# THE GIBBS PHENOMENON, THE PINSKY PHENOMENON, AND VARIANTS FOR EIGENFUNCTION EXPANSIONS

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THE GIBBS PHENOMENON, THE PINSKY PHENOMENON,AND VARIANTS FOR EIGENFUNCTION EXPANSIONSMichael E. Taylor aa Mathematics Department , University of North Carolina , Chapel Hill, NC, 27599, U.S.A.Published online: 07 Feb 2007.

To cite this article: Michael E. Taylor (2002) THE GIBBS PHENOMENON, THE PINSKY PHENOMENON, AND VARIANTS FOREIGENFUNCTION EXPANSIONS, Communications in Partial Differential Equations, 27:3-4, 565-605, DOI: 10.1081/PDE-120002866

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THE GIBBS PHENOMENON, THE PINSKYPHENOMENON, AND VARIANTS FOR

EIGENFUNCTION EXPANSIONS

Michael E. Taylor

Mathematics Department, University of NorthCarolina, Chapel Hill, NC 27599, USA

ABSTRACT

We examine analogues of the Gibbs phenomenon for eigen-function expansions of functions with singularities acrossa smooth surface, though of a more general nature thana simple jump. The Gibbs phenomena that arise still have auniversal form, but a more general class of fractional sineintegrals arises, and we study these functions. We also makea uniform analysis of eigenfunction expansions in the presenceof the Pinsky phenomenon, and see an analogue of the Gibbsphenomenon there. These analyses are done on three classesof manifolds: strongly scattering manifolds, includingEuclidean space; compact manifolds without strongly focus-ing geodesic flows, including flat tori and quotients of hyper-bolic space, and compact manifolds with periodic geodesicflow; including spheres and Zoll surfaces. Finally, we uncovera new divergence phenomenon for eigenfunction expansionsof characteristic functions of balls, for a perturbation of theLaplace operator on a sphere of dimension 5.

565

Copyright & 2002 by Marcel Dekker, Inc. www.dekker.com

COMMUN. IN PARTIAL DIFFERENTIAL EQUATIONS, 27(3&4), 565605 (2002)

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1. INTRODUCTION

The Gibbs phenomenon, for the partial sums of the Fourier series of afunction on the circle with a jump discontinuity, has been seen to havecounterparts in a variety of situations. Multi-dimensional analogues havebeen treated in several recent papers, including (14). These papers examine(amongst other things) the Gibbs phenomenon for a compactly supportedpiecewise smooth function with a simple jump across a smooth surface , ina number of cases, including Fourier inversion on Euclidean space Rn, toriTn, spheres Sn, and other Riemannian manifolds. In this context, by Fourierinversion of a function f we mean taking

SR f Rffiffiffiffiffiffiffiffi

p f 1:1

and passing to the limit R ! 1. Here is an appropriate self-adjointextension of the Laplace operator and R is the characteristic functionof the interval R,R, set equal to 1=2 at the endpoints.

The Gibbs phenomenon is essentially a local effect, but in higherdimensions nonlocal effects also arise in the Fourier inversion of functionswith simple singularities. The following phenomenon was analyzed in (5).Suppose B R3 is a ball of radius a, centered at 0, g 2 C10 R3, andf gB. Then, as R ! 1,

SR f 0 f 0 2

Avg@B g sin aR o1: 1:2

In particular, SR f 0 ! f 0 if and only if the mean value of gj@B vanishes.This result, often called the Pinsky phenomenon, can be understood as afocusing effect.

In (2) the Pinsky phenomenon was treated via an analysis using thewave equation. We can write

SR f x 1

Z 11

sinRt

tut,x dt, 1:3

with

ut, x cos tffiffiffiffiffiffiffiffi

pf x, 1:4

uniquely defined as the solution to the following initial value problem forthe wave equation:

utt u 0, u0, x f x, ut0, x 0: 1:5

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If f is compactly supported on Rn, or in other cases where the singularitiesof f scatter off to infinity and one has a mild control over local energy decay(when the Riemannian manifold is what we call a strongly scattering mani-fold) then the behavior of (1.3) as R ! 1 depends on the behavior ofut, x only for t in a bounded interval (given x restricted to a compact set).In (2) the phenomenon (1.2) is seen to result from a perfect focus caustic.The effects of other caustics on the pointwise behavior of Fourier inversionis also considered in (2).

The papers (1) and (2) also applied wave equation techniques to studythe Gibbs phenomenon, (1) for Fourier inversion on R2 and (2) both for Rn

and other strongly scattering manifolds. While (1) used the formula for thefundamental solution of the wave equation on R R2, (2) used a progres-sing wave expansion.

This paper has the dual purpose of extending the scope of analyses ofGibbs-type phenomena and analyzing the effects of focusing phenomena,particularly the Pinsky phenomenon, not only at points of focus but also ina neighborhood of the focus, thus providing a precise, uniform analysis ofthe behavior of SR f x as R ! 1 for a significant class of functions withsingularities of classical conormal type.

We consider Gibbs-type phenomena for functions having a ya typesingularity on a smooth surface f y 0g, with 0 a < 1. As in the caseof a jump discontinuity, the Gibbs phenomenon takes a universal form. Wehave, as R ! 1,

SR f x Xj0

AjxRaj FajR x, 1:6

on a neighborhood of . Here SR f is a filtered partial Fourier inverse off , defined by inserting a cut-off t into (1.3); see (1.11) below. Its purpose isto isolate the Gibbs effect. Other cut-offs might isolate various focusingeffects. The function vanishes on and has derivative of norm 1. Wemention parenthetically that the analysis giving (1.6) is applicable outsidethe range a 2 0, 1, but the resulting phenomena are not Gibbs phenom-ena. See 4 for further discussion of this point.

The sine-integral function appearing in the classical Gibbs phenom-enon is extended to a class of fractional sine-integrals Fas, appearing in(1.6), whose qualitative properties are explored in 3. We also discuss theefficient numerical evaluation of these functions. It appears that there is aninteresting dichotomy depending on whether a < ac or a > ac, withac 0:79. See 3 for a further discussion of this.

If f has a classical conormal singularity on and the wave ut, x cos t

ffiffiffiffiffiffiffiffi

pf x has a perfect focus at t a, at a point p =2, the uniform

EIGENFUNCTION EXPANSIONS 567

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analysis of SR f x for x in a neighborhood of p, established in 8, takes thefollowing form. In exponential coordinates with center at p (so p 0), wehave, on a neighborhood B of p,

SR f x f x X

eiaRXj0

Rn3=2j j x,Rx, 1:7

where is the order of singularity of f on , e.g., a in the casementioned above. The functions j x, z are determined by the detailednature of the singularities of f on . They are smooth and have the asymp-totic behavior

j x, z jzjn1=2X

eijzj x, z=jzj, 1:8

as jzj ! 1, with smooth on B Sn1.The Pinsky phenomenon is manifested in the critical case

n 3=2, in which case

SR f x f x X

eiaR 0 x,Rx OR1, 1:9

uniformly on a neighborhood of p. Thus the behavior at the focus is given by

SR f 0 f 0 X

eiaR 00, 0; 1:10

compare (1.2). We might have 0, 0 0, e.g., in the context of (1.2) withAvg@B g 0. However, if f actually has a singularity on of ordern 2=2, then 0 0, z will not be identically zero. In such a case, onehas pointwise convergence SR f x ! f x for each x in a neighborhood ofthe focus p, but the convergence will not be uniform. This can be regarded asa shade of the Gibbs phenomenon!

