Solutions to inverse moment estimation problems in dimension 2, using best constrained approximation
Juliette Leblond Elodie Pozzi


 

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    ess supSolutions to inverse moment estimation problems
          in dimension 2, 
       using best constrained approximation
        Juliette LeblondINRIA, Team Factas, 2004 route des Lucioles, 06902 Sophia Antipolis, France Elodie PozziSaint Louis University, Department of Mathematics and Statistics, Ritter Hall, 1N. Grand Blvd., Saint Louis, MO 63103, USA
        
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     We study an inverse problem that consists in estimating the first (zero-order)  moment of 
      some ^2-valued distribution  supported within a closed interval S̅⊂,
      from  partial knowledge
      
      of the solution to the Poisson-Laplace partial differential equation with source term equal to the divergence of  
      on another interval parallel to and located at some distance 
      from S. Such a question coincides with a 2D version of an inverse magnetic “net” moment recovery question that arises in paleomagnetism,  for thin rock samples. We formulate and constructively solve a best approximation problem under constraint in L^2 and in Sobolev spaces involving the restriction of the Poisson extension of the divergence of . Numerical results obtained from the described algorithms for the net moment approximation are also furnished.
     Keywords: Poisson-Laplace partial differential equation, inverse problems, regularization, best constrained approximation and bounded extremal problems, Poisson and Hilbert transforms, Hardy spaces, Maxwell's equations  and issues.
    
    
    
    § INTRODUCTION
     
    The present study concerns the practical issue of estimating the “net” moment (or the mean value) of some ^2-valued distribution (or integrable, or square-integrable function) supported on an interval of the real line, 
    from the partial knowledge of the divergence of its Poisson extension to the upper half-plane, on some interval located on a parallel line at distance h>0 from the first one.
    
    
     This is a two dimensional (2D) formulation of a three dimensional (3D) inverse magnetic moment recovery problem in paleomagnetism, for thin rock samples. There, the magnetization is assumed to be  some ^3-valued distribution (or integrable function) supported on a planar sample (square), of which we aim at estimating the net moment, given measurements of the normal component of the generated magnetic field on another planar measurements set taken to be a square parallel to the sample and located at some distance h to it. Moment and more general magnetization recovery issues are considered in <cit.>. 
    More about the data acquisition process and the use of scanning SQUID (Superconducting Quantum Interference Device) microscopy devices for measuring a component of the magnetic field produced  by weakly magnetized pieces of rocks can be found in the introductory sections of these references.
    
     In both situations, the partial differential equation (PDE) model that drives the behaviour of the magnetic field derives  from  Maxwell equations in magnetostatics, see <cit.>. They ensure that  the magnetic field  derives from a  scalar magnetic potential U which is solution to
    a  Poisson-Laplace elliptic PDE with right-hand side of divergence form, that relates the Laplacian of that potential to the divergence of the magnetization distribution : 
    
        = - μ_0  
          U   ,  Δ  U =    in ^n   ,
    
      with n = 2, 3, where both ,  are ^n-valued quantities, and μ_0 is the permeability of the free space. We stick to situations where   has a  compact  support contained in the hyperplane {x_n = 0}  and measurements of the vertical component b_n of the field  are available on another compact subset of {x_n = h} for some h >0. 
    
     The determination of the magnetic moment (mean value of the magnetization) provides useful preliminary information for the full inversion, i.e. for magnetization estimation, in particular in unidirectional situations where its components are proportional one to the others (the unknown quantity being a -valued distribution, the direction/orientation vector being fixed).
     In  3D situation,  these issues were efficiently analyzed with tools from harmonic analysis,  specifically Poisson kernel and Riesz transforms <cit.>,  in <cit.>, see also references therein, 
    using their links with Hardy spaces of harmonic gradients. 
    The existence of silent sources in 3D, elements of the non-zero kernel of the non injective magnetization-to-field operator (the operator that maps the magnetization to the measured component of the magnetic field)  was established together with their characterization in <cit.>. This makes non unique the solution to the inverse magnetization issue from the corresponding field data. The mean value of the magnetization however is uniquely determined by the field data. 
    
    
     In the present 2D case, we will use similar tools, Poisson kernel, Hilbert transform and harmonic conjugation, links with Hardy spaces of functions of the complex variable <cit.>;  see also <cit.> for related issues. 
    Considering  square-summable magnetizations  supported on an interval S of the real line, 
    we will establish that the magnetization-to-field operator ↦ b_2[] is injective, whence there are no silent sources, and that the mean value ⟨⟩ of the magnetization is yet uniquely determined by the field data b_2[], provided by values of the vertical 
    component on the measurement interval K.
     Our purpose is to establish the existence and to build linear estimators for the mean values ⟨m_i ⟩ on S of the components m_i of  for i=1,2, as was done in <cit.> for the 3D case. Indeed, though an academic version of the related physical issue, the present 2D situation possesses its own mathematical interest and specificity.
    These linear estimators consist in square summable functions ϕ_i
    such that their effect (scalar product) against the data on K is as close as possible to the moment components ⟨m_i ⟩, for all  bounded by some L^2 norm (the functions being built once and for all).
    With _1=(χ_S,0), _2 = (0,χ_S), observe that:
    
    ⟨ m_i ⟩ = ⟨  ,  _i ⟩_L^2(S,^2),
    
    whence, if b_2^* denotes the adjoint operator to b_2, we have:
    
        |⟨ b_2[]   ,  ϕ_i⟩_L^2(K) - ⟨ m_i ⟩|   ≤   ‖‖_L^2(S,^2)‖ b_2^*[ϕ_i]-_i‖_L^2(S,^2),
     where ϕ_iis a square integrable function on K. We will see that the above minimization problem is still ill-posed, even if uniqueness is granted. Specifically, there exist a sequence of functions such that their scalar product on K against b_2[] converges to ⟨ m_i ⟩ as n →∞. Their quadratic norm however diverge, which reflects an 
    unstable behaviour, as is classical in such inverse problems, see <cit.>. Regularization is thus needed (of Tykhonov type), see <cit.> in order to set up and to solve a well-posed problem.
    
     In order to construct such a numerical magnetometer, we then face the best constrained approximation issue (bounded extremal problem, BEP) of finding the function ϕ_i on K satisfying some norm constraint there, such that its scalar product against b_2[] is as close as possible to ⟨m_i ⟩.
    
     Such a norm constraint will be considered both in the Lebesgue space L^2(K) and in the Sobolev space 
    W_0^1,2(K) ⊂ W^1,2(K).
    In those Hilbert spaces, the above problems will be shown to be well-posed, the approximation subsets being closed and convex. The results obtained in L^2(K) and in 
    W_0^1,2(K) are different. In particular, the solutions in W_0^1,2(K) demonstrate less oscillations at the endpoints of K 
    than the ones in L^2(K), 
    and allow to incorporate  vanishing boundary conditions. The results for moment estimation will be compared between them.
    
     Preliminary numerical computations in the 3D setting, with planar squared support and measurement set, were run on appropriate finite elements bases, see  <cit.>; they are actually heavy. 
    In the present 2D setting and on intervals, they are of course much less costly and we perform some of them using expansions on the Fourier bases, in Lebesgue and Sobolev spaces.
     The overview of the present work is as follows. In Section <ref>, we introduce some notation, recall definitions and properties, concerning Lebesgue, Sobolev and Hardy spaces of functions, together with integral Poisson and Hilbert transforms.
    We then study in Section <ref> the properties of the operator b_2 :  → b_2[], which maps the magnetization to the second (vertical) component of the produced magnetic field, and of its adjoint.
    Section <ref> is devoted to the bounded extremal problems of which we consider two versions, in L^2(K) and in W_0^1,2(K), establishing their well-posedness and characterizing their solutions. 
    Computational algorithms together with results of preliminary numerical simulations are provided in Section <ref>, with figures in an appendix. We finally discuss in Section <ref> some concluding remarks and perspectives.
    
    
    
    § PRELIMINARIES
     Let _+ ≃^2_+ be the upper half-plane ℂ_+={x+iy:y>0}⊂,  
    ^2_+={(x,y):y>0}⊂^2. The partial derivatives with respect to x and y will be denoted by ∂_x and ∂_y respectively.
      
    
    
     §.§ Lebesgue and Sobolev spaces
     Let I ⊆ a non empty open interval. For 1≤ p<∞, we will denote by L^p(I,𝕂) with 𝕂=,  or ^2  the space of 𝕂-valued functions f such that | f|^p is integrable on I; we will simply write L^p(I)=L^p(I,).
    
     
    We equip the space L^2(I, ℝ^2) with the following inner product: for f=(f_1,f_2) and g=(g_1,g_2)∈ L^2(I, ℝ^2), 
    
        ⟨ f,g⟩_L^2(I, ℝ^2) =⟨ (f_1,f_2),(g_1,g_2)⟩_L^2(I, ℝ^2)
        = ⟨ f_1,g_1⟩_L^2(I)+⟨ f_2,g_2⟩_L^2(I)  ,
    
    while L^2(I, ) is equipped with: 
    
    
        ⟨ f,g ⟩_L^2(I, )=⟨  f ,  g ⟩_L^2(I)+⟨  f , g ⟩_L^2(I) for  f  ,  g ∈ L^2(I, )   .
    
