,β ( R e i θ )as R → ∞ . We distinguish three cases for the angle θ : the sector { z : − π − π/α < arg z < } ,the sector { z : 0 < arg z < π − π/α } , and their boundaries. Case 1: − π − π/α < θ < . In this case we have the following expansion for F α,β ( R e i θ ), uniformly on closed subsectors:(4.1) F α,β ( R e i θ ) + X m,nβ + nα + m +1=0 i n e i( θ − π/ m R m n ! m ! (log R + (cid:0) θ + π/ (cid:1) i) ∼ ∞ X n =0 exp (cid:0) i( nπ/ − ( θ + π/ β + nα + 1)) (cid:1) Γ ∗ ( β + nα + 1) n ! R β + nα +1 , where Γ ∗ ( z ) equals the Euler gamma function Γ( z ) when z / ∈ − N , and otherwise Hadamardfinite part values are used in the case that z ∈ − N , that is,Γ ∗ ( − k ) := F . p . Z ∞ e − t t − k − d t = ( − k k ! (cid:18) − γ + k X j =1 j (cid:19) . Here γ is the Euler-Mascheroni constant.In order to deduce (4.1), we consider two overlapping subcases: the case where − π < θ < y < − π − π/α < θ < − π/α .In the first subcase, we have the following expression for F α,β (which is the original formof the Fourier transform, before shifting the contour):(4.2) F α,β ( R e i θ ) = F . p . Z ∞ t β exp (cid:0) i t α + R e i( θ − π/ t (cid:1) d t. F. BROUCKE, G. DEBRUYNE, AND J. VINDAS
The idea is now to relate the above expression to an evaluation h g ( Rt ) , f α,β ( t ) i for a distri-bution g which is distributionally small at infinity, so that g satisfies the moment asymp-totic expansion. Making this precise, we consider the space P{ t β + nα } from Section 2 (with α n = β + nα ). Clearly, we have that f α,β ∈ P{ t β + nα } , with c n = i n /n ! in its asymptoticexpansion (2.2) as t → + . The distribution g = g θ ∈ P ′ { t β + nα } will be defined as a reg-ularization of the function exp(e i( θ − π/ t ). If − π < θ <
0, then cos( θ − π/ < i( θ − π/ t ) ψ ( t ) is integrable away from the origin for every test function ψ ∈ P{ t β + nα } ;this product might however not be integrable near the origin. We choose the regularizationcorresponding to the expression (4.2). For ψ ∈ P{ t β + nα } , h g ( t ) , ψ ( t ) i := F . p . Z ∞ exp (cid:0) e i( θ − π/ t (cid:1) ψ ( t ) d t = Z ∞ exp (cid:0) e i( θ − π/ t (cid:1) ψ ( t ) d t + Z (cid:18) exp (cid:0) e i( θ − π/ t (cid:1) ψ ( t ) − X m,nm + nα +Re β ≤− c n e i( θ − π/ m m ! t β + nα + m (cid:19) d t + X ′ m,nm + nα +Re β ≤− c n e i( θ − π/ m m ! 1 β + nα + m + 1 . This defines a continuous linear functional on P{ t β + nα } , and one readily sees that h g ( Rt ) , f α,β ( t ) i = F α,β ( R e i θ ) + X m,nm + nα + β +1=0 i n e i( θ − π/ m R m n ! m ! log R. We remark that for fixed α the last sum is non-empty only for countably many values of β ,namely, the poles of the vector-valued meromorphic function β f α,β .One verifies via contour integration that the generalized moments of g are given by h g ( t ) , t β + nα i = e − i( θ + π/ β + nα +1) Γ ∗ ( β + nα + 1) − δ m, − nα − β − e i( θ − π/ m m ! ( θ + π/ . Here δ m, − nα − β − stands for the Kronecker delta, that is, 1 if − nα − β − m , and 0 otherwise. Since g satisfies the generalized moment asymptoticexpansion (2.3), we readily obtain the expansion (4.1). Upon inspecting the error terms insuch an expansion , one sees that they are uniform when − π + ε ≤ θ ≤ − ε , with arbitrary ε > F α,β . In (3.1)we rotate the contour of integration once again