Open access peer-reviewed chapter

A Survey on Hilbert Spaces and Reproducing Kernels

By Baver Okutmuştur

Submitted: August 1st 2019Reviewed: February 3rd 2020Published: February 28th 2020

DOI: 10.5772/intechopen.91479

Downloaded: 100

Abstract

The main purpose of this chapter is to provide a brief review of Hilbert space with its fundamental features and introduce reproducing kernels of the corresponding spaces. We separate our analysis into two parts. In the first part, the basic facts on the inner product spaces including the notion of norms, pre-Hilbert spaces, and finally Hilbert spaces are presented. The second part is devoted to the reproducing kernels and the related Hilbert spaces which is called the reproducing kernel Hilbert spaces (RKHS) in the complex plane. The operations on reproducing kernels with some important theorems on the Bergman kernel for different domains are analyzed in this part.

Keywords

  • Hilbert spaces
  • norm spaces
  • reproducing kernels
  • reproducing kernel Hilbert spaces (RKHS)
  • operations on reproducing kernels
  • sesqui-analytic kernels
  • analytic functions
  • Bergman kernel

1. Framework

This chapter consists of introductory concept on the Hilbert space theory and reproducing kernels. We start by presenting basic definitions, propositions, and theorems from functional analysis related to Hilbert spaces. The notion of linear space, norm, inner product, and pre-Hilbert spaces are in the first part. The second part is devoted to the fundamental properties of the reproducing kernels and the related Hilbert spaces. The operations with reproducing kernels, inclusion property, Bergman kernel, and further properties with examples of the reproducing kernels are analyzed in the latter section.

2. Introduction to Hilbert spaces

We start by the definition of a vector space and related topics. Let Cbe the complex field. The following preliminaries can be considered as fundamental concepts of the Hilbert spaces.

2.1 Vector spaces and inner product spaces

Vector space. A vector space is a linear space that is closed under vector addition and scalar multiplication. More precisely, if we denote our linear space by Hover the field C, then it follows that

  1. if x,y,zH,then

    x+y=y+xH,x+y+z=x+y+zH;

  • if kis scalar, then kxH.

  • Inner product. Let Hbe a linear space over the complex field C. An inner product on His a two variable function

    :H×HC,satisfying

    1. fg=gf¯forf,gH.

    2. αf+βgh=αfh+βghand fαg+βh=α¯fg+β¯fhforα,βCandf,g,hH.

    3. ff0forfHandff=0f=0.

    Pre-Hilbert space. A pre-Hilbert spaceHis a linear space over the complex field Cwith an inner product defined on it.

    Norm space or inner product space. A norm on an inner product space Hdenoted by is defined by

    f=ff1/2orfH=ffH1/2

    where fHand =Hdenote the inner product on H. The corresponding space is called as the inner product space or the norm space.

    Properties of norm. For all f,gH, and λC, we have

    • f0. (Observe that the equality occurs only if f=0).

    • λf=λf.

    Schwarz inequality. For all f,gH, it follows that

    fgfg.E1

    In case if fand gare linearly dependent, then the inequality becomes equality.

    Triangle inequality. For all f,gH, it follows that

    f+gf+g.E2

    In case if fand gare linearly dependent, then the inequality becomes equality.

    Polarization identity. For all f,gH, it follows that

    fg=14f+g2fg2+if+ig2fig2forf,gH.E3

    Parallelogram identity. For all f,gH, it follows that

    f+g2+fg2=2f2+2g2.E4

    Metric. A metric on a set Xis a function d:X×XRsatisfying the properties.

    • dxy0and dxy=0only if x=y;

    • dxy=dyx;

    • dxydxz+dzy;

    for all x,y,zX. Moreover the space Xdis the associated metric space. If we rearrange the metric with its properties for the inner product space H, then it follows that for all f,g,hHand for all λC,where dsatisfies all requirements to be a metric, we have

    • dfg0and equality occurs only if f=g.

    • dfg=dgf.

    • dfgdfh+dhg.

    • dfhgh=dfg.

    • dλfλg=λdfg.

    Note. The binary function dgiven in the metric definition above represents the metric topology in Hwhich is called strong topology or norm topology. As a result, a sequence fn0in the pre-Hilbert space Hconverges strongly to fif the condition

    fnf0whenevern

    is satisfied.

    2.2 Introduction to linear operators

    Linear operator. A map Lfrom a linear space to another linear space is called linear operator if

    Lαf+βg=αLf+βLg

    is satisfied for all α,βCand for all f,gH.

    Continuous operator. An operator Lis said to be continuous if it is continuous at each point of its domain. Notice that the domain and range spaces must be convenient for appropriate topologies.