    For I ⊊, for ϕ∈ L^2(I), we will write ϕ=ϕ∨ 0 ∈ L^2(ℝ), the function equal to ϕ on I and equal to 0 on \ I. Note that we identify L^2(I) to χ_I    L^2() and to the subspace of L^2(ℝ) of functions supported on I. We also have ϕ_|_I = R_I  ϕ= ϕ for ϕ∈ L^2(I) and the adjoint of the restriction operator R_I from L^2() to L^2(I) is the extension operator . on L^2(I).
    
     When I is bounded, L^2(I)⊂ L^2(ℝ)∩ L^1(ℝ). Moreover, functions in L^2(I) are compactly supported on ℝ.
    
     For I ⊆,
     the Sobolev space W^1,2(I) is the space of real-valued functions f ∈ L^2(I) such that their (distributional) derivative f' belongs to L^2(I); W^1,2(I) is equipped with the norm (see <cit.>):
    
    
        ‖ f‖_W^1,2(I)=‖ f‖_L^2(I)+‖ f'‖_L^2(I)  ,   f∈ W^1,2(I)  .
    
     A function in W^1,2(I) can be extended 
    to a function in W^1,2(ℝ),
    see the extension operator in <cit.>. The space C(I̅) denotes the space of continuous functions on I with the norm
    
    
        ‖ f‖_C(I̅)=sup_x∈I̅| f(x)|  ,     f ∈ C(I̅)  .
    
     Note that W^1,2(I)⊂ C(I̅), the space of continuous real-valued functions on I̅, with compact (hence continuous) injection, see <cit.>. 
    If I is bounded, C^1(I̅) ⊂ W^1,2(I) will denote the space of functions in C(I̅) with (strong) derivative in C(I̅). The space is equipped with the norm 
    
        ‖ f‖_C^1(I̅)=sup_x∈I̅| f(x)|+sup_x∈I̅| f'(x)|  ,     f ∈ C^1(I̅)  ,
     where the (strong) derivative coincides with the distributional derivative. The space C^∞() will denote the space of functions such that derivatives at all order are continuous. The subspace W_0^1,2(I) of W^1,2(I) is the collection of functions f∈ W^1,2(I) such that f=0 at the endpoints of I. A consequence is that f∈ W^1,2(ℝ) for f∈ W_0^1,2(I); in other words, . is the extension operator from W_0^1,2(I) to W^1,2() (see <cit.>.
    We equip W_0^1,2(K) with the norm:
    
    
        ‖ϕ‖_W_0^1,2(K)=‖ϕ'‖_L^2(K)  ,   ϕ∈W_0^1,2(K)  ,
    
    which is 
    equivalent to ‖ϕ‖_W^1,2(K)
    by Poincaré's inequality (see <cit.>).
    
    
    
     §.§ Poisson kernel, conjugate Poisson kernel and Hilbert transform
     For y>0, we denote by P_y the Poisson kernel of the upper half-plane and by Q_y the conjugate Poisson kernel. For x ∈:
    
      
        P_y(x)=1/πy/x^2+y^2,   Q_y(x)=1/πx/x^2+y^2.
     see <cit.>. Let x ∈, y>0. Then
    
        P_y⋆ u(x)=1/π∫_-∞^∞y/(x-t)^2+y^2u(t) dt,    u∈ L^2(ℝ),
    
      and 𝒫_y:u ↦ P_y⋆ u is a bounded (real linear) operator from L^2(ℝ)   to L^2(ℝ) (<cit.>).
    
     The map, x ↦ P_y(x)  belongs to L^1() (where its norm is equal to 1). Moreover,  (x,y) ↦ P_y(x)  belongs to C^∞(_+^2). Therefore, its partial derivatives  are also C^∞(_+^2)-smooth functions, whence for y>0,  x ↦ (∂_x P_y)(x) and x ↦ (∂_y P_y)(x) belong to C^∞(). The functions P_y and Q_y satisfy the Cauchy-Riemann equations on _+: ∂_yP_y=-∂_xQ_y, ∂_xP_y=∂_yQ_y.
     It can be checked by direct computations that x ↦ (∂_x P_y)(x) and x ↦ (∂_y P_y)(x) belong to L^1(), and ∂_x P_y_L^1()=∂_y P_y_L^1()=2 / (π  y). Hence, u → (∂_x P_y)⋆ u and u → (∂_y P_y)⋆ u are  bounded operators from L^2(ℝ)   to L^2(ℝ). Further, for u ∈ L^2(ℝ), (x,y) ↦𝒫_y [u](x)=(P_y⋆ u) (x) is also harmonic  in _+^2, hence a C^∞(_+^2)-smooth function. Thus, its partial derivatives x ↦∂_x( P_y ⋆ u )(x) and x ↦∂_y( P_y ⋆ u )(x) belong to C^∞(), for y>0.
     Using the Fourier transform as in <cit.>, we see that convolution by the Poisson kernel and differentiation commute: for y>0 and u ∈ L^2(ℝ), we have that ∂_x (P_y⋆ u) = (∂_x P_y) ⋆ u.
     
    In particular, the operator 𝒫_y maps continuously L^2(ℝ) onto W^1,2(ℝ). By Cauchy-Riemann equations, Q_y satisfies the  properties of P_y described above. Indeed, u↦∂_x(Q_y⋆ u) and u↦∂_y(Q_y⋆ u) are bounded on L^2(ℝ) and for y>0, x↦∂_x(Q_y⋆ u) and x↦∂_y(Q_y⋆ u) are C^∞().
     In order to simplify the notations, we may not use any parentheses in expressions like ∂_x P_y ⋆ u, ∂_y P_y ⋆ u, ∂_x Q_y ⋆ u and ∂_y Q_y ⋆ u for u ∈ L^2() and y>0.
     Let I,   J ⊊ be non empty open bounded interval. 
    Then, for y >0, R_I   P_y is a Hilbert-Schmidt operator on L^2(J,)⊂ L^2(,) (see <cit.>). 
    Indeed, for u∈ L^2(J), we have
    
        R_I   P_y[u](x) = R_I  (P_y⋆ u)(x)=1/π∫_J h/(x-t)^2+y^2u(t) dt,  x∈ I   ,
    
    and R_I  P_y is an integral operator on L^2(J,) with kernel k(x,t)=P_y(x-t), k∈ L^2(I × J), since P_y(x) is a smooth function for y>0. 
    Thus, it is compact. The same conclusion holds for the operator u ↦ R_I  (∂_x   P_y ⋆ u) (and u ↦ R_I  (∂_y   P_y⋆ u)) on L^2(J,).
     For u∈ L^2(ℝ), the functions 𝒫_y [u]=P_y⋆ u and Q_y⋆ u have non-tangential limits as y tends to zero and 
    
    
        lim_y→ 0P_y⋆ u(x)=u(x)   and   lim_y→ 0Q_y⋆ u(x)=ℋu(x),
    
     where ℋ is the Hilbert transform from L^2(ℝ) to L^2(ℝ) defined by 
    
    
        ℋu(x)=1/πlim_ε→ 0∫_| x-t|>εu(t)/x-tdt.
    
     The Hilbert transform is bounded and isometric on L^2(ℝ) and satisfies ℋ^2=-I, whence its adjoint ℋ^* = - ℋ. The Hilbert transform commutes with the Poisson operator (which can be proved using the Fourier transform): 
    
    
        (P_y⋆ u)= (𝒫_y[u])=P_y⋆ u =𝒫_y[ u ]   .
    
      For y>0, x∈ℝ and u∈ L^2(ℝ), one can prove using <cit.> that:
    
    
        Q_y⋆ u =P_y⋆ℋ u=𝒫_y[ u ]  .
    
     Hence, from (<ref>) and (<ref>), we get that Q_y =ℋ P_y on L^2(). We will also use in the sequel the following equality, for g ∈ L^2():
    
    
        ∂_y 𝒫_y[g]  = -ℋ(∂_x 𝒫_y [g] ).
     It can be obtained using the Fourier transform 
    as in <cit.>. Actually,  for g∈ L^2(), ∂_y  𝒫_y[g] =-[∂_x   Q_y ⋆ g], and ∂_x  𝒫_y[g]= ∂_y  Q_y ⋆ g. And as a consequence of (<ref>), and the isometric character of ℋ, we get ∂_y 𝒫_y[g]_L^2()= ∂_x 𝒫_y[g]_L^2().
     We also have the following uniqueness result, from <cit.>:
    
      Let h>0 and g ∈ L^2() such that ∂_x  𝒫_h[g] = 0 on a non-empty open subset 
      of . Then g ≡ 0 on .
    
    
    As a corollary to Lemma <ref>, we get that if g∈ L^2() is such that 𝒫_h [g] = P_h⋆ g=0 on a non-empty open subset of  for some h>0, then g≡ 0 on .
    
    
    
     §.§ Hardy space H^2
    
     For u∈ L^2(), the complex-valued function P_y⋆ u+iQ_y⋆ u belongs to H^2(_+)=H^2(_+, ), the space of analytic functions F on _+ such that
    
    
        ‖ F‖_H^2(ℂ_+):=sup_y>0(∫_-∞^∞| F(x+iy)|^2 dx )^1/2<∞  ,
    
     see <cit.>. A function F∈ H^2(_+) admits a non-tangential limit F^*∈ L^2(, ) as y tends to 0, almost everywhere on . 
    The function F^* belongs to the space H^2()=H^2(, ) of 
    functions f ∈ L^2() such that 
    
    
        ∫_-∞^∞f(t)/t-zdt=0,   z∈_+.
    