over an angle π/ (2 α ), and after some com-putations one gets the following expression for F α,β , F α,β ( R e i θ ) = e i π ( β +1) /α F . p . Z ∞ t β exp (cid:0) − i t α +e i( θ − π/ π/α ) Rt (cid:1) d t + X m,nm + nα + β +1=0 i n R m e i( θ − π/ m n ! m ! πα i . One can now proceed in the same way as in the discussion of the first subcase, and one againfinds the expansion (4.1). So, we have established that this asymptotic series expansionholds in the range − π − π/α < θ < Case 2: < θ < π − π/α . See e.g. [12, Eq. (3.41), p. 116] for an explicit expression of the error term, which carries over to otherdistribution spaces where the generalized moment asymptotic expansion holds (cf. [12, Sections 3.4 and3.7]). The error terms only depend on a dual seminorm of the g θ and they are uniformly bounded in theranges under consideration. HE FOURIER-LAPLACE TRANSFORMS OF CERTAIN OSCILLATORY FUNCTIONS 9
In this case we get the asymptotic series (4.5) stated below. The first order approximationis(4.3) F α,β ( R e i θ ) ∼ e i η ,θ s π ( α − α − / − βα − R β +1 − α/ α − exp (cid:0) e i η ,θ α − / ( α − (1 − /α ) R αα − (cid:1) , where η ,θ := π − α − β − α − θ, η ,θ := αα − θ − π . Notice that cos( η ,θ ) > R Γ e Rf ( ζ ) g ( ζ ) d ζ as R → ∞ , where Γ is a contour in some region Ω where f and g areholomorphic. The basic idea is to shift the contour to one that passes through a saddle point ζ of f , that is, a point for which f ′ ( ζ ) = 0. If the new contour passes through this point insuch a way that Re f reaches a maximum at ζ on this new contour, then one can use someform of the Laplace asymptotic formula to obtain the asymptotics for the integral. We referto [12, Section 3.6] for more details.Starting from expression (3.1), we will use the method of steepest descent on the integralfrom 1 to ∞ and this will give the main contribution; the other two terms are O (e R R | β | )and O ( R | β |− ) respectively, and as we will see they are negligible with respect to the maincontribution. Set κ := 1 / ( α −
1) , ϕ := θ − π/ π/ (2 α ) and perform the substitution t = R κ s to get Z ∞ t β exp (cid:0) − t α + R e i ϕ t (cid:1) d t = R κ ( β +1) Z ∞ /R κ s β exp (cid:0) R κ +1 (e i ϕ s − s α ) (cid:1) d s. The function(4.4) h ( ζ ) := e i ϕ ζ − ζ α is holomorphic in C \ ( −∞ ,
0] and has a saddle point at ζ = α − κ e i κϕ . We shift the contourof integration to Γ = S j Γ j , whereΓ := [ R − κ , r ] , some small r > := { r e i η : η ranging from 0 to κϕ } ;Γ := [ r e i κϕ , ρ e i κϕ ] , some large ρ ;Γ := { ρ e i η : η ranging from κϕ to 0 } ;Γ := [ ρ, ∞ ) . The main contribution will come from the integral over Γ ; for the other integrals we have: R κ ( β +1) Z Γ + Z Γ ! ≪ e εR κ +1 ; R κ ( β +1) Z Γ + Z Γ ! ≪ e − CR κ +1 . Here, ε is a number depending on r which can be made arbitrarily small by choosing r arbitrarily small, and C is a number depending on ρ which can be made positive by choosing ρ sufficiently large. On Γ , Re h ( ζ ) reaches its maximum at the saddle point; applying [12,Eq. (3.172), p. 137] gives R κ ( β +1) Z Γ ζ β exp (cid:0) R κ +1 (e i ϕ ζ − ζ α ) (cid:1) d ζ ∼ i R κ ( β +1) exp (cid:0) α − κ (1 − /α )e i ακϕ R κ +1 (cid:1) ∞ X n =0 ( − n Γ( n + 1 / h δ (2 n ) (cid:0)p h ( ζ ) − h ( ζ ) (cid:1) , ζ β i (2 n )! R ( κ +1)( n +1 / . The