At this point we need to say more about the introduction of a cut-offinto (1.3) and the analysis of the difference. As mentioned earlier, for com-pactly supported f on Rn and certain other noncompact manifolds satisfyinga strong scattering condition, the contribution to (1.3) from large jtj isnegligible. Clearly, for compact manifolds, the strong scattering conditionfails. As shown in (2), the case M Sn can be treated, making use of theperiodicity of cos tA, where A n 1=221=2. In that case one canexpress SR f x as an integral over a circle, and one obtains results quiteparallel to the case of Rn. We will say more about that case in 9. In anycase, such a trick is not applicable to most compact Riemannian manifolds;notably it does not work for tori. Wave equation techniques can still yield a

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good bit of information about Fourier inversion, but it is convenient to usea cut-off. Pick an even function 2 C10 R, with t 1 for t close to 0,and set

SR f x 1

ZsinRt

tt ut, x dt: 1:11

It is elementary to show that SR f ! f in L2-norm, for any f 2 L2M. We

call this filtered Fourier inversion. The wave equation techniquesmentioned above are effective in analyzing the pointwise behavior ofSR f x in a very general setting. Then it remains to analyze the difference

TR f SR f SR f R

ffiffiffiffiffiffiffiffi

p f , 1:12

where

R R R: 1:13

An attack on (1.12) initiated in (3) takes

TR f x X

R ff x, 1:14

where f: 2 specffiffiffiffiffiffiffiffi

pg is an orthonormal basis of L2M consisting of

eigenfunctions of with eigenvalue 2, and ff f , L2 , and writes

jTR f xj X

jRjj ff j2( )1=2 X

jRjjxj2( )1=2

: 1:15

Applying Hormanders estimateXRR1

jxj2 CRn1, 1:16

given in (6), with n dimM, one obtains

kTR f kL1 CRn3=2 k f k , 1:17

where

M f 2 D0M:X

RR1j ff j2 CR22g:

(1:18

This led to a number of results on pointwise Fourier inversion on compactmanifolds, in (3), (7), (4), and (8).

EIGENFUNCTION EXPANSIONS 569

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Part of the work in applying (1.17) has been to establish that variousfamiliar function spaces embed into M. In particular there are embed-ding results for spaces of conormal distributions. We define I1 M, toconsist of distributions of the form Pf , where f is piecewise smooth witha simple jump across the smooth surface and P is a pseudodifferentialoperator of order and type 1, 0; we write P 2 OPS1, 0M. This containsthe space IM, of classical conormal distributions, defined as above butrequiring that P be a classical pseudodifferential operator of order :P 2 OPSM. It was shown in (4) that when is a smooth hypersurfacein a flat torus Tn,

ITn, Tn: 1:19

In (7) the more general result

I1 M, M 1:20

was established, for a general compact Riemannian manifold M. The result(1.20) was then extended in (8) to

I M, M, 1:21

given 1=2 1, for any smooth conic Lagrangian TMn0.The exponent of R in (1.17) vanishes at n 3=2, the point

where the Pinsky phenomenon arises. This should not be surprising, but itmeans further work is required to show that the Pinsky phenomenon isvisible on compact manifolds. In fact, if we take t 1 on an intervalT0,T0 and know that ut, x cos t

ffiffiffiffiffiffiffiffi

pf x has no caustics as strong

as the perfect focus type for jtj T0, and if a mild geometrical restriction isplaced on M, one can strengthen (1.17) to

limR!1

Rn3=2 kTR f kL1 0, 1:22

for f 2 IM,. We demonstrate this in 5; in fact the proof is a slightmodification of the proof of the basic case n 3=2 done in (8).

Our work in 24 on the Gibbs phenomenon for filtered Fourierinversion deals with a variety of cases in which f 2 IM, with > n 3=2. In such cases it is desirable to supplement (1.17) with Lp-estimates on TR f , which involve lower powers of R. For example, it is clearfrom the definition (1.18) that

kTR f kL2 CR1 k f k : 1:23

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One can interpolate between (1.17) and (1.23), but this does not produceoptimal results. Sharper estimates are produced in 5, using work of (9).Following (9), one obtains an Lpn-estimate on TR f , for pn 2n 1=n 1, and interpolates that with (1.17) and with (1.23). As seen later, in8, examples involving (1.7) explicitly demonstrate the sharpness of theseestimates on TR f in the range pn p 1, therefore giving a proof of thesharpness of the Sogge estimates in this range, different from that given in (9).

These estimates on TR f together with the Gibbs phenomenon analysisyield sharp estimates on SR f in the weak-L

p space LpwM, given f 2IM,, 0 < < 1. In 6 it is shown that fSR f :R 1g is bounded inL1=w M as long as either n 3 or n 3=2n 2. The Pinsky focusanalysis done in 8 also demonstrates the sharpness of this result.

Having described the scope of this paper, we briefly sketch the struc-ture of the following sections. Sections 24 deal with the generalized Gibbsphenomenon and associated special functions Fas. In 5 we deriveLp-estimates on the remainder TR f SR f S

R f , and in 6 we draw

conclusions on Lpw-estimates. Section 7 gives general results on perfectfocus caustics, in preparation for the detailed analysis in 8 of SR f in aneighborhood of a perfect focus, elucidating the Pinsky phenomenon.

In 9 we discuss SR Rffiffiffiffiffiffiffiffi

p for the Laplace operator on

spheres, Zoll surfaces, and variants. The mild geometrical restrictionmentioned before (1.22) fails dramatically here, and we discuss a mechanismfor producing results that are just as sharp in this case as they are forEuclidean space, though with subtle differences in the answers, reflectingthe effect of the special geometry on the analysis. These results are producedfor SRf as R ! 1, avoiding the intervals in which the eigenvalues of

ffiffiffiffiffiffiffiffi

p

cluster. In 10 we consider an example of a perturbation of this operator forwhich the partial sums within clusters exhibit wilder divergence phenomena.

We close with one remark on notation. Above (1.19) we alluded tovarious classes of pseudodifferential operators. Our terminology here, forOPSm1, 0M, etc., follows that used in (10) and in Chapter 13 of (11),terminology essentially derived from (12). The symbol class Sm is denotedSmphg in (13).

2. FILTERED FOURIER INVERSION OF FUNCTIONSWITH CONORMAL SINGULARITIES

Here we analyze

SR f x 1

Z 11

sinRt

tt ut, x dt, 2:1

EIGENFUNCTION EXPANSIONS 571

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where ut, x cos tffiffiffiffiffiffiffiffi

pf x and f x has the form

f x gxxa , 0 a < 1: 2:2

Here g, 2 C1M and we assume > 0 on O, < 0 on MnO, anddx 6 0 for x 2 @O. The cut-off 2 C10 R is assumed to havesupport in an interval I c, c sufficiently small that ut, x develops nocaustics for t 2 I . We assume t 1 for t 2 I1 c1, c1.