     The space H^2()  coincides with the space of boundary values  on  (also “traces”, if smoothness allows) of H^2(_+) functions.
     
    Indeed, the map f∈ H^2()⟼ P_y⋆ f is an isometric isomorphism from H^2() onto H^2(_+).
    
    It is not difficult to see that (I+iℋ) L^2()=H^2(). As a consequence, the map u⟼ P_y⋆(I+iℋ)u is an isomorphism from L^2() onto H^2(_+). A function f∈ H^2(_+) satisfies the boundary uniqueness Theorem <cit.>: if F∈ H^2() is such that F=0 on I⊂ with |I|>0 then F≡ 0 on .
     Note that if  I⊂, the space (I+iℋ)(L^2(I)) coincides with the space of functions F^* of H^2() such that supp  F^*⊂ I.
             
     Observe that for f∈ L^2(,),
    
    
        (I+i)   f =2 (I+i)  [  P_+(f)] =2   P_+(f),
    
     where P_+(f) is the orthogonal projection from L^2(,) onto H^2(), see <cit.>.
     Finally, an amazing property of the Hilbert transform is given by the  following Lemma (of which a proof can be found in <cit.>): 
    
    
    If I, J⊂ with |I|, |J|>0 such that I ∩ J =∅, then for any g∈ L^2() such that (χ_Jg)=0 on I, we have g=0 on J. 
    
    Lemma <ref> can be seen as a consequence of the boundary uniqueness Theorem in H^2(). 
    Indeed, if (χ_J g)=0 on I then the function in H^2() given by F=χ_J g+i(χ_J g) is equal to 0 on I (as J∩ I=∅). By the boundary uniqueness Theorem, it follows that F=0 on  and g=0 on J.
    
     When I∩ J≠∅ and I ⊈J, Lemma <ref> still holds true in some cases:
    
    
      ∙J⊊ I: indeed, the function F=χ_J g+i(χ_J g) is equal to zero on I\ J and the boundary uniqueness Theorem implies that g=0 on J, whenever |I\ J| >0;
    
    
      ∙I∩ J≠∅ and I⊄J: the same function F is equal to zero on I\ (I∩ J) and g=0 on J by the boundary uniqueness Theorem, provided that |I\ (I∩ J)|>0. 
    
     However, if I ⊆ J are bounded intervals, then there exists a function g∈ L^2(), g≠ 0, such that (χ_Jg)≠ 0. 
    With I=(a,b), the (non-zero) function defined on  by g(x)=χ_I(x)/√((x-a)(b-x)) is such that (χ_I g)=(g)=0 on I. Indeed, the function x ↦1/√((x-a)(b-x)) is shown in <cit.> to be an eigenfunction of the operator (χ_I ·)_|_I associated to the eigenvalue 0.
    
    
      ∙ If  I=J, the function g furnishes a solution to the issue.
    
    
      ∙ If I⊊ J, one can write 
    
    (χ_Jg)=(χ_Ig)+(χ_J\ I g).
    
    
    The above function g vanishes outside I hence on J\ I.
    Thus, (χ_J\ I g)=0 on . 
    Further, (χ_Ig)=0 on I, whence (χ_J g)=0 on I, 
    but g≠ 0 on J.
    
    
    
    § MAIN OPERATORS
    
     Let S and K ⊂ be two nonempty open bounded intervals.   Fix h > 0. Let ∈ L^2(S, ℝ^2). 
    
     Taking the convolution of  the PDE Δ  U = by the fundamental solution to Laplace equation in dimension n=2 and applying Green formula, we obtain, for X ∈ℝ^2 ∖supp:
    
    
        U[](X) = -1/2  π ∬_^2(_Y  ) (Y) log |X-Y|   d   Y
        = 1/2  π ∬_^2 (Y)  · _Y log |X-Y|   d   Y
    
        = 1/2  π ∬_^2 (Y)·(X-Y)/|X-Y|^2  d   Y   .
    
    Since the support of  is a subset of S ×{0}, we get, at X = (x,y)∈ℝ^2 ∖ S ×{0}, with Y = (t,0):
    
        U[](x,y) = 1/2  π ∫_Sm_1(t) (x-t) + m_2(t)y/(x-t)^2 + y^2  d   t = 1/2 (Q_y ⋆m_1+P_y ⋆m_2)(x,y)  .
     Let then  b_2[] = -(∂_y U[])_|_K×{h} (from now, we will ignore  on the multiplicative constant μ_0/2): 
    
        b_2:     L^2(S, ℝ^2)    ⟶ L^2(K)
           =(m_1,m_2)   ⟼ -∂_y (Q_y ⋆m_1+P_y ⋆m_2)_|_K×{h}  ,
    
    with the use of Cauchy-Riemann equations, which also allow to get 
    
        b_2[]   =     -(∂_x 𝒫_y [m_1] +∂_y 𝒫_y[ m_2])_|_K×{h}= -[∂_x (P_y ⋆m_1 -Q_y⋆m_2)]_|_K×{h}
        
           =   -∂_x 𝒫_y[m_1-ℋm_2]_|_K×{h}  ,
    
     where the second equality follows from (<ref>). Using the boundedness of the convolution by P_y from L^2(ℝ) to W^1,2(ℝ), the boundedness of the Hilbert transform and the equality ‖m_i‖_L^2(ℝ)=‖ m_i‖_L^2(S), for i∈{1,2} (see Section <ref>), one can prove that the operator b_2 is bounded L^2(S,ℝ^2)→ L^2(Q).  
    
    
    The adjoint operator b_2^*  to b_2 is given by 
    
    
        b_2^*:     L^2(K)    ⟶  L^2(S,ℝ^2)
           ϕ   ⟼ (∂_x   (P_y⋆ϕ)_|_S×{h},∂_x   (Q_y⋆ϕ)_|_S×{h})
    
    and is such that
    
        b_2^*[ϕ]=(((∂_x   P_y)⋆ϕ)_|_S×{h},((∂_x   Q_y)⋆ϕ)_|_S×{h}) 
        =∂_x   (P_y⋆(I,ℋ)ϕ)_|_S×{h}.
    
     Let y>0 fixed. For ∈ L^2(S, ℝ^2) and ϕ∈ L^2(K), we have that 
    
    
        ⟨ b_2[],ϕ⟩_L^2(K) = -⟨∂_x(P_y⋆m_1),ϕ⟩_L^2(K)+⟨∂_x(Q_y⋆m_2),ϕ⟩_L^2(K)
                =-∫_∫_∂_x(P_y(x-t))m_1(t)ϕ(x) dtdx+∫_∫_∂_x(Q_y(x-t))m_1(t)ϕ(x) dtdx
               =∫_(∫_∂_t(P_y(x-t))ϕ(x) dx) m_1(t)dt+∫_(∫_∂_t(Q_y(x-t))ϕ(x)dx)m_1(t)dt
               =∫_S ∂_t(∫_ P_y(t-x)ϕ(x) dx) m_1(t)dt+∫_S∂_t(∫_ Q_y(t-x) ϕ(x) dx) m_1(t)dt
                =⟨ m_1,∂_t(P_y⋆ϕ)⟩_L^2(S)+⟨ m_2,∂_t(Q_y⋆ϕ)⟩_L^2(S)
                =⟨ (m_1,m_2),(∂_t(P_y⋆ϕ),∂_t(Q_y⋆ϕ))⟩_L^2(S,^2),
     with 
    
    
    These operators have null kernel, as follows from the next results.
    
    The operator  b_2 is injective. 
    
    Let ∈Ker  b_2. 
    Then, ∂_x P_y ⋆(m_1-ℋm_2) = 0 on K×{h}, which implies that  m_1-ℋm_2 = 0 on  in view of Lemma <ref>. 
    Hence, m_1=ℋm_2 which implies that ℋm_2 = 0 outside S whence so does m_2+iℋm_2. Because it coincides with the boundary value of a function in H^2(ℂ_+), namely P_y ⋆ (m_2+iℋm_2), this implies that  m_2=0 (see <cit.>). Therefore, m_1=0 and Ker  b_2 = {0}. 
    
     In particular, contrarily to the situation in dimension 3 (and in higher dimensions) see <cit.>, there are no non-vanishing “silent” sources (the corresponding so-called “null-space” is reduced to {0}).
    
    
    The operator b_2^* is injective. 
    
     Let f ∈Ker  b_2^*. Then, we have that ∂_x(P_y⋆ (I,)f)=(0,0) on S×{h} which implies that ∂_x(P_h⋆f)=∂_x(P_h⋆f)=0 on S. By Lemma <ref>, it follows that f=0 on  and f=0 on S.
     
    
     Moreover:
     
    The operators b_2 and b_2^* have a dense range. 
    
     Since b_2^* is continuous (because so is b_2), the following orthogonal decomposition holds true:
    
        L^2(S, ^2) =  Ker  b_2 ⊕Ran  b_2^*  .
    