branch of p h ( ζ ) − h ( ζ ) is chosen here in such a way that Im p h ( ζ ) − h ( ζ ) is increasingin a neighborhood of ζ on Γ .Since all the other contributions are of lower order than every term in the above asymptoticseries, we have the same asymptotic relation (up to a multiplicative constant) for F α,β :(4.5) F α,β ( R e i θ ) ∼ e i(( β +1) π/ (2 α )+ π/ R β +1 − α/ α − exp (cid:0) e i η ,θ α − / ( α − (1 − /α ) R αα − (cid:1) × ∞ X n =0 ( − n Γ( n + 1 / (cid:10) δ (2 n ) (cid:0)p h ( ζ ) − h ( ζ ) (cid:1) , ζ β (cid:11) (2 n )! R nαα − , where h is given by (4.4) and ζ = α − κ e i κϕ . The asymptotic expansion (4.5) holds uniformlyon closed subsectors. Case 3: θ = 0 or θ = − π − π/α .When z crosses the rays θ = 0 and θ = − π − π/α , the asymptotic behavior of F α,β ( z )changes qualitatively from (4.1) to (4.5). On these rays, the asymptotic behavior will be acombination of both (4.1) and (4.5). To fix ideas, assume θ = 0, z = R . The other case isactually treated similarly, as we explain below. We start from the following expression for F α,β , which can be derived as in Section 3, but now only rotating the contour in the integralfrom R κ to ∞ (recall that κ = 1 / ( α − F α,β ( R ) = e i π ( β +1) / (2 α ) Z ∞ R κ t β exp (cid:0) − t α − ie i π/ (2 α ) Rt (cid:1) d t + i R κ ( β +1) Z π α e i η ( β +1) exp (cid:0) R κ +1 (ie i αη − ie i η ) (cid:1) d η + Z R κ t β exp (cid:0) i t α − i Rt (cid:1) d t + Z ( . . . − . . . ) + X ′ . . . =: I + I + I + I + S. (4.6)We will split the integral I into four pieces using partitions of the unity. The splittingwill be done in two steps. In the first step, we split I into two pieces I ,a + I ,e : consider By convention, a change of variables in the space of analytic functionals is done without taking absolutevalue of the Jacobian, e.g., ( − n h δ ( n ) ( ψ ( z )) , f ( z ) i = d n d ω n (cid:18) f ( ψ − ( ω )) ψ ′ ( ψ − ( ω )) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ω =0 . HE FOURIER-LAPLACE TRANSFORMS OF CERTAIN OSCILLATORY FUNCTIONS 11 two functions such that φ a + φ e = 1 on [1 , R κ ] and 0 < ε < φ a ∈ C ∞ [0 , ∞ ) , supp φ a ⊆ [0 , ε ] , φ a = 1 on [0 , ε/ φ e ∈ C ∞ ( −∞ , R κ ] , supp φ e ⊆ [1 + ε/ , R κ ] , φ e = 1 on [1 + ε, R κ ] . The sum I ,a + I + S = F . p . Z ∞ t β exp(i t α ) φ a ( t ) exp( − i Rt ) d t can be treated analogously as in Case 1 , with one modification. It is no longer the casethat the distribution g θ = exp(e i( θ − π/ t ) = exp( − i t ) belongs to P ′ { t β + nα } . To remedy this,we consider the space K{ t β + nα } from Section 2 (with α n = β + nα ). We have that our testfunction t β exp(i t α ) φ a ( t ) is indeed an element of K{ t β + nα } , as it has compact support. Thefunction e − i t can be regularized to yield an element of K ′ { t β + nα } : the divergence at the originis resolved in the same way as in Case 1 , while the divergence of the integral away from theorigin is resolved by formally integrating by parts enough times so that one gets an absolutelyconvergent integral. More precisely, given ψ ∈ K q { t β + nα } with ψ ( t ) ∼ c t β + c t β + α + · · · ,one regularizes the divergent integral R ∞ e − i t ψ ( t ) d t as Z (cid:18) e − i t ψ ( t ) − X m,nm + nα +Re β ≤− c n ( − i) m m ! t β + nα + m (cid:19) d t + X ′ m,nm + nα +Re β ≤− c n ( − i) m m ! 