Exponential instability in the fractional Calderón problem
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This is an electronic reprint of the original article. This reprint may differ from the original in pagination and typographic detail. Author(s): Title: Year: Version: Please cite the original version: All material supplied via JYX is protected by copyright and other intellectual property rights, and duplication or sale of all or part of any of the repository collections is not permitted, except that material may be duplicated by you for your research use or educational purposes in electronic or print form. You must obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or otherwise to anyone who is not an authorised user. Exponential instability in the fractional Calderón problem Rüland, Angkana; Salo, Mikko Rüland, A., & Salo, M. (2018). Exponential instability in the fractional Calderón problem. Inverse Problems, 34(4), Article 045003. https://doi.org/10.1088/1361- 6420/aaac5a 2018
Inverse Problems PAPER • OPEN ACCESS Exponential instability in the fractional Calderón problem To cite this article: Angkana Rüland and Mikko Salo 2018 Inverse Problems 34 045003 View the article online for updates and enhancements. This content was downloaded from IP address 130.234.74.68 on 23/02/2018 at 12:29
1 Inverse Problems Exponential instability in the fractional Calderón problem AngkanaRüland1 and MikkoSalo2 1 Max-Planck-Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany 2 University of Jyvaskyla, Department of Mathematics and Statistics, PO Box 35, 40014 University of Jyvaskyla, Finland E-mail: [email protected] and [email protected] Received 24 November 2017, revised 29 January 2018 Accepted for publication 1 February 2018 Published 20 February 2018 Abstract In this paper we prove the exponential instability of the fractional Calderón problem and thus prove the optimality of the logarithmic stability estimate from Rüland and Salo (2017 arXiv:1708.06294). In order to infer this result, we follow the strategy introduced by Mandache in (2001 Inverse Problems 17 1435) for the standard Calderón problem. Here we exploit a close relation between the fractional Calderón problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in Rüland and Salo (2017 arXiv:1708.06294). Finally, in one dimension, we show a close relation between the fractional Calderón problem and the truncated Hilbert transform. Keywords: fractional Calderón problem, exponential instability, Runge approximation 1. Introduction 1.1. Motivation Inverse problems are typically ill-posed, and studying the degree of ill-posedness (or instability) gives insight into the types of reconstructions that one may expect. Inverse problems range from mildly ill-posed ones, such as x-ray computerized tomography (CT), to severely ill-posed ones such as the Calderón problem in electrical impedance tomography. One way to describe the stability properties is to study the modulus of continuity of the inverse of the A Rüland and M Salo Exponential instability in the fractional Calderón problem Printed in the UK 045003 INPEEY © 2018 IOP Publishing Ltd 34 Inverse Problems IP 1361-6420 10.1088/1361-6420/aaac5a Paper 4 1 21 Inverse Problems IOP Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 2018 1361-6420/18/045003+21$33.00 © 2018 IOP Publishing Ltd Printed in the UK Inverse Problems 34 (2018) 045003 (21pp) https://doi.org/10.1088/1361-6420/aaac5a
2 forward map. For x-ray CT in R2 , the forward map that takes an attenuation coefficient f to its Radon transform Rf is smoothing of order 1/2 and has a Lipschitz continuous inverse, showing that this inverse problem is mildly ill-posed [Nat01]. The situation is different for the Calderón problem: given the forward map that takes a conductivity coefficient γ (in a suitable compact space) to the boundary measurement operator Λγ , its inverse is continuous but only with a logarithmic modulus of continuity [Ale88]. Moreover, logarithmic stability is in general optimal and cannot be improved to Lipschitz or Hölder type stability [Man01, DCR03]. In this paper we consider inverse problems where the underlying models involve fractional differential equations. These models could arise in nonstandard diffusion processes that involve memory effects, such as diffusion in viscoelastic materials or heterogeneous media including soil, or long range interactions including