    Hence, using Proposition <ref>, b_2^* has dense range in L^2(S, ^2).
    Similarly, 
    
        L^2(K) =  Ker  b_2^* ⊕Ran  b_2  .
    
    Hence, using Proposition <ref>, b_2 has dense range in L^2(K). 
    
    
     Using the following equality for f ∈ L^2(, ):
    
    
        (I+i) f=(I+i) (  f -   f).
    
    with ℋ u = ℋ [- i^2 u] = - i ℋ [i u] whence  ℋ [i u] = i ℋ u, we could consider the analytic versions of b_2 and b_2^* denoted by B_2 and B_2^* respectively (see <cit.>):
    
    
        B_2:L^2(S, )   ⟶   L^2(K)
           m=m_1+i m_2   ⟼   -∂_x(P_y⋆(I+iℋ) m)_|_K×{h},
    
    
        B_2^*:L^2(K)   ⟶   L^2(S, )
          ϕ   ⟼   ∂_x(P_y⋆(I+iℋ) ϕ)_|_S×{h}.
    
     Let J be the isomorphism from L^2(S, ) onto L^2(S, ^2) defined by J(ϕ_1+iϕ_2)=(ϕ_1,ϕ_2) which is unitary when L^2(S, ) is equipped with the inner product (<ref>).
    
    
    
    We  have that B_2=b_2∘ J and b_2^*[ϕ]=J∘ B_2^*[ϕ], ϕ∈ L^2(K).
    
    
    
    
    
    
    As a consequence, B_2 and B_2^* are injective and Ran B_2=L^2(K), Ran B_2^*=L^2(S,). 
    
    
    The ranges of b_2 (respectively B_2) and b_2^* (respectively B_2^*) are not closed: Ran b_2⊊ L^2(K), Ran b_2^*⊊ L^2(S,^2). 
     If we assume that the range of B_2^* is closed, then Corollary <ref> and Remark <ref> are to the effect that B_2^* is surjective. Hence, for I⊊ S, with |I|>0, there exists ϕ∈ L^2(K) such that 
    
    
        ∂_x(P_h⋆(I+i)ϕ)_|_S=χ_I   ,
     It follows that ∂_x(P_h⋆(I+i)ϕ)=0 on S\ I whence (I+i)ϕ=0 on  by Lemma <ref>. As in the proof of Proposition <ref>, this leads to ϕ=0 which contradicts Equation (<ref>). We conclude that B_2^* is not surjective. As b_2^*=J∘ B_2^*, we get that b_2^* is not surjective.
    
     Now, if the range of B_2 is closed, then for any g∈ L^2(K), there would be ϕ∈ L^2(S,) such that 
    
    
        g=-∂_x(P_h⋆ (I+i)ϕ)_|_K×{h}.
     Let I⊊ K with |I|>0 and g=χ_I. Then, there exists ϕ∈ L^2(S,) such that 
    
    
        ∂_x(P_h⋆ (I+i)ϕ)_|_K=∂_x(P_h⋆ (I+i)ϕ)_|_K=0  on  K\ I.
     By Lemma <ref>, we get that 
    
    (I+i)ϕ=0  on,
    
    and ϕ-ϕ=0 on . This means that ϕ=ϕ on  whence ϕ=0 on \ S, which proves that ϕ=0 on  by Lemma <ref>, and thus ϕ=0 on . We conclude that ϕ=0 on  which contradicts Equation (<ref>). It follows that the range of B_2 is not closed and thus b_2=J^-1∘ B_2 is not surjective.  
    
    
    The ranges of  b_2 (and  B_2) and b_2^* (and B_2^*) are actually made of C^∞-smooth functions on K and S, respectively, see Section <ref>.
    
    The operators b_2, B_2, b_2^* and B_2^* are compact. 
     
    Because u ↦(∂_x   P_y ⋆ u)_|_S and u ↦(∂_y   P_y⋆ u)_|_S are compact operators on L^2(K,) for y>0 (see Section <ref>), so is B_2^*=(∂_x   P_h   (I+iH))_|_S,
    and thus also B_2.
     As b_2^*=J∘ B_2^* and b_2=B_2∘ J^-1, we get that b_2 and b_2^* are compact.  
    
    
    We will also make use of the operator  a_2  defined by
    
        a_2:L^2(S,ℝ^2)   ⟶    L^2(K)
        
          =(m_1,m_2)   ⟼   [P_y  ⋆(m_1-ℋm_2) ]_|_K×{h}  ,
     
    which is such that b_2[]= -∂_x (a_2[]).
    
    
    We deduce from the proof of Proposition <ref> that a_2^*[ϕ]=(P_y⋆ϕ, P_y⋆ϕ)_|_S×{h}, for ϕ∈ L^2(K), and from the one of Proposition <ref>  that a_2 and a_2^* are Hilbert-Schmidt operators.  
    Further, whenever ϕ∈ W_0^1,2(K), then b_2^*[ϕ] = -a_2^*[∂_x ϕ].
    
    
    § BOUNDED EXTREMAL PROBLEMS FOR MOMENTS ESTIMATES
    
    
     We are interested in solving the following bounded extremal problem (BEP), with ∈ L^2(S, ^2) and M≥ 0: find  ϕ_o∈L^2(K),  ‖ϕ_o‖_L^2(K)≤ M that satisfies
    
    
        ‖ b_2^*[ϕ_o]-‖_L^2(S,^2)=min_ϕ∈ L^2(K)‖ϕ‖_L^2(K)≤ M‖ b_2^*[ϕ]-‖_L^2(S,^2)BEP_mo
     Our motivation is as follows.  Let _1=(χ_S,0), _2 = (0,χ_S). A solution ϕ_o to (<ref>)  furnishes a linear estimator in L^2(K) for net moment estimates: indeed, for i∈{1,2},  
    
        |⟨ b_2[]   ,  ϕ_i⟩_L^2(K) - ⟨ m_i ⟩|=|⟨  ,   b_2^*[ϕ_i] - _i ⟩_L^2(S,^2)|
        ≤‖‖_L^2(S,^2)‖ b_2^*[ϕ_i]-_i‖_L^2(S,^2)  ,
    
     where ϕ_i=ϕ_o[_i] is the solution of (<ref>) with =_i and ⟨ m_i ⟩=∫_S m_i(t) dt.
     Below, we  consider BEP both in the Lebesgue space L^2(K) and in the Sobolev space W_0^1,2(K) in order to control the derivative and the oscillations of the solution.
    
    
    
     §.§ BEP in L^2(K)
    
    
      §.§.§ Well-posedness
    
      Let  ∈ L^2(S, ^2) and M≥ 0. 
     Problem (<ref>) 
     admits a unique solution ϕ_o ∈ L^2(K). Moreover, if
     ∉Ran b_2^*, then ϕ_o saturates the constraint: ‖ϕ_o ‖_L^2(K)= M. 
    
    Note that ‖ϕ_o ‖_L^2(K)= M  also if = b_2^*[ϕ] for some  ϕ∈ L^2(K) such that ‖ϕ‖_L^2(K)≥ M.
    
    
     
    Since Ran  b_2^* is dense in L^2(S, ^2), the assumptions of <cit.> (see also <cit.> and <cit.>) are satisfied, to the effect that if M >0, Problem 
      (<ref>) admits a solution, which is unique and saturates the constraint if ∉Ran  b_2^* (or if =b_2^*[ϕ] for ϕ∈L^2(K) such that ‖ϕ‖_L^2(K)≥ M). 
      
      Observe further that because b_2^* is injective (from Proposition <ref>), 
      uniqueness still holds when ∈Ran  b_2^*.
     
     Existence and uniqueness could also be established by projection onto the closed convex set <cit.>b_2^*[B_M],
    for B_M={ϕ∈ L^2(K)   ,  ‖ϕ‖_L^2(K)≤ M}. 
    Indeed, B_M is weakly compact as a ball in L^2(K) and 
    the weak closedness property of b_2^*[B_M] follows from the continuity of b_2^*, see <cit.>. By convexity of B_M and continuity of b_2^*, b_2^*[ϕ_n] strongly converges to b_2^*[ϕ] in L^2(S, ^2) whenever ϕ_n weakly converges to ϕ in L^2(K). Note that the compactness of b_2^* is enough to get that b_2^*[B_M] is closed. That the constraint is satisfied if ∉b_2^*[B_M] could be established by differentiation of the squared criterion.
    
    
      The density property of b_2^*[L^2(K)] in L^2(S,^2) implies that without the norm constraint, Problem (<ref>) above is ill-posed. Indeed, if ∉b_2^*[L^2(K)] the density gives the existence of a sequence (ϕ_n) of functions in L^2(K) such that ‖ b_2^*[ϕ_n]-‖_L^2(S,^2)→ 0. We claim that ‖ϕ_n ‖_L^2(K)→∞ as n →∞. This can be seen by assuming ‖ϕ_n ‖_L^2(K) to remain bounded and extracting a weakly convergent subsequence, say (ϕ_n) again, to some ϕ∈ b_2^*[B_M] (the later being weakly closed as we saw in the proof of Proposition <ref>). By convexity of b_2^*[B_M], it follows that (ϕ_n) converges strongly to ϕ in L^2(S, ^2) and this implies that = b_2^*[ϕ] ∈Ran  b_2^* which leads to a contradiction. 
    