1 β + nα + m + 1+ q +1 X j =0 ( − j +1 e − i ( − i) j +1 ψ ( j ) (1) + ( − q +2 Z ∞ e − i t ( − i) q +2 ψ ( q +2) ( t ) d t. Using the moment asymptotic expansion on this regularization will give that the asymptoticsof I ,a + I + S are exactly like (4.1) in Case 1 with 0 substituted for θ .Our second step is to deal with the integral I ,e . We first perform the substitution t = R κ s to get I ,e = R ( β +1) κ Z R − κ s β exp (cid:0) − i R κ +1 h ( s ) (cid:1) φ e ( R κ s ) d s, where(4.7) h ( s ) := s − s α . We will estimate I ,e using the stationary phase principle. The function h has a uniquestationary point s = α − κ ; h ′ ( s ) = 0. This stationary point is contained in [ R − κ (1 + ε ) , R is sufficiently large, say R > /κ α . In order to single out the contributionsfrom the endpoints and the interior stationary point, we further split the integral I ,e intothree pieces using φ with φ b + φ c + φ d = 1 on [0 ,
1] and ε ′ with 0 < ε ′ < s / φ b ∈ C ∞ ( R ) , supp φ b ⊆ ( −∞ , s / , φ b = 1 on [0 , s / − ε ′ ]; φ c ∈ C ∞ ( R ) , supp φ c ⊆ [ s / − ε ′ , − ε ′ / , φ c = 1 on [ s / , − ε ′ ]; φ d ∈ C ∞ ( −∞ , , supp φ d ⊆ [1 − ε ′ , , φ d = 1 on [1 − ε ′ / , . This yields three integrals I ,e = I ,b + I ,c + I ,d ; the stationary point s is contained inthe support of φ c if ε ′ is sufficiently small (say ε ′ < (1 − s )). Furthermore, the function φ e ( R κ s ) is 1 on the integration intervals of I ,c and I ,d if R is sufficiently large (say R κ ≥ (1 + ε ) / ( s / − ε ′ )). For the integral I ,b we have: I ,b = R κ Z ∞−∞ ( R κ s ) β exp (cid:0) i R κ +1 ( s α − s ) (cid:1) φ e ( R κ s ) φ b ( s ) d s. We now show that I ,b ≪ n R − n for any n ∈ N . Perform the substitution u = h ( s ) andintegrate by parts n times to obtain I ,b = R κ (i R κ +1 ) n Z J exp( − i R κ +1 u ) d n d u n (cid:18) ( h − ( u ) R κ ) β φ b ( h − ( u )) φ e ( R κ h − ( u )) 1 h ′ ( h − ( u )) (cid:19) d u, where the integration interval is J = [ h ( R − κ (1 + ε/ , h ( s / j d u j h ′ ( h − ( u )) ≪ j R − κ ( α − ( j +1)) ≪ j R jκ , so I ,b ≪ n R κ (1+ | β | ) R nκ R − n ( κ +1) = R κ (1+ | β | ) − n .The integral I ,c equals R κ ( β +1) Z − ε ′ / s / − ε ′ exp (cid:0) − i R κ +1 h ( s ) (cid:1) s β φ c ( s ) d s. The integrand is a smooth function whose support is compact and contains the stationarypoint s . An asymptotic formula can thus be obtained via the stationary phase principle.Employing [12, Eq. (3.212), p. 146] we get I ,c ∼ R κ ( β +1) exp( − i R κ +1 h ( s )) ∞ X n =0 exp(i π (2 n + 1) / n + 1 / n )! R ( κ +1)( n +1 / × (cid:10) δ (2 n ) (cid:0) sgn( s − s ) p h ( s ) − h ( s ) (cid:1) , s β (cid:11) . Finally, I ,d will give a contribution from its endpoint 1, but this will be cancelled by thecontribution from the endpoint 0 of I : I + I ,d ≪ n R − n for every n ∈ N . Also I ≪ n R − n for every n ∈ N .Collecting all terms, we get the asymptotic expansion (1.4), with c n,α,β = 1 n ! exp (cid:18) − i π (cid:0) β + 1 + n ( α − (cid:1)(cid:19) Γ ∗ ( β + nα + 1) , (4.8) d n,α,β = 1(2 n )! exp (cid:0) i π (2 n + 1) / (cid:1) Γ( n + 1 / (cid:10) δ (2 n ) (cid:0) sgn( s − s ) p h ( s ) − h ( s ) (cid:1) , s β (cid:11) . (4.9)where h is given by (4.7) and s = α − κ . Explicitly, we have the following expression for d ,α,β : d ,α,β = e i π/ r πα − α − / − βα − . The case θ = − π − π/α is similar, but starting from equation (3.1) we rotate the contourfrom 0 to R κ over an additional angle of π/ (2 α ), as in the second subcase of Case 1 . Onegets: There are some typos there, one should replace n by 2 n in the phase of the complex exponential and inthe factorial, and n + 1 by n + 1 / λ . HE FOURIER-LAPLACE TRANSFORMS OF CERTAIN OSCILLATORY FUNCTIONS 13 F α,β ( R e − i( π + π/α ) ) + X m,nβ + nα + m +1=0 i n exp( − i m (3 π/ π/α )) n ! m ! R m (log R − i( π/ π/α )) ∼ ∞ X n =0 exp (cid:0) i( nπ/ π/ π/α )( β + nα + 1) (cid:1) Γ ∗ ( β + nα + 1) n ! R β + nα +1 + e i π ( β +1) /α exp (cid:0) i α − / ( α − (1 − /α ) R αα − (cid:1) R β +1 − α/ α − × ∞ X n =0 exp( − i π (2 n + 1) / n + 1 / n )! R nαα − (cid:10) δ (2 n ) (cid:0) sgn( s − s ) p h ( s ) − h ( s ) (cid:1) , s β (cid:11) . Bounds in an hourglass-shaped region near the real line
In this last section we deduce polynomial bounds for F α,β in an hourglass-shaped regionnear the real axis. Given C >
0, consider the closed regionΩ C := (cid:26) z = x + i y ∈ C : | y | ≤ C log(2 + | x | )(1 + | x | ) κ (cid:27) , where κ = 1 / ( α − x negative and sufficiently large in absolute value, z = x + i y lies in the sector treatedin Case 1 , and by the uniformity of the expansions (4.1) there, we have F α,β ( z ) ≪ | x | − − Re β if − − β / ∈ N , | x | − − β log | x | if − − β ∈ N , when z ∈ Ω C and x ≤ . When x is positive, we use a similar contour as in Case 3 : set ρ := Ax κ for a parameter A (to be determined below) and rotate the contour in the integral from ρ to ∞ . We keep x >
1. For the “rotated” integral we have Z ∞ ρ t β exp (cid:0) − t α + e − i π ( − α )( x + i y ) t (cid:1) d t ≪ Z ∞ ρ t β exp (cid:0) − t ( t α − − x − | y | ) (cid:1) d t ≪ e − ρ , since t α − − x − | y | ≥ A α − x − x − C ( x + 1) − κ log( x + 2) ≥ A > x is sufficientlylarge. For the integral over the circle arc we have (using the bounds 2 η/π ≤ sin η ≤ η for0 ≤ η ≤ π/ ρ β +1 Z π α e i βη exp (cid:0) i ρ α e i αη − i( x + i y ) ρ e i η (cid:1) ie i η d η ≪ A x κ (Re β +1) exp( ρ | y | ) Z π α exp( − αρ α η/π + ρxη ) d η ≪ x κ (Re β +1)+ AC π αρ α − πρx (cid:0) − exp( − ρ α + ρxπ/ (2 α )) (cid:1) ≪ x AC + κ Re β − , whenever A > ( π/ (2 α )) κ so that − ρ α + ρxπ/ (2 α ) < The remaining terms in the expression for F α,β ( z ) can be written as Z t β (cid:18) exp(i t α − izt ) − X m,nm + nα +Re β ≤− i n ( − i z ) m n ! m ! t nα + m (cid:19) d t + X ′ m,nm + nα +Re β ≤− i n ( − i z ) m n ! m ! 1 β + nα + m + 1+ Z ρ t β exp (cid:0) i( t α − xt ) (cid:1) exp( yt ) d t. (5.1)Suppose first that Re β ≤ −
1. By Taylor’s theorem, the integrand of the first integralis bounded by ct − ε (cid:12)(cid:12) ( − i z ) ⌊− − Re β ⌋ +1 exp( − i zt ) (cid:12)(cid:12) for some constant c , some positive ε , andsome t ∈ [0 , ≪ x ⌊− − Re β ⌋ +1 . The sum is ≪ x ⌊− − Re β ⌋ .The last integral is ≪ x AC if Re β < − ≪ x AC log x if Re β = −
1. If Re β > − ρ , yielding the bound ≪ A x AC + κ (Re β +1) .In conclusion, for z ∈ Ω C , and any fixed constant A > max(1 , ( π/ (2 α )) κ ), we have F α,β ( z ) ≪ | x | ⌊− − Re β ⌋ +1 + | x | AC , if Re β < − | x | + | x | AC log x, if Re β = − | x | AC + Re β +1 α − , if Re β > − Remark 1.
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