diffusion in turbulent gases or fluids, population dynamics (Lévy flight foraging hypothesis), and financial modelling. These models lead to time-fractional or space-fractional derivatives in the underlying PDE, and such equationsare an active topic of research. See [MK00, JR15, BV16] for further information and references. The mathematical study of inverse problems for fractional equationsgoes back at least to [CNYY09]. By now there are a number of results, most of them for time-fractional models and including many numerical works. It has been suggested that the stability properties of a fractional inverse problem may be different from the corresponding classical model. For example, as shown in [SY11], in the time-fractional heat equation ∂α tu−∆u=0inΩ×(0, T),u| ∂Ω × (0,T) =0, where 0<α<1 and ∂α t is the Caputo derivative, u(0) is determined by u(T) in a mildly illposed way (whereas for α=1 this problem is severely ill-posed). We refer to [JR15] for a detailed discussion and many further references. The work [GSU16] initiated the study of space-fractional Calderón type inverse problems, based on a strong approximation property for fractional equations(see [Sal17] for some further developments). This suggests that space-fractional inverse problems may be more manageable than the classical Calderón problem, and one could ask whether their stability properties could also be different. However, this turns out not to be the case. In this article we show that the space-fractional inverse problem has exponential instability, and thus the logarithmic stability estimate in [RS17] has optimal character. In particular, the stability properties of the spacefractional Calderón problem are similar to those of the classical Calderón problem. 1.2. Results We consider the fractional Calderón problem, which was introduced in [GSU16]. More precisely for s∈(0, 1) , we consider the mapping Λ q : Hs ( B 3\ B 2)→ H−s ( B 3\ B 2) , f → ( − ∆)su | B 3\ B 2 , (1) where u is a solution to ((−∆)s+q)u=0inB 1 , u=fin R n \ B 1 , (2) with q∈L∞(B1) and n⩾1 . Here we denote by Hs the standard L2 based Sobolev spaces, and Hs ( U) is the closure of C∞ c(U) in Hs(Rn) . A Rüland and M Salo Inverse Problems 34 (2018) 045003
3 We are interested in the (in)stability properties of the inverse problem of determining q from the knowledge of Λq , with Λq as in (1). Due to the results from [GSU16], it is known that, if the potentials q1,q2 are such that (−∆)s+qi does not have zero as an eigenvalue, then Λq1=Λ q2=⇒q1=q2. This has been quantified in [RS17], where a logarithmic stability estimate was proved. It is the purpose of this note to show that this logarithmic estimate is optimal and that the problem indeed has an exponential instability. Our main result in this context can be formulated as follows: Theorem 1. Let B1⊂Rn with n⩾1 . For any m∈N={1, 2, 3, ...} there exist constants r0 > 0 and β>0 such that for all sufficiently small >0 and for all q0∈L∞(B1) with q0L∞(B1)⩽r0 2 there are potentials q1,q2 with qi−q0∈Cm 0(B1) such that Λq1−Λq2L2(B3\B2)→L2(B3\B2) ⩽8 exp(− −n (2n+3)m) , q1−q2L∞(B1)=, qi−q0Cm(B1)⩽β, qi − q0 L ∞ (B1) ⩽ ,i ∈{ 1, 2 } . (3) In contrast to the instability examples for the classical Calderón problem, we de not need q0 to be compactly supported in B1. Remark 1.1. We note that although Hs ( B3\B2) and H−s ( B3\B2) are the natural function spaces in the context of the fractional Calderón problem (see the well-posedness discussion in lemma 2.3 in [GSU16]), the disjointness of the domains B1 and B3\B2 allows us to also define the exterior problem (2) for L2(B3\B2) , see remarks 2.2 and 2.5 for more details on this. For simplicity we state our results in terms of these L2 norms instead of the norms used in [RS17]. As a corollary of theorem 1, we directly obtain the exponential instability of the fractional Calderón problem in a neighbourhood of a suitable potential q: Corollary 1.2. Let n⩾1 and B1⊂Rn , assume that m∈N . There is a potential q∈L∞(B1) , a sequence of errors {k}k∈N with k→0 and a sequence of potentials {qk}k∈N⊂Cm(B1) such that (i) q−q k L∞(B1)⩾k 2 and Λq − Λqk L 2 (B3\B2)→L 2 (B3\B2)⩽ Cexp( − −n (2n+3)m k) , (ii) for all 0⩽m<m we have qk→q in Cm ( B1) . Both of these results rely on a combination of ideas introduced by Mandache in the context of the classical Calderón problem, and a close relation between the fractional Calderón problem and the Poisson problem for