For t 2 I , the method of geometrical optics produces a progressingwave expansion of ut, x, as a sum of two functions, with conormal singu-larities across surfaces of the form

t x, jd j 1, j 0, 2:3

on a neighborhood O of . We may assume j xj < c1 for x 2 O. Oneobtains ut, x as the even part of

vt, x A0t, x x ta A1t, x x t1a ANt, x x tNa RNt, x, 2:4

where Aj 2 C1I M are obtained from certain transport equationsand RN 2 CNI M. One can simplify (2.4) as follows. Replace A0t, xby A0 x, x, and absorb the difference into the second term in (2.4). Simi-larly alter the second term and continue. We obtain an expansion of the form

vt, x A0x x ta A1x x t1a ANx x tNa RNt,x, 2:5

with Aj 2 C1M and a new RN 2 CNI M.Thus

SR f x XNj0

AjxIjx,R JNx,R, 2:6

with

Ijx,R 1

Z 11

sinRt

tt x t ja dt, 2:7

and

JNx,R 1

Z 11

sinRt

ttRNt, x dt: 2:8

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Note that if we rewrite (2.2) as

f x ~ggx xa , 2:9

then

A0x ~ggx, x 2 : 2:10

To tackle I0x,R, we first note that omitting t in the integrand justchanges the integral by a rapidly decreasing quantity in R, so we have

I0x,R I#0 x,R OR1, 2:11

with

I#0 x,R 1

Z 11

sinRt

t x ta dt

1

Z 11

sin t

t x t

R

a

dt

Ra FaR x, 2:12

where, for a 2 0, 1,

Fas 1

Z 11

sin t

ts ta dt

1

Z s1

sin t

ts ta dt: 2:13

For a 0 this is the sine-integral one sees in treatments of the classicalGibbs phenomenon. We will make a detailed study of Fas for generala < 1 in the next section.

For j 1, the terms Ijx,R given by (2.7) are amenable to a similaranalysis. As long as j xj < c1, we have thatZ 1

1

sinRt

t x t ja 1 t dt OR1, 2:14

as R ! 1. Now, for j 1, a 2 0, 1, this is not an absolutely convergentintegral, but it is well defined as an oscillatory integral, and the validity ofEq. (2.14) is established by a standard integration by parts argument. Thus,parallel to (2.11)(2.12), we have, for j 1,

Ijx,R I#j x,R OR1, 2:15

EIGENFUNCTION EXPANSIONS 573

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with

I#j x,R 1

Z 11

sinRt

t x t ja dt

Raj FajR x, 2:16

where Faj is given as in (2.13), with now Fb defined as an oscillatoryintegral, for b 0.

3. QUALITATIVE BEHAVIOR OF THEFRACTIONAL SINE-INTEGRALS

The functions Fas, for a < 1, arose in the last section to describe thefiltered Fourier inversion of functions with a singularity of type xaacross a smooth surface S fx: x 0g, assuming d 6 0 on S. Thefunctions are given by

Fas 1

Z s1

sin t

ts ta dt

1

Z 11

sin t

ts ta dt: 3:1

If a 2 0, 1 this integral is absolutely convergent. For a 0 it is interpretedas an oscillatory integral. The case a 0 gives the familiar sine-integralone sees in treatments of the Gibbs phenomenon for Fourier inversion offunctions with a simple jump discontinuity. We want to establish a numberof properties of Fas and discuss its numerical evaluation.

Note that (3.1) is a convolution. Since convolution by sin t=tcoincides with Fourier multiplication by 1, 1, we see that

Fas CaF1, 1tt1a

s, 3:2

where t1a is taken to be the boundary value on R of a functionholomorphic on Re t < 0. That is,

t1a ei1a jtj1a, t < 0: 3:3

If a 2 0, 1, t1a is locally integrable, and for a 0 it is a principal-valuedistribution. More generally, we define it by inductive application oftb1 b1d=dttb. Since (3.2) gives Fa as the Fourier transform of adistribution with compact support, we see that Fas is a smooth functionof s for each a < 1.

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One can compute the factor Ca by integrating both ta and t1aagainst et

2=2 and using the Plancherel identity. One obtains the identity

2Caei1a=2 21=2a

1 a

2

1 a2

: 3:4

An examination of (3.2) allows us to specify the asymptotic behaviorof Fas as s ! 1.

Proposition 3.1. Given a < 1, there exists real-valued Ba such that

Fas sa Basins 1 a=2

sOs2, jsj ! 1: 3:5

Proof. By (3.2), the asymptotic behavior of Fas for large jsj arises from thethree singularities of 1, 1tt1a. The power singularity at t 0 givesback the term sa in (3.5). There are two jumps, of magnitude Ca at t 1and of magnitude Caei1a at t 1. One verifies that these give rise tothe second term on the right side of (3.5), with

Ba 2Caei1a=2, 3:6

which in turn is given by (3.4). &

We next explore the properties of Fas for s of moderate size, at leastfor a not too negative.

Proposition 3.2. For 1 < a < 1, we have

Fa0 > 0: 3:7

Proof. This follows readily from the representation

Fa0 1

Z 10

sin t

tta dt:

One sees that

I Z 1

sin t

tta dt, 2 Z

is an alternating sequence whose absolute value is monotonically decreasing.(In fact, one can read off Fa0 from (3.15) below.) &

EIGENFUNCTION EXPANSIONS 575

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It is well known that the sine-integral F0s has the property

F0s > 0, 8 s 0: 3:8

This is not true for all Fas, a 2 0, 1. In fact, we have the following.

Proposition 3.3. Assume s 2 2k 1, 2k 2, k 2 Z. Then

lima%1

Fas 1: 3:9

Proof. Write ta f1a f2a with f1a supported on 0, 1 and f2a on 1,1.Then convolution by 1 a f1a is an approximate identity as a % 1.Meanwhile one verifies directly that

lima!1

1 a

Zsin t

tf2as t dt 0,

so we have

lima%1

1 aFas sin s

s: 3:10

&

On the other hand, using (3.8) and the asymptotic behavior (3.4), wehave the following:

Proposition 3.4. There exists ac 2 0, 1 such that

0 a < ac, s 0 ) Fas > 0: 3:11

Numerical evidence suggests that the optimal value is

ac 0:79: 3:12

For an efficient numerical evaluation of Fas, with a 2 0, 1, it is convenientto use the representation (3.2), i.e.,

Fas Caffiffiffiffiffiffi2

pZ 10

t1aheits ei1aeits

idt: 3:13

Writing the expression in square brackets as

2 ei1a=2 cos ts 1 a2

, 3:14

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we have

Fas Baffiffiffiffiffiffi2

pZ 10

t1a cos ts 1 a2

dt, 3:15

with Ba as in (3.6). Evaluating at s 0 and using (3.7), we verify thatBa > 0, consistent with (3.4). We write the integral on the right side of(3.15) as

cos1 a

2F1as sin

1 a2

F2as, 3:16

where

F1as Z 10

t1a cos ts dt,

F2as Z 10

t1a sin ts dt:

3:17

Now we can expand cos ts and sin ts into power series and integrate term byterm, obtaining

F1as X1k0

1k

2k!s2k

2k a , F2as X1k0

1k

2k 1!s2k1

2k 1 a :

3:18

If one wishes to graph Fas over s 2 8, 8, it is adequate to sum theseseries over 0 k 40.