    
      §.§.§ A constructive solution
    
     Assume that ∉Ran b_2^*. From Proposition <ref>, the solution of (BEP_mo) is a minimum on L^2(K) of the functional ϕ↦‖ b_2^*[ϕ]-‖_L^2(S,^2)^2 which satisfies  the constraint ‖ϕ‖^2_L^2(K)= M^2. Differentiating the criterion and the constraint w.r.t. ϕ, we get that such a critical point ϕ_o is given by the equation:
    
        ⟨ b_2b_2^*[ϕ_o],ψ⟩_L^2(K)-⟨ b_2[],ψ⟩_L^2(K) = -λ ⟨ϕ_o,ψ⟩_L^2(K)  ,
    
    with γ∈ℝ and for all ψ∈ L^2(K).
    It follows that: 
    
        ⟨ b_2   b_2^*[ϕ_o]+ λ ϕ_o,ψ⟩_L^2(K)=⟨ b_2(),ψ⟩_L^2(K),   for all ψ∈ L^2(K)   ,
     whence: 
    
        CPE_mo
          b_2   b^*_2[ϕ_o] +λ ϕ_o = b_2[]   .
    
     We have λ > 0. Indeed, taking ψ=ϕ_o in Equation (<ref>), we get 
    
    
        -λ ⟨ϕ_o,ϕ_o⟩_L^2(K)=-λ ‖ϕ_o‖_L^2(K)^2=⟨ b_2^*(ϕ_o)-,b_2^*(ϕ_o)⟩_L^2(S,ℝ^2).
    
     By <cit.>, for any g∈ L^2(S,^2),
    
    
        ⟨ b_2^*(ϕ_o)-,b_2^*(ϕ_o)- g⟩_L^2(S,ℝ^2)≤ 0.
    
     Taking g=0, we obtain that ⟨ b_2^*(ϕ_o)-,b_2^*(ϕ_o)⟩_L^2(S,ℝ^2)≤ 0. Thus, -λ ≤ 0. As λ=0 if and only if ∈Ran b_2^*, we conclude that -λ  < 0.
     Assume that ∉Ran b_2^*. Then there exists a unique λ > 0 such that (<ref>) holds true and ‖ϕ_o‖_L^2(K)=M. In this case,
    
        λ ‖ϕ_o‖_L^2(K)^2= λ  M^2=- ⟨ b_2^*(ϕ_o)-,b_2^*(ϕ_o)⟩_L^2(S,ℝ^2)  .
     From now on,  when there is no ambiguity, we will write f^' = ∂_x   f for the derivative of real-valued differentiable functions f defined on  or on intervals of .
     For f∈ L^2(K), at y =h >0, we have that:
    
        b_2   b_2^*[f] = [
          - P_h ⋆  (χ_S   (P_h ⋆f)^')
          + Q_h ⋆ ( χ_S   (Q_h^'⋆f))
          ]^'_|_K
    
        = [
          - P_h^'⋆  (χ_S   (P^'_h ⋆f))
          + Q_h^'⋆(χ_S   (Q^'_h ⋆f))
          ]_|_K  .
    
     The critical point equation (<ref>) can be written as:
    
        [ - P_h^'⋆  (χ_S   (P^'_h ⋆ϕ_o))
          + Q_h^'⋆(χ_S   (Q^'_h ⋆ϕ_o))
          ]_|_K+λϕ_o=-P_h'⋆(e_1-e_2)_|_K,
    
     where (e_1,e_2)=.
     Observe that, 
    if the  function η∈ L^2(K) represents the error on the measurements of b_2[], then 
    
    
        |⟨ b_2[]+η  ,  ϕ_i⟩_L^2(K) - ⟨ m_i ⟩|   =   |⟨  ,   b_2^*[ϕ_i] - _i ⟩_L^2(S,^2)+⟨η,ϕ_i⟩_L^2(K)|
           ≤   ‖ ‖_L^2(S,^2)‖ b_2^*[ϕ_i]-_i‖_L^2(S,^2) +‖η‖_L^2(K) ‖ϕ_i‖_L^2(K)
           ≤   ‖ ‖_L^2(S,^2)‖ b_2^*[ϕ_i]-_i‖_L^2(S,^2) +M ‖η‖_L^2(K).
    
     For a choice of M>0, the solution of (<ref>) guarantees that ‖ b_2^*[ϕ_i]-_i‖_L^2(S,^2) is minimal and the term ‖η‖_L^2(K) is controlled by M=‖ϕ_i‖_L^2(K). The challenge of the numerical implementations is to choose a right M>0 such that ϕ_i satisfies the two conditions mentioned above. We will observe in numerical implementations that the solution ϕ_i can oscillate at the endpoints. 
    However, since Ran  b_2 ⊂ W^1,2(K), then  (<ref>) is to the effect that ϕ_i∈ W^1,2(K) ⊂ C(K).
    
    
    
     §.§ BEP in W_0^1,2(K)
     An alternative is to formulate the BEP in a space of functions such that these oscillations are controlled.
    It thus seems natural to search for a solution in the Sobolev space W_0^1,2(K).
    
     We denote by b^*_2|_W the restriction of b_2^* to W_0^1,2(K). For ϕ∈ W_0^1,2(K),
    
    
        b_2^*[ϕ]=b^*_2|_W[ϕ]=((P_h⋆ϕ)'_|_S,(Q_h⋆ϕ)'_|_S) 
          =((P'_h⋆ϕ)_|_S,(Q'_h⋆ϕ)_|_S)
          =((P_h⋆ϕ')_|_S,(Q_h⋆ϕ')_|_S),
    
     where (ϕ)^'=ϕ^'=ϕ^'∨ 0. Notice that ϕ∈ W^1,2() only when ϕ∈ W^1,2_0(K) (see  <cit.>).  Let  ∈ L^2(S, ^2) and M≥ 0. The problem can thus be stated as the one of finding  ϕ_o∈ W_0^1,2(K),  ‖ϕ_o‖_W_0^1,2(K)≤ M such that:
    
    
        ‖  b^*_2|_W[ϕ_o] -‖_L^2(S,^2)= min_ϕ∈ W_0^1,2(K)‖ϕ‖_W_0^1,2(K)≤ M‖ b^*_2|_W[ϕ] -‖_L^2(S,^2)  .BEP_mo,W
    
    
      §.§.§ Well-posedness
    
      Let ∈ L^2(S, ^2) and M≥ 0. Problem (<ref>) 
      admits a unique solution ϕ∈ W^1,2_0(K).
      
      If
     ∉b^*_2|_W[W_0^1,2(K)], then ϕ_o saturates the constraint: ‖ϕ_o ‖_W_0^1,2(K)= M.
     
    We apply again <cit.> in order to get  existence and uniqueness of a solution ϕ_o∈ W^1,2_0(K)  to (<ref>) saturating the constraint if ∉b^*_2|_W[W_0^1,2(K)]. Yet, because b_2^* is injective (from Proposition <ref>),
      uniqueness still holds when ∈ b^*_2|_W[W_0^1,2(K)]. Note that in this case, the constraint is saturated also  if =b_2^*[ϕ] for ϕ∈L^2(K) such that ‖ϕ‖_W_0^1,2(K)≥ M.
     Let ψ∈ L^2(K) with zero mean on K. There exists a unique ϕ∈ W_0^1,2(K) such that ϕ^'=ψ
      (and ‖ϕ‖_W_0^1,2(K) = ‖ϕ^'‖_L^2(K)). 
      Then, (<ref>) is equivalent to the following problem: let  ∈ L^2(S, ^2) and M≥ 0, find  ψ_o∈L^2(K) such that ∫_K ψ_o(x) dx=0, ‖ψ_o‖_L^2(K)≤ M that satisfies
    
    
        ‖ a_2^*[ψ_o]-‖_L^2(S,^2)= min_ψ∈ L^2(K)∫_K ψ(x) dx=0
        
        ,   ‖ψ‖_L^2(K)≤ M‖ a_2^*[ψ] -‖_L^2(S,^2)  .
    
     Proposition <ref> is to the effect that a unique solution ϕ_o to (<ref>) does exist, whence also a unique solution ψ_o to (<ref>), which is such that ϕ_o^'=ψ_o.
    
     We also have the following density result.
    b_2^*[W_0^1,2(K)] is dense in L^2(S,^2).
       First, observe that W_0^1,2(K) is dense in L^2(K) since the set C_c^∞(K) of C^∞(K) functions with compact support ⊂ K is dense in L^2(K) (see <cit.>). As W^1,2(K)⊂ L^2(K), we use the continuity of b_2^* to get:
        
        b_2^*[W_0^1,2(K)] ⊂  b_2^*(L^2(K))=b_2^*[W_0^1,2(K)^L^2(K)] ⊂b_2^*[W_0^1,2(K)]^L^2(S,ℝ^2)  ,
    
        Recall that b_2^*[L^2(K)] is dense in L^2(S, ^2) (for the L^2(S, ^2) norm), 
        from Corollary <ref>. We therefore conclude to the density of b_2^*[W_0^1,2(K)] in L^2(S, ^2).
    
     Yet, Lemma <ref> 
    implies that without the norm constraint, Problem (<ref>) is ill-posed. The minimum is in this case an infimum and is equal to 0 while minimizing sequences diverge.
    