the Laplacian. Let us explain this in more detail: In the article [Man01] Mandache showed that the Calderón problem has an exponential instability, and that thus the stability results of Alessandrini (see [Ale88, Ale90]) are optimal. To achieve this, Mandache used two main ingredients in his proof, which were later also exploited by Di Cristo and Rondi [DCR03], who systematically derived similar estimates to prove exponential instability in a variety of inverse problems. • Firstly, he deduced very strong smoothing properties for the local analogue of the operator Γ(q):=Λ q−Λ0 . • Secondly, he made use of the ‘separating’ properties of the space L∞ . These are to be understood in the sense of lemma 3.4 and definition 3.1. A Rüland and M Salo Inverse Problems 34 (2018) 045003
4 The latter property is a general property of the space L∞ and hence does not depend on the problem at hand. As a consequence, we can also rely on this in the context of the fractional Calderón problem. It is the first property, i.e. the smoothing properties of Γ(q) , which is based on the specific problem. It is thus one of the main contributions of this note to show that the strong smoothing properties still hold for the fractional Calderón problem. Here, as had been observed in [DCR03], the key step consists of the construction of a suitable orthonormal basis (lemma 2.1), which interacts well with the fractional Laplacian. In order to deduce the regularizing properties of Γ(q) , we rely on an identity that relates the Poisson operator for the fractional Laplacian to the Poisson operator for the Laplacian (see lemma 2.1). As a further consequence of the construction of the orthonormal basis from lemma 2.1, we deduce the (almost) optimality of the Runge approximation property for the fractional Laplacian, which was given in theorem 1.3 in [RS17] and which quantified the approximation results from [DSV17] and [GSU16]: Theorem 2. There is a sequence {vp}p∈N⊂L2 ( B1) , v p L 2 (B1) = 1 , with the following property: If u∈Hs(Rn) satisfies (i) (−∆)su=0in B1, u|R n \ B 1=f for some f∈L2 ( B3\B2) , (ii) u−v p L 2 (B1)⩽p−1 , we have that for some constant c0, which only depends on n, s, f L2(B 3\ B 2 ) ⩾c 0 2p. Remark 1.3. We refer to theorem 2 as almost showing optimality in our approximation result from [RS17], as on the one hand the result of theorem 3 gives evidence of the exponential dependence of the control on the admissible error. But on the other hand, this exponential behaviour is formulated with respect to L2 norms only, instead of using the combination of L2 and Hs norms as in the statement of theorem 1.4 in [RS17]. Finally, we remark that in one dimension a very precise relation between the fractional Laplacian and the truncated Hilbert transform can be obtained (see lemma 5.1 in section5). This in particular allows us to exploit the well-studied properties of the truncated Hilbert transform and its singular value decomposition [Kat10, KT12] in order to prove exponential instability results in one-dimension. We illustrate this by presenting a second variant of theorem 2 and its proof in section5 (see theorem 3). The remainder of the article is organized as follows: In section 2 we present the main novelty of the article and deduce the central smoothing estimates for the fractional Calderón problem. This is achieved by constructing a suitable orthonormal basis adapted to the fractional Laplacian (see lemma 2.1). In section3 we combine the smoothing estimate with the general properties of L∞ . Here we follow the analogous reasoning of Mandache [Man01]. The combination of the results from sections2 and 3 then provide the proofs of theorem 1 and of corollary 1.2. Using the orthonormal basis from section2 we present the argument for theorem 2 in section4. In order to illustrate the explicit connection between the truncated Hilbert transform (and its well-studied singular value basis), we discuss the one-dimensional set-up in more detail in section5. Exploiting the singular value decomposition of the truncated Hilbert transform, we there derive another variant of theorem 2. Finally, in section6 we show that the classical Hadamard example of exponential instability also holds in the context of the Caffarelli–Silvestre extension with s∈(0, 1) . A Rüland and M Salo Inverse Problems 34 (2018) 045003