    Lipschitz constant of a linear operator. If Lis a linear operator from Hto Gwhere Hand Gare pre-Hilbert spaces, then the Lipschitz constant for Lis its norm Land it is defined by

    L=supLfG/fH:0fH.E5

    Theorem 1. Let Lbe a linear operator from the pre-Hilbert spaces Hto G.Then the followings are mutually equivalent:

    1. Lis continuous.

    2. Lis bounded, that is,

      supLfG:fHk<

    for 0k<.

  • Lis Lipschitz continuous, that is,

    LfLgGλfgH,

    where 0λ<and f,gH.

  • Some properties of linear operators. Let BHGbe the collection of all continuous linear operators from the pre-Hilbert spaces Hto G. Then

    • BHGis a linear space with respect to the natural addition and scalar multiplication satisfying

    αL+βMf=αLf+βMf,

    where Land Mare linear operators, fHand α,βC.

    • Whenever H=G, then BHGis denoted by BH.

    • If Kis another pre-Hilbert space, LBHGand KBGK. Then the product

    KLf=KLfforfHBHK.

    In addition,

    1. KξL+ζM=ξKL+ζKM

    2. ξL=ξL

    3. L+ML+Mand

    4. KLKL.

    are also satisfied.

    2.3 Hilbert spaces and linear operators

    Linear form (or linear functional). A linear operator from the pre-Hilbert space Hto the scalar field Cis called a linear form (or linear functional).

    Hilbert spaces. A pre-Hilbert space His said to be a Hilbert space if it is complete in metric. In other words if fnis a Cauchy sequence in H, that is, if

    fnfm0whenevern,m,

    then there is fHsuch that

    fnf0whenevern.

    Note. Every subspace of a pre-Hilbert space is also a pre-Hilbert space with respect to the induced inner product. However, the reverse is not always true. For a subspace of a Hilbert space to be also a Hilbert space, it must be closed.

    Completion. The canonical method for which a pre-Hilbert space His embedded as a dense subspace of a Hilbert space H˜so that

    fgH˜=fgHforf,gH

    is called completion.

    Note. If Lis a continuous linear operator from a dense subspace Mof a Hilbert space Hto a Hilbert space G, then it can be extended uniquely to a continuous linear operator from Hto Gwith preserving norm.

    Theorem 2. Let Mand Nbe dense subspaces of the Hilbert spaces Hand G, respectively. For fH,gMand 0λ<, if a linear operator Lfrom Mto Gsatisfies

    LfgGλfHgG,E6

    then Lis uniquely extended to a continuous linear operator from Mto Gwith norm λwhere the norm coincides with the minimum of such λ.

    Theorem 3. Let Ωμdenotes a measure space so that Ωis the union of subsets of finite positive measure and L2Ωμconsists of all measurable functions fωon Ωsuch that

    Ωfω2ω<.E7

    Then L2Ωμis a Hilbert space with respect to the inner product

    fgΩfωgω¯ω.E8

    Theorem 4 (F. Riesz). For each continuous linear functional φon a Hilbert space H, there exists uniquely gHsuch that

    φf=fgforfH.E9

    Theorem 5. Let Mbe a closed subspace of a Hilbert space H.Then the algebraic direct sum relation

    H=MM

    is satisfied. In other words, fHcan be uniquely written by

    f=fM+fMwithfMM,fMM.E10

    In addition, fMcoincides with the distance from fto M

    fM=minfg:gM.E11

    Remark. In a Hilbert space, the closed linear span of any subset Aof a Hilbert space Hcoincides with A.

    Total subset of a Hilbert space. A subset Aof a Hilbert space His called total in Hif 0is the only element that is orthogonal to all elements of A. In other words,

    A=0.

    As a result, Ais total if and only if every element of Hcan be approximated by linear combinations of elements of A.

    Orthogonal projection. If Mis a closed subspace of H, the map ffMgives a linear operator from Hto Mwith norm 1. We call this operator as the orthogonal projection to Mand denote it by PM.

    Note. If Iis the identity operator on H, then IPMdenotes the orthogonal projection to M, and the relation

    f2=PMf2+IPMf2E12

    is satisfied for all fH.

    Weak topology. The weakest topology that makes continuous all linear functionals of the form ffgis called the weak topology of a Hilbert space H.

    Note. If fH, then with respect to the weak topology, a fundamental system of neighborhoods of f is composed of subsets of the form

    UfAϵ=h:fghg<ϵforgA,

    where Ais a finite subset of Hand ϵ>0. Then a directed net fλconverges weakly to fif and only if

    fλgλfgforallgH.

    Operator weak topology. The weakest topology that makes continuous all linear functionals of the form

    LLfgforfH,gG

    is called the operator weak topology in the space BHGof continuous linear operators from Hto G. In addition, a directed net Lλconverges weakly to Lif

    LλfgλLfg.