4. THE GIBBS PHENOMENON FOR FILTEREDFOURIER INVERSION

We bring together results of the last two sections to analyze conver-gence of SR f to f near the singular set . We recall our set-up. We assume

f x ~ggx xa , 0 a < 1, 4:1

with as in (2.3) and ~gg 2 C1. By (2.5) we also have

f x XNj0

Ajx x ja RN0, x, 4:2

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with smooth Ajx, while (2.6) and (2.15)(2.16) give

SR f x XNj0

AjxRajFajR x JNx,R OR1, 4:3

with Fas analyzed in 3 and JNx,R given by (2.8), i.e.,

JNx,R SRxN0, 4:4

the partial Fourier inversion of the following function on R:

xNt tRNt, x: 4:5

Since xN 2 CN0 R, we have jxNj ChiN , so a crude estimate yields

jJNx,Rj CNRN1, N 2: 4:6

A sharper estimate will be given below.To analyze the sums in (4.2), we use (3.4) to write

RajFajR x xja RajajR x, 4:7

where, for 0 a < 1,

jasj Csa

1 jsj1a C1 jsj , 4:8

and, for j 1,

jajsj C

1 jsj , 4:9

so, for j 1,RajajR x CRaj1 jR xj CR

aj: 4:10

Thus comparing (4.2) and (4.3) gives

SR f x f x

A0xhRaFaR x xa

iXNj1

RajAjxajR x JNx,R OR1: 4:11

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So far we have the estimate (4.6) on jJNx,Rj, but taking the expansion outfor a few more terms and using estimates of the form (4.10) yields thesharper estimate

jJNx,Rj CNRa1N , N 0: 4:12

Taking the N 0 case of (4.11) now gives the following.

Proposition 4.1. Assume f is given by (4.1), i.e.,

f x ~ggx xa , 0 a < 1, 4:13

with ~gg 2 C1. Then

SR f x f x A0xhRaFaR x xa

iORa1, 4:14

uniformly for x 2 O, a neighborhood of .

Recalling that A0x ~ggx on , we can write

f x A0x xa hx x1a , h 2 C1O, 4:15

and rewrite (4.14) as

SR f x A0xRaFaR x hx x1a ORa1, 4:16

uniformly for x 2 O.One useful feature of (4.14) is that we can go from

jFas sa j Csa

1 jsj1a C1 jsj 4:17

to

jSR f x f xj C xa

1 jR xj1a CR

a

1 jR xj CRa1: 4:18

In particular this is ORa1 on any compact subset of O on which x isbounded away from zero. But of course the greatest interest in (4.14) and(4.16) arises from the detailed behavior of Fas, as exposed in 3.

Remark. While we have phrased the results in this section for functions f ofthe form (4.1) with 0 a < 1, the analytical techniques apply to generalf 2 IaM,, even for a =2 0, 1. If a < 0, almost no alterations are neededin the presentation. However, the phenomenon then captured by (4.3)

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involves uniform convergence, and it is unremarkable compared to theGibbs phenomenon. General f 2 I0M, might have a logarithmicsingularity, leading to a slight variation in the a 0 case of (4.3), whichwe leave to the reader.

The behavior of SR f for f 2 IaM, with a 1 is also un-Gibbs-like,

but in the opposite way, exhibiting extreme non-localization. To illustratethis in the case a 1, we consider the case where f is a smooth multipleof surface measure on . In that case, we replace (2.5) by vt, x A0x x t , so we have an analysis of (2.6) with

I0x,R 1

Z 11

sinRt

tt x t dt 1

sinR x x : 4:19

This exibits the same type of non-localization phenomenon as Fourierinversion of 2 E0R. For f 2 IaM, with a > 1, matters can evidentlybe worse.

5. Lp-LOCALIZATION

If f is more singular than the critical degree at which the Pinskyphenomenon arises, for example if f 2 IM, with > n 3=2,then we do not have L1-localization results. Here we will obtain estimatesthat imply Lp-localization results for between n 3=2 and 1. We beginby noting the following operator properties, noted in (1.17) and (1.23):

kTR f kL1 CRn3=2 k f k ,

kTR f kL2 CR1 k f k ,

5:1

One could apply interpolation to this, but a stronger bound is obtained byapplying results of (9). For this, take

pn 2n 1n 1 , n

n 12n 1 : 5:2

It is shown in (9) that

kPR f kLpn CRn k f kL2 , 5:3

where

PR R,R1ffiffiffiffiffiffiffiffi

p: 5:4

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We hence have

kTR f kLpn CRn k f k : 5:5

Interpolating (5.5) with each part of (5.1) yields the following.

Proposition 5.1. Given even 2 C10 R, we have

kTR f kLp CRn5=4n1=2p k f k , 2 p pn, 5:6

and

kTR f kLp CRn3=2n=p k f k , pn p 1: 5:7

Let f 2 IM,. Since IM, M, the results (5.6)(5.7) areapplicable to f . The estimates (5.6)(5.7) yield Lp-localization as long as theexponent of R is negative. As we will discuss in Proposition 5.3 below, undercertain conditions we can say that

limR!1

Rn3=2kTR f kL1 0: 5:8

In such a case, we can interpolate (5.5) with (5.8), to get the followingrefinement of what one could obtain from (5.7):

Proposition 5.2. If f 2 IM, and (5.8) holds, then, for pn < p 1,

limR!1

Rn3=2n=p kTR f kLp 0: 5:9

We now give a general condition under which (5.8) holds. This result isbasically a corollary of Proposition 2.1 of (8).

Proposition 5.3. Assume that, for R 1, " 2 0, 1,XRR"

jxj2 C"Rn1 C",RRn1, 5:10

uniformly for x 2 M, with

limR!1

",R 0, 8" > 0; lim"!0

" 0: 5:11

Let M be a smooth n 1-dimensional surface. Assume that there existT0 2 0,1 and > 0 such that for the geodesic flow-out of the unit normalbundle of any caustics that occur for jtj T0 have order n 1=2 .

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Assume

f 2 IM,: 5:12

Then, for each even 2 C10 R such that 1 for jj T0 1, we have(5.8).

Proof. As in the proof of Proposition 2.1 in (8), we pick 1 2 C10 R, equalto 1 on 1, 1, and to 0 outside 2, 2, and set "t 1"t, S"R S"R .Following (2.7)(2.11) of (8), and using the inclusion IM, M,we obtain from (5.10)(5.11) that

kSR f S"R f k2L1 C" ",RR2n3: 5:13

Next we have

S"R f x SR f x 1

ZsinRt

t"t t ut, x dt, 5:14

where ut, x cos tffiffiffiffiffiffiffiffi

pf x. Whenever " < 1=T0 1, " is sup-

ported on T0 < jtj < T" for some T" 2 T0,1. Hence our hypothesis onpossible caustics produced by the flow-out of the unit conormal bundle of for jtj T0 gives

jS"R f x SR f xj C" Rn3=2, 5:15

uniformly in x. Combining (5.13) and (5.15), we have

lim supR!1

R2n3 kSR f SR f k2L1 C", 8" 2 0, 1=T0 1,

5:16

which gives (5.8). &

As discussed in 3 of (8), an argument in the spirit of (14) showsthat the hypothesis (5.10) holds provided the geodesic flow-out of theunit cotangent spaces SxM have only stable caustics, for t 6 0. Thisholds for generic compact Riemannian manifolds M. Clearly it holdsfor M Tn, a flat torus; in this case (5.8) was established in (5.25)(5.30)of (4).

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6. THE GIBBS PHENOMENON ANDWEAK-Lp ESTIMATES

Let M be a compact, n-dimensional Riemannian manifold, M asmooth n 1-dimensional surface. Our goal in this section is to prove thefollowing.

Proposition 6.1. Assume 0 < < 1 and f 2 IM,. Then

SR f is bounded in L1=w M, for R 1, 6:1

provided that either

n 3 6:2

or n 4 and

n 32n 2 : 6:3

Remark. As we will see in 8, boundedness in (6.1) can fail for n 4 and < n 3=2n 2.