5 2. Smoothing In this sectionwe construct an orthonormal basis {fm,k,l} of L2 ( B3\B2) such that the corresponding solutions um,k,l to the fractional exterior value problem (−∆)su m,k,l =0inB 1 , um,k,l=χ B3\B2 fm,k,lin Rn \ B1 , (4) satisfy exponential decay bounds. We will then translate this decay into smoothing properties for the operator Γ(q):=Λ q−Λ0 (see proposition 2.4), which will play a central role in our instability argument. Lemma 2.1. Let n⩾1 and B1⊂Rn . For m⩾0 denote by {hm,l}0⩽l⩽lm the spherical harmonics of degree m on ∂B1 . Let s∈(0, 1) and consider the mapping A0 defined by L2 ( B3\B2 ) f→ rB1u∈L2 ( B1 ) , where u is the solution of (−∆)su=0 in B1 with u=χB3\B2f in Rn\B1 , and rB1 denotes the restriction onto the unit ball. Then there exists an orthonormal basis {fm,k,l}m,k⩾0,0⩽l⩽lm of L2 ( B3\B2) , where fm,k,l(x)=gm,k(|x|)hm,l(x/|x|), such that for constants C,c > 0 which only depend on n, s, A0fm,k,l L 2 (B1)⩽ Ce −c(m+k). Remark 2.2 (Well-posedness with exterior L2 ( B3\B2) data). As already pointed out in remark 1.1, a natural set-up for the exterior problem (4) for the fractional Laplacian is to consider exterior Dirichlet data in Hs ( B3\B2) (see lemma 2.3 in [GSU16]). If, as in lemma 2.1 and in various other places of the article, the exterior data are however localized onto a subset which is disjoint from the domain in which the fractional Schrödinger equationis considered, then it is also possible to deduce well-posedness of the fractional Schrödinger equationwith less regular exterior data. Let us explain this in the present context of data in L2 ( B3\B2) with a fractional Schrödinger equationposed on B1 (as in lemma 2.1). If f∈L2 ( B3\B2) , we may find u∈Hs(Rn) solving ((−∆)s+q)u=0 in B1 with u|R n \B1=f by writing u = E0 f + v, where E0 denotes extension by zero and v solves ((−∆)s+q)v=F in B1 with v|R n \ B 1=0 , and F=−(−∆)s(E0f)|B1 . But since f is supported in B3\B2 , one has F∈C∞ ( B1) by the pseudolocal property of Fourier multipliers, and one can find a solution v∈Hs(Rn) by standard well-posedness theory. The previous argument shows well-posedness with exterior Dirichlet data f in L2 ( B3\B2) , and also the estimate u L2( R n) ⩽f L2(B 3\ B 2 ) +CF H−s(B1) ⩽Cf L2(B 3\ B 2) which again follows from the pseudolocal property. Proof. The proof relies on the explicitly known Poisson formula for the fractional Laplacian in B1 (see for instance [Buc16]) and its close relation to the Poisson formula for the Laplacian. In constructing the desired basis, we argue in two steps: First we deal with the spherical and then with the radial contributions of the basis functions. A Rüland and M Salo Inverse Problems 34 (2018) 045003
6 Step 1: Poisson formula. By the explicit formula for the Poisson operator for the fractional Laplacian (see for instance [Buc16]) we have that for x∈B1 and some constant c=c(n,s)>0 , u ( x ) c (1−|x|2)s= Rn\B1 1 |x−y|n f ( y ) (|y|2−1)sdy= ∞ 1 ∂B1 rn−1 |x−rω|n f ( r ω) (r2−1)sdωd r = ∞ 1∂B1 1 | x/r − ω | n f(rω) r(r2 −1 )sdωdr, (5) where we introduced polar coordinates y=rω with |ω|=1 in Rn\B1 . Now we recall that the Poisson kernel for the standard Laplacian on the ball B1 is given by K (x,ω)=cn(1−|x| 2 ) |x− ω | n. We also assume that the exterior data in (4) have the form f(rω)=r(r 2−1 ) s g(r)h(ω) . Combining these observations with (5), it follows that for some constant ˜ c=˜ c(n,s)>0 u (x)=˜ c(1−|x|2)s ∞ 1 g ( r ) 1 −| x | 2/r2 ∂B1 K(x/r,ω)h(ω)dωdr . If h = hm,l is a spherical harmonic of degree m, then ∂B1K ( x / r, ω) h m,l(ω) d ω = | x/r | mhm,l x | x | , since this is the corresponding solution of the Laplace equation. This implies that u (x)=˜ c(1 −| x | 2)s | x | mhm,l x |x|∞ 1 g ( r ) r m (1−|x| 2 /r 2 ) dr. (6) The expression (6) already displays the first decay properties of u(x) in dependence of m∈N , if we assume that g is supported in B3\B2 and is bounded. We next seek to specify g in order to obtain the full decay properties, i.e. the decay in m and k, where the latter will be a consequence of the choice of the radial functions g(r) . Step 2: Construction of the radial component. We seek to specify the radial part of the functions f in order to obtain an orthonormal basis {f m,k,l }∞ m,k=0 of L2 ( B3\B2) , where fm,k,l(rω)=r(r2−1)sgm,k(r)hm,l(ω). Here hm,l are spherical harmonics of degree m, and the functions gm,k∈L1 ([ 2, 3]) are to be determined. We aim for decay properties for the corresponding solutions um,k,l in the form u m,k,lL 2 (B1)⩽ C(n,s)e− c ( m + k ). (7) Noting that um,k,l(˜ rω)=cn,s(1−˜ r2)s˜ rmhm,l(ω)Fm,k(˜ r) (8) and by invoking the orthogonality of the spherical harmonics, the decay (7) would follow from (6), if the function F m,k(˜ r):= 3 2 gm,k(r) r m (1−˜ r 2 /r 2 ) dr A Rüland and M Salo Inverse Problems 34 (2018) 045003