    Operator strong topology. The weakest topology that makes continuous all linear operators of the form

    LLfforfH

    is called the operator strong topology. Moreover a directed net Lλconverges strongly to Lif

    LλfLfλ0forallfH.

    Theorem 6. Let Hand Gbe Hilbert spaces and BHGbe a continuous linear operator from Hto G. Then

    • the closed unit ball Uf:f1of His weakly compact;

    • the closed unit ball L:L1of BHGis weakly compact.

    Theorem 7. Let Hbe a Hilbert space and AH.Then if Ais weakly bounded in the sense

    supfAfg<forgH,E13

    then it is strongly bounded, that is, supfAf<.

    Theorem 8. If Hand Gare Hilbert spaces and Lis a linear operator from Hto G, then the strong continuity and weak continuity for Lare equivalent.

    Theorem 9. Let Hand Gbe Hilbert spaces. Then the following statements for LBHGare mutually equivalent:

    1. Lis weakly bounded; that is, for fH,gG, we have

      supLLLfg<

    2. Lis strongly bounded; that is, for fH,we have

      supLLLf<.

    3. Lis norm bounded (or uniformly bounded); that is,

      supLLL<.

    Theorem 10. A linear operator Lfrom the Hilbert spaces Hto Gis said to be closed if its graph

    GLfLf:fHE14

    is a closed subspace of the direct sum space HG, that is, whenever n,

    fnf0inHandLfng0inGg=Lf.

    Theorem 11. If Lis a closed linear operator with a domain of a Hilbert space Hto another Hilbert space G, then it is continuous.

    Sesqui-linear form. A function Φ:H×GCis a sesqui-linear form (or sesqui-linear function) if for f,hH,g,kGand α,βC,

    iΦαf+βhg=αΦfg+βΦhgE15
    iiΦfαg+βk=α¯Φfg+β¯ΦfkE16

    are satisfied where Hand Gare Hilbert spaces.

    Remark. If LBHG, then the sesqui-linear form Φdefined by

    Φfg=LfgGE17

    is bounded in the sense that

    ΦfgλfHgGforfH,gG,E18
    whereλL.

    Remark. If a sesqui-linear form Φsatisfies the condition (18), then for fH,the linear functional

    gΦfg¯

    is continuous on G.If we apply the Riesz theorem, then there exists uniquely fGsatisfying

    fGλfHandΦfg=fgGforgG.

    Hence ffbecomes linear, and as a result we obtain

    Φfg=fgG=LfgG.

    Adjoint operator. If LBHG,then the unique operator LBGHsatisfying

    Φfg=fLgHforfH,gGE19

    is called the adjoint of L.

    Remark. By the definitions of Land L,it follows that

    LfgG=fLgHforfH,gG.E20

    Isometric property. The adjoint operation is isometric if

    L=Lissatisfied.E21

    Remark. Let H,G, and Kbe Hilbert spaces and KBGKand LBHGbe given. Then

    KLBHKandKL=LKE22
    KerL=RanLandKerL=ClosRanLE23

    where KerLis the kernel of Land RanLis the range of L.

    Theorem 12. If L,MBHG,then the following statements are mutually equivalent.

    1. RanMRanL.

    2. There exists KBHsuch that M=LK.

    3. There exists 0λ<such that

    MgλLgforgG.

    Quadric form. Let Hbe a Hilbert space. A function

    φ:HC

    is a quadratic form if for all fHand ζC,

    φζf=ζ2φfE24

    and

    φf+g+φfg=2φf+φgE25

    are satisfied.

    Note. If LBH,the quadratic form φon His defined by

    φf=LffforfH,E26

    and it is bounded

    φfλf2forfH,E27

    where λL.

    Remark. The sesqui-linear form Φassociated with Lcan be recovered from the quadratic form φby the equation

    Φfg=14φf+gφfg+φf+igφfigE28

    for all f,gH.

    Self-adjoint operator. A continuous linear operator Lon a Hilbert space His said to be self-adjoint if L=L.

    Remark.Lis self-adjoint if and only if the associated sesqui-linear form Φis Hermitian.

    Remark. If Lis self-adjoint, then the norm of Lcoincides with the minimum of λgiven in (27) for the related quadratic form

    Theorem 13. If Lis a continuous self-adjoint operator, then

    L=supLff:f1.E29

    Positive definite operator. A self-adjoint operator LBHis said to be positive (or positive definite) if

    Lff0forallfH.

    If Lff=0only when f=0,then Lis said to be strictly positive (or, strictly positive definite).

    Note. For any positive operator LBH, the Schwarz inequality holds in the following sense

    Lfg2LffLgg.E30

    Theorem 14. Let Land Mbe continuous positive operators on Hand G, respectively. Then a continuous linear operator Kfrom Hto Gsatisfies the inequality

    KfgG2LffHMggGforfH,GE31

    if and only if the continuous linear operator

    LKKM

    on the direct sum Hilbert space HGwith

    fgLf+KgKf+Mg

    is positive definite.