Proof of Proposition 6.1. Pick 2 C10 R as in 2. Then the results of24 give

SR f bounded in L1=w M, for R 1: 6:4

It remains to consider TR f . We set p 1= 2 1,1, and consider threecases.

First suppose pn p

But the right side of (6.7) is seen to be pn 2n 2=n 1, so in fact(6.7) holds whenever p pn.

Finally suppose 1 < p 2, i.e., 1=2 < 1. In this case we can use(5.1), which gives

kTR f kL2 Ck f k1 : 6:8

This finishes the proof.

7. PERFECT FOCUS CAUSTICS

Here we study the asymptotic behavior as ! 1 of

vx, Z

eitut, x dt, 7:1

where ut, x is a conormal distribution whose Lagrangian TR Rnn0 is that of the fundamental solution of the (flat-space) waveequation. Note that if

Rt, x sin tffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffi

p x, 7:2

then a general such ut, x can be written

ut, x pt, x,DxRt, x qt,x,Dx@tRt, x, 7:3

where pt, x,Dx and qt, x,Dx are smooth families of pseudodifferentialoperators on Rn.

Let us first consider the case where pt, x,Dx and qt, x,Dx are inde-pendent of t. Let us assume > 0. We have

v1x, Z

eitpx,DxRt, x dt

ZZ

eitpx, sin tjjjj eix d dt

Z

px, jjjj eix d

n2Z

Sn1

px, !eix! dS!: 7:4

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If we assume px, 2 Sm, with asymptotic expansion

px, ! Xj0

mjpjx,!, 7:5

we have

v1x, mn2Xj0

jjx, x, 7:6

where

jx, z Z

Sn1

pjx,!eiz! dS!: 7:7

Note that j is smooth and satisfies the estimate

jjx, zj Chzin1=2: 7:8

More precise asymptotics for jx, z as jzj ! 1 follow via the stationaryphase method:

jx, z X

jzjn1=2 eijzjX0

x, z=jzj jzj, 7:9

with smooth on Rn Sn1. A paradigm is given by p0x,! 1, n 3,

in which case

0x, x sin jxjjxj : 7:10

Similarly, given qx, 2 Sm1, with expansion

qx, ! Xj0

m1jqjx,!, 7:11

we have (for > 0)

v2x, Z

eitqx,Dx@tRt, x dt mn2Xj0

jjx, x, 7:12

where

jx, z Z

Sn1

qjx,!eiz! dS!, 7:13

which also satisfies an estimate of the form (7.8).

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Next we treat the case pt, x,Dx tpx,Dx, with 2 C10 R. Theanalysis of this case will suffice for the treatments of the Pinsky focus on Rn

and Tn in 8. In this case, (7.4) is replaced by

v3x, Zteitpx,DxRt, x dt v1x, , 7:14

where v1x, is given by (7.4) and the convolution is with respect to .Consequently

v3x, Xj0

ZSn1

pjx,! mn2j eix!

dS!, 7:15

where we can cut off mn2j for 1. It is elementary to produce theasymptotic expansion

mn2j eix!

mn2jeix!

Xk0

kx !k, 7:16

where

0x ! x !: 7:17

Hence we have

v3x, Xj, k0

mn2jkjkx, x, 7:18

with

jkx, z Z

Sn1

pjx,!kx ! eiz! dS!, 7:19

functions having estimates and asymptotic expansions similar to (7.8)(7.9).One has similar results when t is inserted into the integrand in (7.12).

Now we are ready to consider

v4x, Z

eit pt, x,DxRt, x dt, 7:20

given pt, x, 2 Sm, compactly supported in t. We also assume pt, x, 0for jj 1. We have

v4x, Z

eit pt, x, sin tjjjj eix d dt

cZ

pp jj, x, eix

jj d, 7:21

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mod O1, as ! 1. Here pp, x, is an element of Smcl , rapidlydecreasing as jj ! 1. We can write

jj1pp, x, Xj0

mj1 qj, x, , 7:22

and then

v4x, mn2Xj0

jZqj

1 jj, x,

eix d: 7:23

Each integral here is changed only by O1 if we insert a smooth cut-off, supported on 1=2 jj 2. We can then switch to spherical polarcoordinates and apply the following result.

Lemma 7.1. Given q, 2 SR, there exists an asymptotic expansionZqrr dr

Xk0

akk, ! 1: 7:24

Proof. Set t 1= and write the left side of (7.24) as

At Z

qsts ds, t > 0:

This is easily seen to extend continuously to t 2 0,1, with A0 given as0

Rqs ds. Similarly we extend

A0t Z

sqs0ts ds,

and so on. Hence A 2 C10,1. Then the expansion (7.24) follows fromthe formal power series expansion of At about t 0. &

An analogue of (7.24) also works with extra parameters attached to qand . In (7.23), we set jj!, r 1 jj, and hence eix eix!eirx!,and obtain

v4x, mn2Xj, k0

jk jkx, x, 7:25

with

jkx, z Z

Sn1

qjkx,!eiz! dS!, 7:26

having the same qualitative properties as (7.18)(7.19).

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Finally, a similar analysis works onZeit qt, x,Dx @tRt, x dt: 7:27

8. FOURIER INVERSION NEAR A PINSKY FOCUS

Here we analyze the behavior of SR f x uniformly for x in a neighbor-hood of p, when f 2 IM, @B and B is a ball centered at p. We begin in theclassical context of Fourier inversion on Rn.

Thus consider a ball B Rn, n 2. We may as well assume its centeris p 0. We take f 2 IRn, @B, compactly supported, and analyze

SR f x 1

Z 11

sinRt

tut, x dt, 8:1

for x in a neighborhood of 0. As usual, ut, x cos tffiffiffiffiffiffiffiffi

pf x. Let a be the

radius of B. Then ut, x has two perfect focus caustics, at x 0, t a.Fix b 2 0, a. We desire to analyze the behavior of (1) as R ! 1, uniformlyfor x 2 Bb fx: jxj bg. To get started on this, fix c 2 0, a b, andintroduce a smooth partition of unity jt, 2 j 2, satisfying

supp 0 c, c, supp1 c=2, a 2,supp 2 a 1,1: 8:2

We can arrange that jt jt. With the jt inserted into the inte-grand in (8.1), denote the resulting term by SjR f x, 2 j 2.

Since 0tut, x is smooth on R Bb, it is easy to see that

S0R f x f x OR1, uniformly on x 2 Bb: 8:3

Also the functions 2tut, x are smooth on R Bb. If n is odd, they arecompactly supported in t. If n is even they have asymptotic expansions ast ! 1 of the form

2tut, x Xj0

vjxjtjn1j , 8:4

so we have

S2R f x OR1, uniformly on x 2 Bb: 8:5

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It remains to examine S1R f x. Note that, for t 2 supp1, we have,mod C1,

ut, x cost affiffiffiffiffiffiffiffi

ppx,Dx sint a

ffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffi

p qx,Dx,

8:6

with

px,D 2 OPSn1=2, qx,D 2 OPSn1=2: 8:7

In fact here we can take px,D pD and qx,D qD. A similaranalysis holds for t 2 supp 1. Hence the results of 7 give the following:

Proposition 8.1. Given the ball B Rn and f 2 IRn, @B, compactly sup-ported, we have, for

S1R f x S1R f x 1

ZsinRt1t

tut, x dt 8:8

an asymptotic expansion of the form

S1R f x X

eiaRXj0

Rn3=2jj x,Rx, 8:9

uniformly on Bb, where the functions j x, z are smooth and satisfy the

estimates

jj x, zj Cjhzin1=2, 8:10

and indeed have the asymptotic expansion

j x, z X

jzjn1=2 eijzjX0

jx, z=jzj jzj, 8:11

as jzj ! 1, with j smooth on Bb Sn1. (Here .)