7 satisfied (1 − ˜ r 2 ) s ˜ r m ˜ r( n − 1 )/ 2 Fm,k(˜ r) L 2 ([0,1]) ⩽ C(n,s)e− c ( k + m) . We seek to deduce conditions which guarantee this. We remark that in this context, the exponential decay in m can be obtained comparatively easily and would already follow from the assumption gm,k∈L1 ([ 2, 3]) with g m,k L 1 ([2,3]) ⩽C , where C > 1 is independent of m⩾0 , as in this case F m,r L∞([0,1]) ⩽2−mg m,k L 1 ([2,3]) ⩽C2−m . The derivation of the decay in the parameter k is more subtle and will be explained in the sequel. To this end, we rewrite the expression for Fm,k ( ˜ r) by spelling out the series representation of (1− ˜ r2 / r2 ) −1 : Fm,k(˜ r)= ∞ j=0 ˜ r2j 3 2 g m,k( r ) rm+2jdr. (9) Here we used the absolute (even geometric) convergence of the sum (and the assumption that gm,k∈L1 ([ 2, 3]) ) in order to exchange integration and summation. In order to infer the desired order of vanishing, we will arrange that for k⩾1 one has 3 2 g m,k( r ) r m+2jdr=0, 0 ⩽j⩽k −1. (10) In addition, the desired orthonormality of the basis fm,k,l ( r ω ) requires that 3 2 rn+1(r2 − 1)2sgm,k(r)gm,l(r)dr=δkl . (11) We symmetrize the conditions (10) and (11) slightly, in order to deal with the same weighted scalar product in both equations. Setting ˜ gm,k ( r ) : = rn+1 ( r2−1 ) 2sgm,k ( r) then turns (10) and (11) into (˜ g m,k ,r−(m+2j) )s= 0 for 0 ⩽j⩽k − 1 when k⩾1, (˜ g m,k, ˜ g m,l ) s =δ lk (12) where ( f,g) s := 3 2 r−n−1(r2 − 1)−2sf(r)g(r)d r . In order to satisfy the conditions in (12), we inductively choose ˜ g m,0 (r)∈V 0 :=span{r−m}, ˜ gm,k(r)∈Vk:=span{r− m ,r− m − 2 ,...,r− m − 2k }for k⩾1. To this end, we set ˜ gm,0(r)=cr−m for some c∈R\{0} (which is chosen such that ˜ gm,0 satisfies the desired normalization (˜ gm,0,˜ gm,0)s=1 ) and assume that for k⩾1 the functions ˜ gm,j∈Vj are already defined for j⩽k−1 and {˜ gm,0,...,˜ gm,k−1} forms an orthonormal basis of Vk−1 with respect to (·,·)s . Then, the conditions in (12), which have to be satisfied by ˜ gm,k , can be formulated as ˜ gm,k⊥sVk−1,(˜ gm,k,˜ gm,k)s=1. (13) By Gram–Schmidt orthonormalization in the finite dimensional Hilbert space (Vk,(·,·)s) , we have that Vk=Vk−1⊕⊥,sWk, where Wk is a one-dimensional vector space. Choosing ˜ gm,k∈Wk with ˜ gm,ks=1 (up to a choice of a sign this is unique) implies the conditions from (13). Moreover, we note that A Rüland and M Salo Inverse Problems 34 (2018) 045003
14 Since for ˜ ψ∈Z it holds that ˜ ψC m ⩽(N√n)mψC m, we will have ˜ ψC m ⩽β if N: = max {k∈N:k⩽ (β(µ) −1 ) 1/m}. Since also ˜ ψ L∞⩽ , we infer that Z⊂X,β,m by our choice of N∈N . The estimate on the cardinality for Z follows by noting that |Z| = 2Nn and by plugging in the estimate N ⩾1 2 ( β µ ) 1 m. □ Combining lemmas 3.2 and 3.4 and applying the pigeonhole principle finally yields the argument for theorem 1. Proof of theorem 1. Let r0 be as in lemma 3.2, and let µ>0 be as in lemma 3.4. We choose β>0 such that β/µ =(C 12n + 1 ) m / n , where C1 > 1 is a sufficiently large constant that is to be determined. Let ∈(0, min{r0/2, 1, µ−1β}) and q0∈L∞(B1) with q0L∞⩽r0/2 be arbitrary. By lemma 3.4 the set q0+Xm,,β has an ε-discrete subset q0 + Z with |Z|⩾ exp( 2−n−1 (β(µ) −1 ) n/m ) ⩾ exp( C1 −n/m ) . As q 0 +X m,,β ⊂B∞ r0 , the set Y from lemma 3.2 is a δ-net for Γ(q0+Xm,,β) . Choosing δ= exp(−−n m 1 2n+3 ) thus yields that |Y|⩽ exp( C (log(δ −1 )) 2n+3 ) ⩽ exp( C −n/m ) . If C1 > C, we thus infer that |Z|>|Y| and therefore also |q0+Z|>|Y| . This implies that there exist q1,q2∈q0+Z such that Γ(q1) and Γ(q2) are in the same ball of the δ-net Y. Hence, from (20) we obtain that Λ q1 −Λ q2 L2(B 3\ B 2 ) → L2(B 3\ B 2 ) ⩽4Γ(q 1 )−Γ(q 2 ) X ⩽8δ, which yields the first estimate in (3). The remaining ones follow by the definition of X,β,m and of Z (in fact z1−z2L∞= for any distinct z1,z2∈Z ). □ We remark that the proof for corollary 1.2 follows along exactly the same lines as the proof of corollary 2 in [Man01], if the operator −∆ is replaced by the operator (−∆)s . As it does not involve any new ideas, we omit its proof and refer to the argument in [Man01]. 