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

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

Post on 16-Mar-2017

212 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

<ul><li><p>This article was downloaded by: [York University Libraries]On: 13 November 2014, At: 13:26Publisher: Taylor &amp; FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK</p><p>Communications in Partial Differential EquationsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lpde20</p><p>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.</p><p>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</p><p>To link to this article: http://dx.doi.org/10.1081/PDE-120002866</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (the Content) containedin the publications on our platform. However, Taylor &amp; Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor &amp; Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.</p><p>This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms &amp; Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions</p><p>http://www.tandfonline.com/loi/lpde20http://www.tandfonline.com/action/showCitFormats?doi=10.1081/PDE-120002866http://dx.doi.org/10.1081/PDE-120002866http://www.tandfonline.com/page/terms-and-conditionshttp://www.tandfonline.com/page/terms-and-conditions</p></li><li><p>THE GIBBS PHENOMENON, THE PINSKYPHENOMENON, AND VARIANTS FOR</p><p>EIGENFUNCTION EXPANSIONS</p><p>Michael E. Taylor</p><p>Mathematics Department, University of NorthCarolina, Chapel Hill, NC 27599, USA</p><p>ABSTRACT</p><p>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.</p><p>565</p><p>Copyright &amp; 2002 by Marcel Dekker, Inc. www.dekker.com</p><p>COMMUN. IN PARTIAL DIFFERENTIAL EQUATIONS, 27(3&amp;4), 565605 (2002)</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Yor</p><p>k U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rari</p><p>es] </p><p>at 1</p><p>3:26</p><p> 13 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>1. INTRODUCTION</p><p>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</p><p>SR f Rffiffiffiffiffiffiffiffi</p><p>p f 1:1</p><p>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.</p><p>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,</p><p>SR f 0 f 0 2</p><p>Avg@B g sin aR o1: 1:2</p><p>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.</p><p>In (2) the Pinsky phenomenon was treated via an analysis using thewave equation. We can write</p><p>SR f x 1</p><p>Z 11</p><p>sinRt</p><p>tut,x dt, 1:3</p><p>with</p><p>ut, x cos tffiffiffiffiffiffiffiffi</p><p>pf x, 1:4</p><p>uniquely defined as the solution to the following initial value problem forthe wave equation:</p><p>utt u 0, u0, x f x, ut0, x 0: 1:5</p><p>566 TAYLOR</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Yor</p><p>k U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rari</p><p>es] </p><p>at 1</p><p>3:26</p><p> 13 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>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).</p><p>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</p><p>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.</p><p>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.</p><p>We consider Gibbs-type phenomena for functions having a ya typesingularity on a smooth surface f y 0g, with 0 a &lt; 1. As in the caseof a jump discontinuity, the Gibbs phenomenon takes a universal form. Wehave, as R ! 1,</p><p>SR f x Xj0</p><p>AjxRaj FajR x, 1:6</p><p>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.</p><p>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 &lt; ac or a &gt; ac, withac 0:79. See 3 for a further discussion of this.</p><p>If f has a classical conormal singularity on and the wave ut, x cos t</p><p>ffiffiffiffiffiffiffiffi</p><p>pf x has a perfect focus at t a, at a point p =2, the uniform</p><p>EIGENFUNCTION EXPANSIONS 567</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Yor</p><p>k U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rari</p><p>es] </p><p>at 1</p><p>3:26</p><p> 13 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>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,</p><p>SR f x f x X</p><p>eiaRXj0</p><p>Rn3=2j j x,Rx, 1:7</p><p>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</p><p>j x, z jzjn1=2X</p><p>eijzj x, z=jzj, 1:8</p><p>as jzj ! 1, with smooth on B Sn1.The Pinsky phenomenon is manifested in the critical case </p><p>n 3=2, in which case</p><p>SR f x f x X</p><p>eiaR 0 x,Rx OR1, 1:9</p><p>uniformly on a neighborhood of p. Thus the behavior at the focus is given by</p><p>SR f 0 f 0 X</p><p>eiaR 00, 0; 1:10</p><p>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!</p><p>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</p><p>568 TAYLOR</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Yor</p><p>k U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rari</p><p>es] </p><p>at 1</p><p>3:26</p><p> 13 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>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</p><p>SR f x 1</p><p>ZsinRt</p><p>tt ut, x dt: 1:11</p><p>It is elementary to show that SR f ! f in L2-norm, for any f 2 L2M. We</p><p>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</p><p>TR f SR f SR f R</p><p>ffiffiffiffiffiffiffiffi</p><p>p f , 1:12</p><p>where</p><p>R R R: 1:13</p><p>An attack on (1.12) initiated in (3) takes</p><p>TR f x X</p><p>R ff x, 1:14</p><p>where f: 2 specffiffiffiffiffiffiffiffi</p><p>pg is an orthonormal basis of L2M consisting of</p><p>eigenfunctions of with eigenvalue 2, and ff f , L2 , and writes</p><p>jTR f xj X</p><p>jRjj ff j2( )1=2 X</p><p>jRjjxj2( )1=2</p><p>: 1:15</p><p>Applying Hormanders estimateXRR1</p><p>jxj2 CRn1, 1:16</p><p>given in (6), with n dimM, one obtains</p><p>kTR f kL1 CRn3=2 k f k , 1:17</p><p>where</p><p>M f 2 D0M:X</p><p>RR1j ff j2 CR22g:</p><p>(1:18</p><p>This led to a number of results on pointwise Fourier inversion on compactmanifolds, in (3), (7), (4), and (8).</p><p>EIGENFUNCTION EXPANSIONS 569</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Yor</p><p>k U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rari</p><p>es] </p><p>at 1</p><p>3:26</p><p> 13 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>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,</p><p>ITn, Tn: 1:19</p><p>In (7) the more general result</p><p>I1 M, M 1:20</p><p>was established, for a general compact Riemannian manifold M. The result(1.20) was then extended in (8) to</p><p>I M, M, 1:21</p><p>given 1=2 1, for any smooth conic Lagrangian TMn0.The exponent of R in (1.17) vanishes at n 3=2, the point</p><p>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</p><p>ffiffiffiffiffiffiffiffi</p><p>pf x has no caustics as strong</p><p>as the perfect focus type for jtj T0, and if a mild geometrical restriction isplaced on M, one can strengthen (1.17) to</p><p>limR!1</p><p>Rn3=2 kTR f kL1 0, 1:22</p><p>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).</p><p>Our work in 24 on the Gibbs phenomenon for filtered Fourierinversion deals with a variety of cases in which f 2 IM, with &gt; 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</p><p>kTR f kL2 CR1 k f k : 1:23</p><p>570 TAYLOR</p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Yor</p><p>k U</p><p>nive</p><p>rsity</p><p> Lib</p><p>rari</p><p>es] </p><p>at 1</p><p>3:26</p><p> 13 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>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).</p><p>These estimates on TR f together with the Gibbs phenomenon analysisyield sharp estimates on SR f in the weak-L</p><p>p space LpwM, given f 2IM,, 0 &lt; &lt; 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.</p><p>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</p><p>R f , and in 6 we draw</p><p>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.</p><p>In 9 we discuss SR Rffiffiffiffiffiffiffiffi</p><p>p for the Laplace operator on</p><p>spheres, Zoll surfaces, and variants. The mild geometrical restrictionmentioned bef...</p></li></ul>