Specializing to the critical case n 3=2, we have:

Corollary 8.2. If f 2 In3=2Rn, @B, B a ball of radius a centered at 0, then,uniformly on Bb B, we have

SR f x f x X

eiaR0 x,Rx OR1: 8:12

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Note that matching this with (1.2) we have for f 2 I0R3, @B, piece-wise smooth with simple jump across @B,

X

eiaR00, 0 2

Avg@B f

sin aR, 8:13

where f denotes the jump of f across @B (from R3nB to B). In particularwe have the result of (5) that SR f 0 ! f 0 for such a piecewise smooth f ,if and only if the average value of the jump is zero. The uniform analysisprovides a counterpoint to this result on the behavior of SR f 0. Namely,suppose f is piecewise continuous with a simple jump across @B and that theaverage of the jump is zero, so SR f 0 ! f 0. If there is a nontrivial jump,then the functions 0 0, z are not identically zero. As a consequence, (8.12)shows that SRf does not converge to f uniformly on a neighborhood ofzero, even though SRf x ! f x for each x 2 Bb. In fact, (8.12) records theprecise way in which uniform convergence fails. See (8.26) below for anexplicit example. This phenomenon is evidently somewhat parallel to theGibbs phenomenon.

We record some more explicit computations for specific examples offunctions singular on the boundary of the unit ball B B10 R3. Forbrevity of notation we use O0Rk to denote a remainder with asymptoticexpansion of the formX

Xjk

eiRj x,RxRj,

where the asymptotic behaviors as in (8.11) hold. Being O0Rk impliesbeing ORk, but it is much more precise. Note that in (8.12) we couldreplace OR1 by O0R1.

Our first example is

f1x Bx: 8:14

By rotational symmetry, in the analysis of the perfect focus causticsproduced by cos t

ffiffiffiffiffiffiffiffi

pf1, one has amplitudes pjx,! and qjx,! in (7.3)

and (7.13) that are independent of !, at x 0, with a similar observationapplying to (7.19). Hence in this case (8.12) can be given the more explicitform

SRBx 12

sinRsx,Rx cosRcx,Rx O0R1,

8:15

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uniformly for jxj b < 1, where sx, z and cx, z are smooth andbehave like (8.11), and

s0, z sin jzjjzj , c0, z 0: 8:16

The factor 2=sinR arises from (8.13). Next, take

f2x jxj2B: 8:17

Since this differs from f1 by an element of I1R3, @B, we have

SRf2x jxj2 2

sinRsx,Rx cosRcx,Rx O0R1,

8:18

uniformly for jxj b < 1, with the same s and c as in (8.15). We use thisas a tool in the analysis of SR f3x, where

f3x 1 jxj2B, 8:19

an element of I1R3, @B.To do this, we use the identity

SRD f x DSR= f x, D f x f x,

which gives

SRLf x LSRf x R@

@RSR f x, 8:20

where

Lgx @@

Dgx1

x rgx: 8:21

Now Lf3 2f2, so

R@

@RSR f30

4

sinRO0R1, 8:22

and hence

SRf30 4

cosR

RO0R2: 8:23

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At this point, using the radial symmetry of f3x and the same reasoning asapplied to get (8.15), we obtain

SR f3x 1 jxj2 4

cosR

R

c3x,Rx

sinR

R

s3x,Rx O0R2,

c30, z sin jzjjzj ,

s30, z 0, 8:24

valid uniformly on jxj b < 1. For our fourth example, we take

f4x x1B: 8:25

Note that f4 1=2@f3=@x1. Since SR commutes with @=@x1, we obtainfrom (8.24) that

SRf4x x1 2

cosRc4x,Rx sinRs4x,Rx O0R1,

c40, z @

@z1

sin jzjjzj ,

s40, z 0, 8:26

uniformly on jxj b < 1.We return to more general considerations. If f 2 IRn, @B and

> n 3=2, then S1R f is not bounded on Bb. We can estimate Lp-norms, readily obtaining from (8.9) thatZ

Bb

jS1R f xjp dx CRppn3=2n: 8:27

As long as the functions 0 0,Rx in (8.9) are not identically zero, we alsohave a lower bound of the same form. Note that the right side of (8.27) isbounded for R 2 1,1 if and only if

np n 3

2: 8:28

We could replace Rn by a variety of noncompact Riemannian mani-folds satisfying the strong scattering condition, but rather than dwelling onthat we now consider the case of a compact, n-dimensional, Riemannianmanifold M. We assume B is a ball in M, of radius a, centered at p, and wewill make the standing assumption that the exponential map takes a ball inTpM diffeomorphically onto B. We will also assume that M and @B satisfythe hypotheses of Proposition 5.3. In particular we assume the geodesicflow starting at a fiber SxM has no perfect focus. This hypothesis holds ifM is a flat torus Tn, or more generally has nonpositive sectional curvature

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everywhere, and in fact it holds generically. It does not hold for the standardspheres Sn, which will be discussed in further detail in 9.

Now take f 2 IM, @B. The analyses of S0Rf x and S1R f x proceedas before, but we need to replace the analysis of S2R f x. Now, with 1 0 1, we have

S2R f x S2R f x TR f x, 8:29

and, by Proposition 5.3,

f 2 IM, @B ) limR!1

Rn3=2 kTR f kL1 0: 8:30

As a result we have the following. As before, we take Bb to be centered at pand have radius b < a.

Proposition 8.3. Given the ball B M, and given f 2 IM, @B, we have (inexponential coordinates)

SR f x f x X

eiaR 0x,RxRn3=2 oRn3=2,

8:31

uniformly for x 2 Bb.

This result holds for all 2 R, but it is most useful when n 3=2. It is useful to record Lp-estimates on the remainder in (8.31).Since (8.30) holds, we can apply Proposition 5.2, to obtain the following.

Proposition 8.4. Given the ball B M and given f 2 IM, @B, we have, forpn < p 1,

SR f f X

eiaR 0x,RxRn3=2

Lp

oRn3=2n=p:

8:32

In particular, if n 3, then for p > 4 the Lp-norm of the remainder isoR3=p. If n 2, then for p > 6 the Lp-norm of the remainder isoR1=22=p.

As noted in (8.27) and the remark following it, the Lp-norm ofX

eiaR 0x,RxRn3=2 8:33

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is on the order of Rn3=2n=p, so this dominates the quantity estimated in(8.32). Hence, we see that for p > pn,

kSR f S0R f kLp CRn3=2n=p, 8:34

for typical f 2 IM, @B. This implies that the estimate (5.7) is sharp(for 0), which in turn implies that the Sogge estimate of kPR f kLpfor pn < p

with ut, x cos tA f x, and then RA SN for N < R < N 1. For neven we can set

SN f x 1

4

Z~SS

sinN 3=4tsin 1=4t ut, x dt, 9:4

and then RA SN for N 1=2 < R < N 3=2. Here S1 R=2Z and~SS R=4Z, and N is taken in Z.