4. Approximation Again exploiting the smoothing properties of the basis from lemma 2.1, in this sectionwe provide the argument for theorem 2. This in particular essentially proves the optimality of the moduli of continuity in the approximation results in [RS17]. Proof of theorem 2. We use the basis which was constructed in lemma 2.1: We recall that the functions fm,k,l(rω):=gm,k(r)hm,l(ω) from lemma 2.1, where m,k⩾0 and 0⩽l⩽lm , form an orthonormal basis of L2(B3\B2) and are associated with the functions um,k,l:=A0(fm,k,l) , which are solutions of (4). Due to the structure of the Poisson operator for the fractional Laplacian (see (6)), these functions are again of the form um,k,l(rω)=qm,k(r)hm,l(ω). (22) Here hm,l(ω) denote the spherical harmonics and qm,k(r) are explicit functions (see lemma 2.1 A Rüland and M Salo Inverse Problems 34 (2018) 045003
15 and its proof), which however are in general not pairwise orthogonal. We note that it is possible to write any function u satisfying condition (i) in theorem 2 with boundary values f = m,k,l αm,k,l f m,k,l ∈L2 ( B 3 \B 2 ) as u= m,k,l αm,k,lum,k,l . (23) This sum converges in L2 ( B1) , since by the orthogonality of the spherical harmonics hm,l and by (16) one has αm,k,lum,k,l 2 L2(B1)= m,l k αm,k,lum,k,l 2 L2(B1) ⩽ c 2 n,s m,l 2 −2m ( k|αm,k,l|2 −k ) 2 ⩽c2 n,s m,k,l |αm,k,l|2. With this preparation at hand, we define the sequence of functions {vp}p∈N that will satisfy the assertions in theorem 2 as vp(rω):=A0 f p,0,0( r ω) u p,0,0L 2 (B1) = u p,0,0( r ω) u p,0,0L 2 (B1) . Thus in particular, v p L2(B 1 ) =1 . For convenience, we also abbreviate α 0 : = u p,0,0 −1 L2(B1 ) . We now proceed in three steps: Step 1: Assuming that a function u satisfies the conditions (i), (ii) in theorem 2 with a decomposition as in (23), we claim that for p⩾2 ∞ k=0 αp,k,0up,k,0L2(B1)⩾ 1 2 . (24) Argument: This follows from orthogonality, the normalization of vp and the structure of the functions um,k,l. Indeed, by orthonormality of the spherical harmonics and by using the notation introduced in (22), we have that p −2⩾ u−vp 2 L2(B1) =∞ k=0 αp,k,0qp,k−α0qp,02 L2 r([0,1]) + (m,l)=( p,0) ∞ k=0 αm,k,lqp,k2 L2 r([0,1]) . Here L2 r ([ 0, 1]) denotes the radial part of the L2 norm after introducing polar coordinates and integrating out the spherical variables, and α 0= 1 up,0,0 L2(B 1 ). In particular, p−1⩾ ∞ k=0 αp,k,0qp,k−α0qp,0L2 r([0,1]) = ∞ k=0 αp,k,0up,k,0 −vpL2(B1 ) ⩾vpL2(B1)− ∞ k=0 αp,k,0up,k,0L2(B1). Choosing p⩾2 , recalling the normalization of vp and rearranging hence implies (24). Step 2: Let u be as in Step 1. We claim that there exists k0∈N∪{0} such that for p⩾2 A Rüland and M Salo Inverse Problems 34 (2018) 045003
16 | αp,k0,0|⩾ 2p−2 c n,s , (25) where cn,s > 0 is the constant in (16). Argument: We argue by contradiction and assume that the claim were wrong. Thus for all k we would have | αp,k,0 | <2 p−2 cn,s . By Step 1, the triangle inequality and the bound (16) this would lead to the following chain of estimates: 1 2 ⩽ ∞ k=0 αp,k,0up,k,0L2(B1)⩽ ∞ k=0|αp,k,0|up,k,0L2(B1 ) <∞ k=0 2p−2 cn,s cn,s2−p−k=1 2. This is a contradiction. Step 3: Using Steps 1 and 2 we conclude the proof. Indeed, by Steps 1 and 2 for any u = A0( f ) with data f = m,k,l αm,k,l f m,k,l ∈L2 ( B 3 \B 2 ) satisfying (i), (ii) in theorem 2 for the function vp with p⩾2 defined above, we on the one hand obtain the bound f L2(B3 \ B2)⩾ | αp,k0,0 | fp,k0,0 L2(B3 \ B2)⩾ 2p−2 cn,s using the orthonormality of the functions fm,k,l and choosing k0∈N∪{0} as in Step 2. On the other hand, by the initial normalization v p L 2( B1 ) =1 . Combining the last two observations and defining c 0 : = 1 4cn,s implies the claim. 5. The 1D Calderón problem and the truncated Hilbert transform In this sectionwe discuss the one-dimensional situation in greater detail: In the one-dimen- sional setting the mapping properties of A0 can be made even more explicit by using a close relation between the fractional Laplacian and the truncated Hilbert transform. To illustrate this, we again consider the Poisson operator (where for simplicity the exterior data is zero in [−3, −2]) Aq :L 2 ([2, 3]) → L 2 ([ − 1, 1]),g → χ [−1,1] u , where u is a solution to ((− ∆) s + q ) u = 0 in ( −1, 1 ) , u=gχ [2,3] in R\ [ − 1, 1]. By the representation formula for the solution to the fractional Laplacian in one dimension (see for instance [Buc16]) we infer that χ [−1,1]A0g(x)=c(s) [2,3] 1−|x| 2 | y | 2 − 1 s g(y) | x − y | dy . (26) A Rüland and M Salo Inverse Problems 34 (2018) 045003