    
    
      §.§.§ A constructive solution
     Observe that b_2=(b_2^*_|_W)^*, since b_2^*_|_W=a_2^*(ϕ'), where
    ϕ' denotes the derivation from W_0^1,2(K) onto the space of L^2(K)-functions with zero mean.
    Thus, for f∈ W_0^1,2(K), we get
    
    
        b_2   b^*_2|_W[f] 
        = [
          - P_h^'⋆ (χ_S   P_h ⋆f^')
          + Q_h^'⋆ (χ_S   Q_h ⋆f^')
          ]_|_K.
    
     Assume that ∉b^*_2|_W[W_0^1,2(K)].
    From Proposition <ref>, we obtain a  critical point equation  somewhat different from (<ref>) for the solution ϕ_o in this case (see also Section <ref>): 
    
        ⟨ b_2 b^*_2|_W[ϕ_o],ψ⟩_L^2(K)-⟨ b_2[],ψ⟩_L^2(K) = γ ⟨ϕ_o',ψ'⟩_L^2(K) = -γ ⟨ϕ_o”,ψ⟩_L^2(K)  ,
     with γ<0 and ∀ψ∈ W_0^1,2(K). The solution ϕ_o∈ W_0^1,2(K) of (<ref>) is thus given by  
    
        CPE_mo, W
        b_2   b^*_2|_W[ϕ_o] + γϕ_o” = b_2[]   ,
    
    with γ <0, such that ‖ϕ_o ‖_W^1,2(K)=‖ϕ_o' ‖_L^2(K)= M if ∉Ran b^*_2|_W. The equation <ref> is considered in the weak sense in L^2(K). 
    Thus, 
    
        [
          - P_h^'⋆  (χ_S   (P_h ⋆ϕ'_o))
          + Q_h^'⋆(χ_S   (Q_h ⋆ϕ'_o))
          ]_|_K+γϕ_o”=-P_h'⋆(e_1-e_2)_|_K.
    
    
      §.§.§ An equivalent problem in H^2
     Let ℂ_+={x+iy∈ℂ: x∈ℝ,y>0} and H^2(ℂ_+) denotes the Hardy space of the upper half-plane defined in Section <ref>. As a function F∈ H^2(_+) can be identified to its boundary function F^*∈ H^2() so that F_|_S×{0} can be defined as F^*_|_S.
     Given a function g∈ L^2(S,ℂ), for ε>0, there is a function F∈ H^2(ℂ_+) such that ‖ g-F^*_|_S‖_L^2(S,ℂ)<ε. Indeed, if f∈(H^2()_|_S)^⊥, then for g∈ H^2(), 
    
        ⟨ f,g_|_S⟩_L^2(S,)=⟨f,g⟩_L^2(,)=0.
     Taking g(t)=1/t-z with z∈ such that z>0, it follows that 
    
        ∫_-∞^+∞f(t)/t-z=0,   z∈_+,
     and f∈ H^2_-(). By the boundary uniqueness Theorem, f=0 on S so H^2()_|_S is dense in L^2(S,ℂ).
     For M ≥ 0, let
    
        ℬ_M={F∈ H^2(ℂ_+):supp  F^*⊂ K   and    ‖ F^*‖_L^2(K)≤ M }⊂ H^2(ℂ_+)   .
    
    
      Let 
      g∈ L^2(S,). 
    The problem (<ref>) is equivalent to the following bounded extremal problem (with L^2(S,) norm):
    
    
        min_F∈ℬ_M‖ F(.+ih)_|_S-g‖_L^2(S,),BEP__+bis  .
    
    
     Let ∈ L^2(S,^2) be such that J(g)=. By Proposition <ref>, there exists a function ϕ_o∈ W_0^1,2(K) such that 
    
    
        ‖b_2^*_|_W[ϕ_o]-‖_L^2(S,^2)=min_ϕ∈ W_0^1,2(K)‖ϕ‖_W_0^1,2(K)≤ M‖ b^*_2|_W[ϕ] -‖_L^2(S,^2)  .
    
     Using the isomorphism J, we get for any ϕ∈ W_0^1,2(K),
    
        ‖b_2^*_|_W[ϕ] -‖_(L^2(S, ^2)=‖ J(B_2^*_|_W[ϕ])-‖_L^2(S,^2)
        = ‖B_2^*_|_W[ϕ] -J^-1‖_L^2(S,)   ,
     where B_2^*_|_W[ϕ]=(P_h⋆(I+iℋ)ψ)_|_S and ψ is the (unique) function of L^2(K) such that ϕ'=ψ. Let F_o ∈ℬ_M defined by P_y⋆ (I+iℋ)ψ_o=F_o with ϕ_o'=ψ_o. It follows that F_o(.+ih)=B_2^*_|_W[ϕ_o]=(P_h⋆(I+iℋ)ψ_o) and (  F_o^*)_|_K=ψ_o. Then, 
    
    
        ‖ F_o(.+ih)_|_S-g‖_L^2(S,)   =   ‖B_2^*_|_W[ϕ_o] -J^-1‖_L^2(S,)
           =   min_ϕ∈ W_0^1,2(K)‖ϕ‖_W^1,2(K)≤ M‖ B^*_2|_W[ϕ] -J^-1‖_L^2(S,^2)
           =   min_ϕ∈ W_0^1,2(K)‖ϕ‖_W^1,2(K)≤ M‖ (P_h⋆(I+iℋ)ϕ')_|_S -J^-1‖_L^2(S,^2)
           =   min_ψ∈ L^2(K)‖ψ‖_L^2(K)≤ M‖ (P_h⋆(I+iℋ)ψ)_|_S -J^-1‖_L^2(S,^2)
           =   min_F∈ℬ_M‖ F(.+ih)_|_S-g‖_L^2(S,),
    
     where we use that P_y⋆ (I+i)ψ∈ℬ_M for any ψ∈ L^2(K). 
    
    
    § ALGORITHMS AND NUMERICAL COMPUTATIONS
    
    
     §.§ Construction of the algorithms
    
    
      §.§.§ Bases functions
       Let S = (-s,s), K=(-q,q), s, q ∈_+. For n∈, let g_n(x)=exp  (i   n  π  x / q), for x ∈ K.
    They are eigenfunctions of the Laplacian on K:
    
        g_n”= - [n π/q]^2   g_n= - μ_n   g_n  ,   where  μ_n= [nπ/q]^2 >0   .
     Up to multiplication by the constant factor 1/(2 q)^1/2 for normalization, the family of functions ( g_n)_n,  n ∈, is the Hilbertian Fourier basis of L^2(K, ) 
    (see <cit.>).
    
     We  use it to expand and compute the solutions to the above bounded extremal problems,  (<ref>) in L^2(K) and (<ref>) in W_0^1,2(K). 
    
    Indeed, functions 
    ϕ 
    in L^2(K) can be expanded in Fourier series on the basis (g_n)_n,  n ∈: 
    
        ϕ = ∑_n ∈ℤ c_n   g_n   ,
    
    with c_n  ∈ l^2(ℤ), c_-n = c̅_n, c_0 ∈ (since ϕ is real-valued). 
     Moreover, ϕ belongs to W^1,2(K)  if, and only if, the coefficients of its expansion on (g_n)_n,  n ∈, are such that  
    n   c_n  ∈ l^2(ℤ) and c_n = - c_-n∈ i.
    
       Another family of appropriate functions in L^2(K) is made of piecewise constant functions on small intervals (say K_l, l = 1, ⋯, L) covering K=(-q,q) (so called P_0 finite elements), see also <cit.>. We use it for the computations of the solutions to the associated forward problem (Section <ref>).
                 
              
    
      §.§.§ Operators, matrices in L^2(K) and W_0^1,2(K)
                 For all ϕ∈ L^ 2(K), we use the expression
      
        b_2^*  [ϕ]= ((∂_x   P_y) ⋆ϕ, (∂_x   Q_y) ⋆ϕ)_|_S×{h}
            = ( P_y' ⋆ϕ, Q_y' ⋆ϕ)_|_S×{h}  .
     Solutions to (<ref>) in L^2(K) (actually in W^1,2(K) ⊂ L^2(K)) are provided by the critical point equation (<ref>).
    
             For moments estimation, we consider the specific functions =_1   , _2, with _1 = (χ_S,0) for ⟨ m_1 ⟩,  _2 = (0,χ_S) for ⟨ m_2 ⟩. 
        From Lemma <ref> and the relation (<ref>), it holds that
        _i ∉Ran b_2^*. Hence, the constraint in the BEP is saturated by the approximant, following Propositions <ref> and <ref>.
    
     Hence, using the Fourier basis  (g_n) in order to express solutions to (CPE_mo) as ϕ_o = ∑_n ∈ℤ c_n   g_n, we get that, ∀ k ∈:
    
        ∑_n ∈ℤ c_n  ⟨b_2^*[g_n],b_2^*[g_k]⟩_L^2(S,ℝ^2)+λ  c_k
        =⟨,b_2^*[g_k]⟩_L^2(S,^2)  ,
    
    for λ > 0 such that ‖ϕ_o‖_L^2(K)^2=∑_n ∈ℤ|c_n|^2 =M^2.
    