We briefly describe how the Gibbs effect shapes up in this case. Sayf is a function on Sn that is smooth away from a smooth surface and has the form (2.9) near . For convenience, assume n is odd. Let usagain bring in the cut-off t. This time one has the following variant of(2.6)(2.7):

SN f x XKj0

AjxIjx,N JK x,N, 9:5

with

Ijx,N 1

ZS1

sinN 1=2tsin 1=2t t

x t

aj

dt: 9:6

If we write

~t tsin1=2t t, 9:7

and

~t ~ x h~t ~ x

i ~ x t xt, x, 9:8

we can inductively trade t in (9.6) for factors that are just smooth func-tions of x. Then we obtain the following variant of (4.3):

SN f x XKj0

eAAjx N 1=2ajFajN 1=2 x JK x,N ON1, 9:9

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where eAAjx are obtained from the functions Ajx of (2.5) in an explicitfashion, which we write down just for j 0:

eAA0x x2 sin x=2 A0x: 9:10

This treats the case of odd n. The analysis for even n is similar.The cut-off t might be supplemented by other cut-offs, to isolate

various focusing effects. However, in place of an analysis of large t effects onthe remainder SN f SN f , here the following simple result suffices.

Proposition 9.1. Let K Sn be compact and take A ft 2 S1 : t 6 1g. Ifut, x is smooth on a neighborhood of A K S1 Sn, then

SNf x SN f x ON1, uniformly on x 2 K : 9:11

The uniform analysis near a Pinsky focus done in 8 extends tospheres, as follows. Let B Sn be a ball; say its center is p1, with antipodep2, so B Ba1p1 and S

nnB Ba2 p2, with a1 a2 . Then if f 2ISn, @B we have, in exponential coordinates centered at p1,

SN f x f xX

eia1N1=20x, N1=2x N1=2n3=2

oNn3=2, 9:12

uniformly on a neighborhood of p1 0, and similarly near p2, if n is odd,with a similar formula if n is even. The functions 0 x, z have the samegeneral form as in 8. However, if for example 0 and f B, the valuesof 0 0, 0 will differ from those that work in (8.13).

To illustrate this point, lets take n 3 and f B. The functionut, x cos tA f x on R S3 is given by

ut, x @tsin t f xjtj cos tf xjtj sin t@t f xjtj, 9:13

where f xr denotes the spherical mean of f , over a 2-sphere of radius rcentered at x 2 S3. For f B we have

@t f p1jtj t a1 t a1,@t f p2jtj t a2 t a2: 9:14

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In concert with (9.3) this yields

SNB p1 11

sin a1sina1=2

sinN 1=2a1 o1,

SNB p2 1

sin a2sina2=2

sinN 1=2a2 o1,9:15

as N ! 1, which can be compared with (1.2).The behaviors of (9.12) and (9.15) contrast with the behaviors of (1.2)

and (8.31), valid for eigenfunction expansions on Riemannian manifolds Msatisfying the hypotheses of Proposition 5.3, in the following significant way.In the context of Corollary 8.2, for example, the values of SR f x at aPinsky focus x p have only small jumps as R crosses points inSpec

ffiffiffiffiffiffiffiffi

p(for large R). However, typically, for f 2 In3=2Sn, @B, the

jumps from SNf p to SN1 p at a Pinsky focus p are not small. Thisillustrates the breakdown of the results of 5 when M Sn, and of courseit is connected with the clustering of the spectrum of the Laplace operatoron Sn.

Phenomena such as described above also hold for other compact rank-one symmetric spaces, such as CPn and HPn; cf. (2) for the analogues of(9.1)(9.4).

There are also extensions of the results on spheres given above to Zollsurfaces, smooth Riemannian manifolds with periodic geodesic flow, whichwe can assume to be of period 2. Consider the Laplace operator onsuch a manifold. More generally than

ffiffiffiffiffiffiffiffi

p, we consider any self-adjoint

operator A of the form

A ffiffiffiffiffiffiffiffi

p B, B 2 OPS1M: 9:16

The following result provides a replacement for (9.1)(9.2). We assume thatM is connected and dim M 2.

Proposition 9.2. Assume all geodesics on M have the minimal period 2. Thereexists a positive, self-adjoint A 2 OPS1M, commuting with A, such that

A A I S, S 2 OPS1M, 9:17

and

e2iA I : 9:18

This result is due to (15); see also 29.2 of (13). We make some remarkson the proof; for more details see (13). To start, we can say that

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e2iA P 2 OPS0M, and the global calculus of Fourier integral operatorsyields that the principal symbol of P is constant, so

P cI R, jcj 1, R 2 OPS1M: 9:19

Hence SpecP has only c as an accumulation point, so we can apply thelogarithm and write

P e2iQ, Q 2 OPS0M, 9:20

where Q is self-adjoint and commutes with A. Furthermore,

Q I S, e2i c, S 2 OPS1M: 9:21

Note the following relation between the principal symbols of R and S:

S 12ic

R: 9:22

Now the desired operator A is given by A AQ. If necessary we can adda negative integer to to ensure that A is positive.

Since SpecA Z and S 2 OPS1M commutes with A, it followsthat SpecA clusters around small neighborhoods of N , for largeN 2 Z. The clusters are contained in intervals

IN N "N ,N "N, "N CN1: 9:23

Here we can take any C > kASk. Hence, for large R,

N "N < R < N 1 "N1)RA N1=2A SN , 9:24

where SN is given by (9.3), with

ut, x cos tA f x: 9:25

This time the equation

utt A2u 0, u0,x f x, ut0, x 0 9:26

is not quite a partial differential equation, but the solution operator cos tAretains enough of the general features of cos t

ffiffiffiffiffiffiffiffi

pthat the results on the

Gibbs phenomena and Pinsky phenomena described above persist in thismore general situation. This provides a description of RA f as R ! 1,avoiding the clusters in IN , given by (9.23). In 10 we will say more aboutwhat can happen when R runs through these clusters.

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The global calculus of Fourier integral operators is useful in identify-ing c and the principal symbol of R in (9.19), but in some cases of interestmore elementary tools suffice for the calculation. For example, consider

A ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V

p A0 B, 9:27

with

A0 n 12

2 !1=2, B 2 OPS1M, M Sn: 9:28

This case was studied in (16) and (17). In this case, P in (9.19) is equal toP2, where

Pt eitAeitA0 : 9:29

We have

P0t iA0,Pt iBPt, P0 I : 9:30

On the symbol level, this gives

@p

@tHap ibp i

Xjj1

ijj

!bp

iXjj2

ijj

!

ap pa

: 9:31

If p p0 p1 , we obtain

@p0@t

Ha1p0 0, p00, x, 1: 9:32

Hence p0 ) 1, so c 1 in (9.18). Given that p0 ) 1, the equation for p1simplifies to

@p1@t

Ha1p1 ib1p0, p10, x, 0, 9:33

so

p1t, x, iZ t0

b1

F sx,

ds, 9:34

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where F s is the Hamiltonian flow on TM TSn generated by Ha1 , i.e.,the geodesic flow in this case. One verifies that

b1x, 1

2jjV1x, V1x Vx n 12

2, 9:35

so we have for the principal symbol of R:

Rx, i

2jj

Z 20

V1F sx, ds, 9:36

with :TSn ! Sn the standard projection. See 29.2 of (13) for general cal-culations of c and R. The calculation above is done much in the spirit of (18).

10. SPHERICAL HARMONIC EXPANSIONS:ANOTHER DIVERGENCE EFFECT

In 9 we considered the Laplace operator on spheres (and Zollsurfaces), and also various self-adjoint perturbations of the Laplace opera-tor, of order zero. We obtained fairly precise results on convergence ofeigenfunction expansions of functions with conormal singularities, as longas all the contributions from a cluster were grouped together. Here we willconsider some examples showing that the partial sums within clusters canbehave somewhat more wildly.