17 Based on this representation formula, it will be possible to relate A0 to the truncated Hilbert transform. This will allow us to transfer results on the well-studied truncated Hilbert transform to the fractional Laplacian. Before formulating this connection precisely, we recall that the Hilbert transform is defined as the singular integral operator H:L2(R)→L2(R),f→ H(f)(t):=p.v. R f ( y ) t − ydy . This principal value integral is immediately seen to be well-defined on Lipschitz functions, but can be extended to the whole class of L2 functions. This can for instance be achieved by noting that H ( f )= F−1 ( Ff (ξ)( −i sgn (ξ ))) , where F:L2 ( R ) →L2 ( R ) ,f→ F ( f )= Rf ( x ) e−2πixξdx denotes the Fourier transform (see for example [Gra08], chapter 4 for more information on singular integrals and the Hilbert transform). We define the truncated Hilbert transforms H [2,3] :L2 ([ 2, 3 ]) → L2 ([− 1, 1 ]) ,f → H ( E 0 f )|[−1,1] , H[ −1,1 ]:L2([ − 1, 1]) → L2([2, 3]),f → H(E0f) | [ 2,3 ] , where E0 denotes extension by zero to R . Lemma 5.1. Let A0:L2([2, 3]) →L2([−1, 1]) be the mapping from above. Then we have that for all x∈[−1, 1] A 0 g ( x )= −c ( s )( 1−x2 ) sH [2,3](( ·2−1 ) −sg )( x ) . We remark that since in this sectionwe are working in (weighted) L2 spaces without derivatives, it does not really matter, whether we consider function spaces on the open or closed intervals (−1, 1) , (2, 3) . Since in the context of the truncated Hilbert transform the literature most commonly uses the notation of closed intervals, we follow this convention in the sequel. Proof. The proof follows immediately by rewriting the expression from (26): For x∈[−1, 1] we have that A0g(x)=c(s) [2,3] 1 −| x | 2 |y|2−1 sg ( y ) |x−y|dy =−c(s)(1−x2)s[2,3] (y2−1)−sg(y) x−yd y = − c(s)(1 − x2)sH[ 2,3 ](( · 2 − 1)−sg)(x).□ Motivated by lemma 5.1, we recall some properties of the singular value decomposition (σl,fl,gl)∈R+×L2([2, 3]) ×L2([−1, 1]) of the truncated Hilbert transform H[2,3] (see [Kat10]). The sequences {fl}l∈N and {gl}l∈N are orthonormal bases of L2([2,3]) and L2([−1,1]), which are generalized eigenfunctions of the truncated Hilbert transform in the sense that H [2,3] f l =σ l g l ,H [ − 1,1] g l =−σ l f l . Here the completeness of the orthonormal system {gl}l∈N⊂L2([−1, 1]) can for instance be obtained by a combination of unique continuation and duality arguments (see for instance lemma 2.7 in [Rül17]). The singular values asymptotically satisfy exponential bounds (see [KT12]): A Rüland and M Salo Inverse Problems 34 (2018) 045003
18 0<σ l⩽e−cl. (27) Moreover, the functions fl are also eigenfunctions of a (degenerate) Sturm-Liouville problem. They are eigenfunctions (with weighted Neumann boundary conditions) of the operator L ( x,dx ) f: =( P ( x ) f ( x )) + 2 ( x− σ) 2f ( x ) , (28) where P (x):=(x − 1)(x+1)(x − 2)(x − 3),σ= 5 4. The associated eigenvalues λn satisfy λn⩾c1n2 (see [Kat10, KT12]). Based on this and lemma 5.1, we define function spaces and a basis adapted to the operator A0: Definition 5.2. Let (σl,fl,gl)∈R+×L2([2, 3]) ×L2([−1, 1]) be the singular value decomposition of H[2,3]. Then we define the weighted functions ˆ fl ( y ) : =( y2−1 ) sfl ( y ) , ˆ gl ( x ) : =( 1−x2 ) sgl ( x ) , and for γ∈R we consider the function spaces L2 γ ([ 2, 3 ]) : = L2 ([ 2, 3]) equipped with the scalar product (f,g )L2 γ ([2,3]) : =( f, ( ·2−1 ) 2γg )L2([2,3]) . Remark 5.3. We remark that for s∈(−1, 1) the spaces L2([2,3]) and L2 ±s ([ 2, 3]) are isomorphic. Indeed, due to the strict disjointness of the intervals [2,3],[−1,1], there exist constants c1,c2∈(1/3, 3) such that for all s∈(−1, 1) c 1 f L2 s ([ 2,3 ]) ⩽f L2([ 2,3 ]) ⩽c 2 f L2 s ([ 2,3 ]) . The advantage of working with the spaces L2 ±s ([ 2, 3]) instead of working with the standard space L2([2,3]) is that the functions ˆ fl are well-adapted to the fractional Laplacian. Indeed, by virtue of lemma 5.1 they satisfy A 0 ˆ f l= −c s( 1−x2 ) sH [2,3]( f l)= −c s( 1−x2 ) s σl g l= −c sσlˆ g l . (29) Moreover, the functions {ˆ fl}l∈N form an orthonormal basis of L2 −s ([ 2, 3]) (as the functions {fl}l∈N are orthonormal with respect to L2([2,3])). As a side product of this relation between the one-dimensional fractional Laplace operator and the truncated Hilbert transform, we derive another version of theorem 2 showing the (almost) optimality of the one-dimensional quantitative approximation result of arbitrary functions by s-harmonic functions (theorem 1.3 in [RS17]): Theorem 3. There exists a sequence {hk}k∈N∈L2([−1, 1]) such that the following holds: For any function ˜ hk with the property that (i) ˜ hk approximates hk in the sense that ( 1−x2 ) −s ( h k −˜ h k) L2([ −1,1 ]) ⩽k−1, (30) (ii) ˜ hk satisfies the equation A Rüland and M Salo Inverse Problems 34 (2018) 045003