     For t_1   ,  t_2 ∈ K, x ∈ S, let:
      
        I(t_1,t_2,x) = P_h'(x-t_1)  P_h'(x-t_2)+Q_h'(x-t_1)   Q_h'(x-t_2)   ,
    
    and
    
        k(t_1,t_2) = ∫_-s^s
        I(t_1,t_2,x)   dx   .
     Following the expressions of the operator b_2^* in Section <ref>, we obtain:
    
        ⟨b_2^*[g_n],b_2^*[g_k]⟩_L^2(S,^2) = 1/π^2∫_-q^q∫_-q^q
           k(t_1,t_2)   e^i π/q(n t_1 + k t_2)  dt_1   dt_2   .
    
    These are the Fourier coefficients of  χ_K(t_1)  χ_K(t_2)   k(t_1,t_2).
     Observe that 
    
    
        ⟨b_2^*[ϕ],b_2^*[ψ]⟩_L^2(S,ℝ^2)   =   ⟨ P_h' ⋆ϕ,P_h' ⋆ψ⟩_L^2(S)+⟨ Q_h' ⋆ϕ,Q_h' ⋆ψ⟩_L^2(S)
           =   ∫_-s^s P_h' ⋆ϕ(x)P_h' ⋆ψ(x) dx+∫_-s^s Q_h' ⋆ϕ(x)Q_h' ⋆ψ(x) dx
    
        =∫_-s^s ∫_-q^q∫_-q^q(P_h'(x-t_1)P_h'(x-t_2)+Q'_h(x-t_1)Q'_h(x-t_2))ϕ(t_1)ψ(t_2) dt_1dt_2dx   .
    
    We have, for ϕ, ψ∈ L^2 (K),
     
        ⟨ b_2^*[ϕ],b_2^*[ψ]⟩_L^2(S, ^2)
        = 1/π^2∫_-s^s∫_-q^q∫_-q^q
         I(t_1,t_2,x)  ϕ(t_1)  ψ(t_2)   dt_1  dt_2  dx    .
    
    We can interchange the integrals, which leads to
    
        ⟨ b_2^*[ϕ],b_2^*[ψ]⟩_L^2(S, ^2) = 1/π^2∫_-q^q∫_-q^q
           k(t_1,t_2)  ϕ(t_1)  ψ(t_2)    dt_1   dt_2   .
    
    Taking ϕ =g_n and ψ = g_k leads to the expression (<ref>). 
     This allows to   compute the left-hand side of (<ref>) and of (<ref>).
      Similarly, solutions to (<ref>) in W_0^1,2(K), are furnished by (<ref>).
    Notice that b_2^*_|_W [ϕ]= b_2^*  [ϕ] if ϕ∈ W_0^1,2(K).
    Using   (g_n)  in order to express the solutions to (CPE_mo,W) as ϕ_o = ∑_n ∈ℤ d_n   g_n,  we get that, ∀ k ∈:
    
        ∑_n ∈ℤ d_n  ⟨b_2^*[g_n],b_2^*[g_k]⟩_L^2(S,^2)- γ μ_k   d_k=⟨,b_2^*[g_k]⟩_L^2(S,^2)  .
    
    for γ <0 such that ‖ϕ_o'‖_L^2(K)^2=∑_n ∈ℤμ_n  |d_n|^2 =M^2, which has to be compared  with (<ref>).
    
    
    
     Note that the above functions I(t_1,t_2,x) and k(t_1,t_2) admit explicit expressions which can be used to more precisely compute the quantities 
    ⟨ b_2^*[ϕ],b_2^*[ψ]⟩_L^2(S, ^2) in Lemma <ref>.
    
    
    
    
    
      §.§.§ Forward problems, b_2[_i], b_2[]
     Concerning the computation of the right-hand sides ⟨,b_2^*[g_k]⟩_L^2(S,^2) of the critical point equations, following the definition of the operator b_2 in Section <ref>,
      
        b_2[]= - (P'_h ⋆m_1 -Q'_h⋆m_2)_|_K  ,   ∀=(m_1,m_2) ∈ L^2(S, ℝ^2)  ,
    
    we first compute, from the above definition of _i for i = 1,2:
    
        ⟨_1,b_2^*[g_k]⟩_L^2(S, ^2) = ⟨ b_2[_1],g_k⟩_L^2(K)
    
        = ∫_-s^s∫_-q^q P_h'(x-t)   e^i k πt/q  dt   dx = ∫_-q^q(P_h(s-t) -P_h(s+t))   e^i k πt/q  dt   ,
    
    and
    
        ⟨_2,b_2^*[g_k]⟩_L^2(S, ^2) = ⟨ b_2[_2],g_k⟩_L^2(K)
    
        = ∫_-s^s∫_-q^q Q_h'(x-t)   e^i k πt/q  dt   dx = ∫_-q^q(Q_h(s-t) -Q_h(s+t))   e^i k πt/q  dt   .
       These expressions are computed by discretization on a family of piecewise constant functions on K = ∪_l K_l, summing up the contributions on K_l, although   they are    linked to the Fourier coefficients of χ_K   P_h and χ_K   Q_h. 
    
       This is all we need in order to solve the linear systems (<ref>) in c_n and (<ref>) in d_n, and to compute the solutions ϕ_i related to _i and M.
     In order to generate synthetic data, the components m_i of magnetizations ∈ L^2(S, ^2) are also modelled as  finite linear combinations of  a family of piecewise constant functions on S= ∪_l S_l, l= 1, ⋯ L.  The quantity  b_2[] is computed from its Fourier series, while its Fourier coefficients (equal to ⟨ b_2[],g_k⟩_L^2(S)) are computed as those of b_2[_i] above, summing up the contributions of the intervals S_l, l= 1, ⋯ L.
    
     We  finally compute (with a quadrature method):
    
        ⟨ b_2[]   ,  ϕ_i⟩_L^2(K)= ∑_n ∈ℤ c_n  ⟨ b_2[]   ,   g_n ⟩_L^2(K)  ,
    
    which is expected to furnish the approximation ⟨ m_i^e⟩=⟨ b_2[]   ,  ϕ_i⟩_L^2(K) to ⟨ m_i ⟩.
    For error estimation,  we will compare ⟨ m_i^e⟩ to the actual moments ⟨ m_i ⟩ (i = 1,2). 
    
    
    
    
    
    
     §.§ Numerical illustrations
     In this section, we discuss preliminary numerical results concerning the computation of  ⟨ m_i^e⟩=⟨ b_2[]   ,  ϕ_i⟩_L^2(K) in order to estimate the mean value ⟨ m_i⟩ of each component of m.
    
      Recall that the solutions to (CPE_mo) in L^2(K), ϕ_i=ϕ_o[_i], i=1,2, 
     are given by:
    
    
        [ - P_h^'⋆  (χ_S   (P^'_h ⋆ϕ_i))
          + Q_h^'⋆(χ_S   (Q^'_h ⋆ϕ_i))
          ]_|_K+λϕ_i={[ -( P_h'⋆χ_S )_|_K  i = 1   ,; ( P_h'⋆ χ_S )_|_K  i = 2   , ].
        for λ >0; we have ‖ϕ_i ‖_L^2(K)= M (λ).
       
       
      The solutions ϕ_i=ϕ_o[_i], i=1,2, to (CPE_mo,W) in W_0^1,2(K) are given by:
     
    
        [
          - P_h^'⋆  (χ_S   (P_h ⋆ϕ'_i))
          + Q_h^'⋆(χ_S   (Q_h ⋆ϕ'_i)) ]_|_K-λϕ_i”=
        {[ -( P_h'⋆χ_S )_|_K  i = 1   ,; ( P_h'⋆ χ_S )_|_K  i = 2   , ].
    
    with λ = - γ >0, and yet ‖ϕ_i ‖_W_0^1,2(K)= M (λ), in the weak sense in L^2(K). 
    
     The quantities (coefficients) ⟨b_2^*[g_n],b_2^*[g_k]⟩_L^2(S,^2) in (<ref>) and (<ref>)
    are computed with  fft (fast Fourier transform), following Lemma <ref>, while for numerical purposes, the series will be truncated at some order N∈_*.
    
     Various situations can be examined, with respect to the distance h between the magnetization support S̅ contained in 
    ×{0 } and the measurement set K ⊂×{h },  and to their length |S|, |K|.
    
    In order to match with the physics of the model, we assume that
    S= (-s,s) and K= (-q,q) are centered at the origin, and that
    |K| > |S| > h >0, with (approximately) |K| ≃ 1.5   |S|, |S| ≃ 20   h. We will take h=0.1, s=1 and q=1.5.
    The order of truncation will be N=250.
    
    
     We will compare the estimates ⟨ m_i^e⟩ of ⟨ m_i⟩ for functions ϕ_i solutions to the BEP in L^2 or in W_0^1,2, associated to particular values of λ, for some magnetizations .
    The computations are performed with Matlab, R2017a.
    
     Tables below 
     furnish the estimated moments ⟨ m_i^e⟩ for i∈{1,2}, the corresponding values of the relative errors ε_i and norm constraints M, ‖ϕ_i‖_L^2(K) or ‖ϕ_i‖_W^1,2_0(K). Figures are provided in the Appendix.
    