We look at operators of the following form: pick b 2 0, 1 and take

A n 12

2ibL12

!1=2, on Sn, 10:1

where L12 generates a one-parameter group of rotations of period 2 in thex1x2-plane in R

n. This is a zero-order perturbation of

A0 n 12

2 !1=2,

SpecA0 kn 12

k: k 0, 1, 2, . . .

, 10:2

and the k-eigenspaces Vk of A0 split into 2k 1 eigenspaces Vk for A,with eigenvalues k 2k b1=2, spread out around the point k, over

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an interval

Jk k , k , k 2k bk1=2 k b=2: 10:3

The example (10.1) does not quite fit into the class of operators studiedin 9, but the following does:

A1 n 12

2 ibL12A10

!1=2, 10:4

with A0 as in (10.2). This has the same eigenspaces as A, but eigenvaluesk 2k b=k1=2, which are clustered about k. Clearly the two opera-tors A and A1 give rise to the same eigenfunction expansions. We will phrasethe results in terms of the operator A, given by (10.1).

Note that, for " "k sufficiently small,

k"A k"A0, k "A k"A0, 10:5

and the analysis of k"A0 has been described in 9. We are now interestedin analyzing RA f as R crosses the various eigenvalues of A in the intervalJk. Note that, for R 2 Jk, R =2 SpecA,

RA f k"A0 f Xk

Then, with gkq gk + Eq, we havehR2A f R1A f

i+ Eq

Xjj

ggkq ei Sjgkq, 10:10

where the last identity defines Sjgkq.Let us take f B, with B Baq. We want to study the series

(10.10), with gk given by (10.7). We first note that the behavior of Sjgkqdepends mainly on gkq only near 0 (and if the antipode q0 iscontained in Cq). This is established as follows.

Lemma 10.1. If f B and gk is given by (10.7), then, for any compactX Snnfq, q0g,

supX

jgkj CX k1, supX

jrgkj CX : 10:11

Proof. We examine fgkg near @B. The behavior elsewhere (away from qand q0) is simpler. By (9.9), for x close to @B, gkx is given by

eAA0xF0k 1=2 x eAA0xF0k 1=2 x, 10:12plus lower order terms. This is equal to eAA0x times

xZ 10

F 00k s x ds, F 00s sin s

s, 10:13

and writing x k1k x, we readily verify the estimates (10.11). &

Corollary 10.2. Let I be any compact subset of T1nf0g if q0=2Cq, or ofT1nf0,g if q0 2 Cq. Then

supI

jgkqj CI k1, supI

jg0kqj CI : 10:14

Now the behavior of gkq near 0 is given by the behavior of gkxnear x q, which in turn is given by (9.12). In exponential coordinatescentered at q, we have

gkx X

eiak1=2 kn3=2

eia0x, k 1=2x

0

x, k 1=2x

n oOkn5=2: 10:15

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Note that the factor eia in front of the first 0 destroys the cancellation onewould have without its presence. This is very important for the analysis thatfollows. It is also significant that the unwritten terms in the remainder in(10.15) have a form similar to the written terms, just with lower powers ofk attached.

Parallel to (8.15), we have

0 0, z cn njzj, nr r1n=2 Jn=21r, 10:16

for certain nonzero constants cn . Furthermore, using

0 x, z 0 0, z xZ 10

@x0 tx, z dt,

we have

j0 x,Rx 0 0,Rxj CjxjhRxin1=2

CR jxj1 hRxin1=2, 10:17

for 0 1. Note that, for n 5,Rh yin1=2 dy is finite for all < 1

(and finite for 1 if n 6). We also recall the classical formulaZ 10

Jrr dr 21=2

1=2 , >1

2, 10:18

so 6 0 for n=2 1, n 3.Putting the results above together, and keeping in mind that

Sjgkq gkq Dj, 10:19

where Dj is the Dirichlet kernel:

Dj 1

2

sin j 1=2sin1=2 , 10:20

we have the following:

Proposition 10.3. Let f B and gk be given by (10.7). Here B Baq. If theantipode q0=2Cq, then, for each fixed j 2 Z, Sjgkq in (10.10) has the followingbehavior as k ! 1, provided n 5:

Sjgkq n=21 kn5=2X

cneiak1=2eia 1

!Dj

okn5=2: 10:21

There is an analogous result if q0 2 Cq.

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This in turn leads to the following:

Proposition 10.4. With f B as in Proposition 10.3, if n 5 we have thepointwise convergence

limR!1

RA f x f x, 10:22

if and only if x =2Cq [ Cq0 . (As usual we define f 1=2 on @B.) If n 5 thedivergence on Cq [ Cq0 is bounded; it is unbounded if n 6.

Proof. The convergence follows from Lemma 10.1 plus the previous materialon RA0 f . The divergence follows from (10.21). &

This result can be compared with a result of (19), treating B Tn,n 5. There the divergence of the Fourier series of B at points other thanthe center of B is derived from a number-theoretic analysis.

One can go further, replacing A in (10.1) by

A2 n 12

2 ib1L12 ib2L34A10 ib3L56A20

!1=2, 10:23

and so on. The eigenspaces of A2 are formed by splitting the eigenspaces ofA, producing eigenvalues in clusters and then in further, tighter clusters.One can have further divergence phenomena for RA2 f , as R crosses thesevarious clusters, particularly in higher dimensions. We will not pursue thedetails.

ACKNOWLEDGMENT

The authors research has been supported by the NSF grant. DMS-9877077.

REFERENCES

1. Colzani, L.; Vignati, M. The Gibbs Phenomenon for Multiple FourierIntegrals. J. Approximation Theory 1995, 80(1), 3979.

2. Pinsky, M.; Taylor, M. Pointwise Fourier Inversion: A Wave EquationApproach. J. Fourier Anal. and Appl. 1997, 3(6), 647703.

3. Brandolini, L.; Colzani, L. Localization and Convergence of Eigenfunc-tion Expansions. J. Fourier Anal. and Appl. 1999, 5(5), 431447.

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4. Taylor, M. Pointwise Fourier Inversion on Tori and Other CompactManifolds. J. Fourier Anal. and Appl. 1999, 5(5), 449463.

5. Pinsky, M. Pointwise Fourier Inversion and Related EigenfunctionExpansions. Comm. Pure Appl. Math. 1994, 47(5), 653681.

6. Hormander, L. The Spectral Function of an Elliptic Operator. ActaMath. 1968, 121(34), 193218.

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10. Taylor, M. Pseudodifferential Operators; Princeton Univ. Press:Princeton, New Jersey, 1981.

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12. Hormander, L. Pseudodifferential Operators and HypoellipticEquations. Proc. Symp. Pure Math. 1997, 10(1), 138183.

13. Hormander, L. The Analysis of Linear Partial Differential Equations,Vol. 4; Springer-Verlag: New York, 1985.

14. Diustermaat, J.; Guillemin, V. The Spectrum of Positive EllipticOperators and Periodic Bicharacteristics. Invent. Math. 1975, 29(1),3979.

15. Colin de Verdiere, Y. Sur le Spectre des Operateurs Elliptiques aBicharacteristiques Toutes Periodiques. Comment. Math. Helv. 1979,54(3), 508522.

16. Weinstein, A. Asymptotics of Eigenvalue Clusters for the LaplacianPlus a Potential. Duke Math. J. 1977, 44(4), 883892.

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Received December 2000Revised May 2001

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