19 (− ∂xx) s˜ h k= 0 in [ −1, 1 ] ,˜ h k=χ[2,3] ˜ f k in R\ [ −1, 1] for some ˜ fk∈L2 ([ 2, 3]) , the associated control function ˜ fk satisfies ˜ f kL 2 ([2,3]) ⩾ Ce ck. The proof is slightly simpler than the higher dimensional analogue in section4, since we can more directly exploit orthogonality. Proof of theorem 3. We consider the functions hk: =ˆ gk∈L2 ([ −1, 1]) . Then by orthogonality of the functions gk with respect to the L2([−1,1]) topology and due to the presence of the weight (1−x2 ) −s in (30), any approximation ˜ hk to hk within an k−1 error threshold (in the weighted norm), has to be of the form ˜ hk=αkˆ gk+¯ g, with ( ¯ g, ( 1−x2 ) −2s ˆ gk ) L 2 ([−1,1]) = 0 and αk⩾1−k−1 . As a consequence of (29), the associated control function ˜ fk has to be of the form ˜ fk=−c−1 sσ−1 kαkˆ fk+¯ f, with (¯ f, ˆ fk)L2 −s ([2,3]) = 0 . Due to orthogonality in the space L2 −s ([ 2, 3]) and due to the bound (27), we therefore obtain that ˜ fk L2 −s ([2,3]) ⩾ Cαkσ −1 k⩾ Ce ck . By virtue of the equivalence of the L2([2,3]) and L2 ±s([2, 3]) norms, this also yields a lower bound for the L2([2,3]) norm of the control. □ 6. Exponential instability in boundary-bulk measurements Finally, we remark that the classical Hadamard example (see for instance [ARRV09]), showing that exponential instability can occur by passing from boundary to interior measurements, can also be directly transferred to the context of the fractional Laplacian (or rather its Caffarelli–Silvestre extension, see [CS07]): Lemma 6.1. Let Is( y ) denote the modified Bessel function of first kind. The function vn(x,y)=Csn−ssin(nx)ysIs(ny) , for a suitable constant Cs=0 , is a solution to ∇·y 1 − 2s ∇v=0inR 2 +, v=0onR×{0}, lim y→0 y1−2s∂yv= sin(nx)on R ×{ 0 }, (31) where R2 +: = R× ( 0, ∞) . For any y0 > 0 this function satisfies as n→∞ vn ( x,y0 ) ∼Csn−s sin( nx ) ys 0 ( 2 π ny0 ) −1/2eny 0 . A Rüland and M Salo Inverse Problems 34 (2018) 045003
20 Proof. The result follows from a direct calculation, e.g. by inserting v(x,y) into (31). Separating variables, the bulk equationfor Is(ny) turns into a modified Bessel equation: For z = ny we have z2I s ( z )+ zI s ( z ) − ( z2 + s2 ) Is ( z )= 0 in ( 0, ∞ ) . As the modified Bessel function satisfies (see [Olv10]) I s(z)∼ 1 2z s1 Γ(s+1)as z→ 0, I s(z)=Is+1(z)+s z Is(z), the boundary conditions in (31) are then satisfied by choosing the constant Cs=0 appropriately. Moreover, we have (see [Olv10]) I s(z)∼ ez √ 2πz as z →∞ . The asymptotics as n→∞ thus follow from the asymptotics of Is(z) as z→∞ . □ The example given in lemma 6.1 shows that there is an exponential instability in controlling bulk by boundary values for solutions to the Caffarelli–Silvestre extension problem. 7. Conclusion Summarizing, by adapting the instability argument due to Mandache [Man01] to the setting of the fractional Calderón problem, we have shown that similarly as in the setting of the classical Calderón problem, the fractional Calderón problem has an exponential instability. Therefore, in spite of the nonlocality of this inverse problem one cannot hope for better than logarithmic stability properties. In particular, the stability estimates from [RS17] are (up to the precise exponents) sharp. This displays a difference between the time and space fractional inverse problems discussed in the introduction, as in several time fractional problems, surprising improvements in the corresponding stability properties were obtained. In addition to this, the constructions in the present article also show the sharpness of the quantitative approximation properties from [RS17] (see theorem 1.4). Acknowledgments MS is supported by the Academy of Finland (Finnish Centre of Excellence in Inverse Problems Research, grant numbers 284715 and 309963) and an ERC Starting Grant (grant number 307023). ORCID iDs Angkana Rüland https://orcid.org/0000-0003-1169-344X Mikko Salo https://orcid.org/0000-0002-3681-6779 A Rüland and M Salo Inverse Problems 34 (2018) 045003
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