    
     
    Below, for the computation of the solutions ϕ_i  to the corresponding BEP, we choose λ=λ_i such that 1 ≤ -log_10(λ_i) ≤ 9 is an integer and such that the relative error for moments estimation given by
     
     
        ε_i =|⟨ m_i⟩ - ⟨ m_i^e  ⟩|/|⟨ m_i⟩|  
          ,
    
      is small enough 
     among this range for the corresponding data , while still providing an acceptable value for the quantity M(λ_i), say between 10 and 20, see Table <ref>.
     Observe that this trade off is made possible by the fact that the data are available for the present simulations. Elsewhere, in general, λ=λ_i has to be chosen in terms of the behaviour of the error ‖ b^*_2[ϕ_i] -_i‖_L^2(S,^2) (and of the associated constraint M(λ_i)), independently of the unknown magnetization .
    
      In all the numerical experiments, the net moment will be kept fixed to: ⟨ m_1⟩=-0.1 and ⟨ m_2⟩=0.1, whence: 
    
        ε_i
        = 10  |⟨ m_i  ⟩- ⟨ m_i^e  ⟩|
        = |(-1)^i-1 + 10  ⟨ m_i^e  ⟩| 
        .
    
    
      §.§.§ Solutions ϕ_i to BEP
    
        The solutions ϕ_i∈ L^2(K) for i=1,2 are plotted in Figure <ref>, <ref> (λ_i=10^-3 and 10^-5). The solutions ϕ_i∈ W^1,2_0(K) can be seen in Figures <ref>,  <ref> (λ_i=10^-8 and 10^-9). Their norms are given in Table <ref>. 
     
      The solutions in L^2(K) show quite many oscillations close to the boundary of S, less in W^1,2_0(K),  as expected. Interestingly, these functions behave as affine or constant in a large interval contained in the interior of S. We also see that as λ_i decreases, the corresponding constraint M(λ_i) grows, which also increases the oscillating phenomenon, mainly in L^2(K). 
    
    
      
    
    
    
      §.§.§ Constant magnetizations
     
    Take m_1 =-0.05 on S=[-1, 1] and m_2 = 0.05 on S (whence = 0.05  (_1-_2) and m_i = 0.05   (-1)^i  χ_S).  See Figure <ref> and Table <ref> for plots of  m_i and of b_2[], computed as explained in Section <ref>. 
    
    
    
      §.§.§ Magnetizations with large support
     
    Take m_1 =-0.1 on [-1, 0], 0 elsewhere in S, and m_2 = 0.1 on [0, 1], see Figure <ref>, together with Table <ref>.
    
    
    
      §.§.§ Other magnetization
     Here we take m_1 and m_2 as below. See Figure <ref> and Table <ref>.
    
    
        m_1=
        -0.05   on  [-0.2, 0],
        
        -0.1   on  [0, 0.2],
        
        -0.2   on  [0.2, 0.4],
        
        -0.1   on [0.4, 0.6],
        
        -0.05   on  [0.6, 0.8],
        
        0   elsewhere in S,
                    m_2=
        0.05   on [-0.8, -0.6],
        
        0.1   on  [-0.6, -0.4],
        
        0.2   on  [-0.4, -0.2],
        
        0.1   on [-0.2, 0]
        
        0.05   on  [0, 0.2],
        
        0   elsewhere in S.
    
    
      §.§.§ Magnetizations with small support
     Here we take m_1 and m_2 as described below. See Figure <ref> and Table <ref>.
    
    
        m_1=
        10   on  [-0.5, -0.49],
        
        -10   on  [0, 0.0.1],
        
        -10   on  [0.2, 0.21],
        
        0   elsewhere in S,
                    m_2=
        10   on  [-0.9, -0.89],
        
        -10   on  [-0.3,-0.29],
        
        10   on  [0.2, 0.21],
        
        0   elsewhere in S.
    
    
    
    
      §.§.§ Comments, discussion
    
    Overall, as presented in the above tables, we obtain quite accurate results for  net moment estimation. The net moments of magnetizations with large support are more precisely estimated than those of magnetizations with small support: the value ε_i of the error decreases whenever the size of the support increases, for a same value of the parameter λ_i. This phenomenon is mostly true in L^2(K) for the present examples. These examples may not always provide smaller estimation errors ε_i in W^1,2_0(K) than in L^2(K), since the behaviour of ε_i depend on the specific data (smoothness properties of , size of its support, ...) and parameters. These properties, together with the influence of the noise in the computations and in the data, will be studied  in a forthcoming work.
    
    
    
    § PERSPECTIVES, CONCLUSIONS
     In order to complete the results of Section <ref>, some properties of b_2 remain to be studied. Even if b_2 is injective, it is not coercive (strongly injective), in the sense that  b_2[]_L^2(K) can  be small even when _L^2(S, ^2) is not small: there may exist such “almost silent” source terms . 
    However, lower bounds for b_2 can be established if we restrict to truncated Fourier expansions, showing that these cannot be “almost silent”.
    Also, one can estimate the constants involved in the upper bounds of b_2 and b_2^*.
    The  spectral study of the operator b_2   b_2^* remains to be fulfilled, in view of the numerical analysis of the BEP.
     The BEP studied in Section <ref> can be stated with slightly more general constraints.
    In particular,  let f∈ L^2(K). Consider the following BEP in L^2(K):
    
    
        min_ϕ∈ L^2(K)‖ϕ-f‖_L^2(K)≤ M‖ b^*_2[ϕ] -‖_L^2(S,^2) BEP_f.
     One can prove 
    existence and uniqueness of the function ϕ_o∈ L^2(K) solution of (<ref>) using the same arguments as in the proof of Proposition <ref>. Moreover, if
     ∉Ran b_2^* (or if = b_2^*[ϕ] for some  ϕ∈ L^2(K) such that ‖ϕ - f‖_L^2(K)≥ M), then the solution ϕ_o still saturates the constraint, i.e. ‖ϕ_o-f‖_L^2(K)=M, and is given by the following implicit equation, with γ  < 0, see (<ref>):
    
    
        b_2b_2^*[ϕ_o]-γϕ_o=b_2[]+γ f   .
     Let now f∈ W_0^1,2(K). The BEP can be stated as follows:
    
        min_ϕ∈ W_0^1,2(K)‖ϕ-f‖_W_0^1,2(K)≤ M‖ b^*_2[ϕ] -‖_L^2(S,^2) BEP_f,W.
     Again, as for (<ref>), a solution ϕ_o∈ W_0^1,2(K) exists, is unique, and saturates the constraint if
    ∉b^*_2|_W[W_0^1,2(K)] (or if =b_2|_W^*[ϕ] for ϕ∈L^2(K) such that ‖ϕ - f‖_W_0^1,2(K)≥ M): ‖ϕ_o-f ‖_W_0^1,2(K)= M. It satisfies the following equation, with γ < 0:
    
    
        b_2b_2^*[ϕ_o]+γϕ”_o=b_2[]+γ f   .
     The above problem will be considered for ϕ_o∈ W^1,2(K) with norm constraint therein, where it is also well-posed and gives promising numerical results. Moreover, W^1,2(K) functions are continuous on K in the present 1-dimensional setting, which is a suitable property of the solution, in view of computing its inner product with the available measurements in a pointwise way. Note however that the continuity of the solution is already granted by the critical point equation (<ref>) above and the smoothness properties of functions in Ran  b_2. 
    
     Some BEP of mixed type, with  constraints in other norms, like uniform, L^1, BMO, could also be formulated, as well as related extremal problems consist in looking for the best bounded extension of a given data b_d ≃ b_2[] from K to .
      Following Section <ref>, further numerical analysis and improvements of the computational schemes will be considered.
    In particular, we expect that refined implementations of the computations in the Fourier domain will provide more efficient and accurate recovery schemes, in the present setting as well as in the 3D one, see <cit.>. Both a quantitative and a qualitative study of the relations λ↦ M(λ) and λ↦ϵ_i(λ) (for the different error terms) remain to be done. 
    This will be the topic of a further work, 
    together with a study concerning the influence of the parameters s, q, h, and of the characteristics of m_i, on the behaviour of the solutions. 
     Both explicit and asymptotic expressions, in terms of the size q of the measurement set K, relating the first moments of  to those of b_2[], could also be derived, as in <cit.> for the 3D situation.  Besides, the quantities b^*_2[x^k] for monomials x^k could be exactly computed.  Other functions  may be considered as well, both for the present net moment estimation problem or for higher order or local moments recovery. One can also take the components e_i to be appropriate band-limited basis functions, of 
    Slepian type  <cit.>.
    
    
     Particularly interesting magnetizations are the unidirectional ones, of which the moment recovery using the above linear estimator process still requires a specific study. Also, more general situations where   belongs to L^1(S, ^2) or is a measure remain to be considered, see <cit.>, together with magnetizations supported on a two-dimensional set.
      Finally, the full inversion problem of recovering  in L^2(S, ^2) itself from b_2[] will be considered in a further work. It can also be stated as a BEP of  which the solution will involve the operator b_2   b_2^*. Again more general magnetization distributions will be studied.
    
    
    
    § ACKNOWLEDGMENTS
     The authors warmly thank Jean-Paul Marmorat for his help in numerical implementations. 
    
    
    
    
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    § APPENDIX: NUMERICAL ILLUSTRATIONS