    Theorem 15. Let Lbe a continuous positive definite operator. Then there exists a unique positive definite operator called the square root of L,denoted by L1/2,such that L1/22=L.

    Modulus operator. The square root of the positive definite operator LLis called the modulus (operator) of Lif Lis a continuous linear operator.

    Isometry. A linear operator Ubetween Hilbert spaces Hand Gis called isometric or an isometry if

    UfG=fHforfHE32

    is satisfied, that is, it preserves the norm.

    Note.Eq. (32) implies that a continuous linear operator Uis isometric if and only if UU=IH;in other words,

    UfUgG=fgHforf,gH,E33

    that is, Upreserves the inner product.

    Unitary operator. A surjective isometry linear operator U:HHis called a unitary (operator).

    Note. Observe that if UBHis a unitary operator, then U=U1.

    Partial isometry. A continuous linear operator Ubetween Hilbert spaces Hand Gis called a partial isometry if

    fKerU=RanUUf=f.

    The spaces KerUand RanUare called the initial space of Uand the final space of U, respectively.

    Note. If Uis a partial isometry, then its adjoint Uis also a partial isometry.

    Theorem 17. Every continuous linear operator Lon Hadmits a unique decomposition

    L=UL˜,E34

    where L˜is a positive definite operator and Uis a partial isometry with initial space the closure of RanL˜.

    3. Reproducing kernels and RKHS

    We continue our analysis on the abstract theory of reproducing kernels.

    3.1 Definition and fundamental properties

    Reproducing kernels. Let Hbe a Hilbert space of functions on a nonempty set Xwith the inner product fgand norm f=ff1/2for fandgH. Then the complex valued function Kyxof yand xin Xis called a reproducing kernel ofHif

    1. For all xX, it follows that Kx=KxH,

    2. For all xXand all fH,

    fx=fKx,E35

    are satisfied.

    Note. Let Kbe a reproducing kernel. Applying (35) to the function Kxat y,we get

    Kxy=Kyx=KyKx,forx,yX.E36

    Then, for any xX, we obtain

    Kx=KxKx1/2=Kxx1/2.E37

    Note. Observe that the subset KxxXis total in H, that is, its closed linear span coincides with H. This follows from the fact that, if fHand fKxfor all xX, then

    fx=fKx=0forallxX,

    and hence fis the 0element in H. As a result, 0=H.

    RKHS. A Hilbert space Hof functions on a set Xis called a RKHS if there exists a reproducing kernel Kof H.

    Theorem 18. If a Hilbert space Hof functions on a set Xadmits a reproducing kernel K, then this reproducing kernel Kis unique.

    Theorem 19. There exists a reproducing kernel Kfor Hfor a Hilbert space Hof functions on X, if and only if for all xX, the linear functional Hffxof evaluation at xis bounded on H.

    Hermitian and positive definite kernel. Let Xbe an arbitrary set and Kbe a kernel on X, that is, K:X×XC. The kernel Kis called Hermitian if for any finite set of points y1ynX, we have

    i,j=1nϵ¯jϵiKyjyiR.

    It is called positive definite, if for any complex numbers ϵ1,,ϵn, we have

    i,j=1nϵ¯jϵiKyjyi0.

    Note. From the previous inequality, it follows that for any finitely supported family of complex numbers ϵxxX, we have

    x,yXϵ¯yϵxKyx0.E38

    Theorem 20. The reproducing kernel Kof a reproducing kernel Hilbert space His a positive definite matrix in the sense of E.H. Moore.

    Properties of RKHS. Given a reproducing kernel Hilbert space Hand its kernel Kyxon X, then for all x,yX, we have

    1. Kyy0.

    2. Kyx=Kxy¯.

    3. Kyx2KyyKxx(Schwarz inequality).

    4. Let x0X. Then the following statements are equivalent:

      1. Kx0x0=0.

      2. Kyx0=0for all yX.

      3. fx0=0for all fH.

      Theorem 21. For any positive definite kernel Kon X, there exists a unique Hilbert space HKof functions on Xwith reproducing kernel K.

      Theorem 22. Every sequence of functions fnn1that converges strongly to a function fin HKXconverges also in the pointwise sense, i.e., for any point xX,

    limnfnx=fx.

    In addition, this convergence is uniform on every subset of Xon which xKxxis bounded.

    Theorem 23. A complex valued function gon Xbelongs to the reproducing kernel Hilbert space HKXif and only if there exists 0λ<such that,

    gygx¯λ2KyxonX.E39

    gcoincides with the minimum of all such λ.

    Theorem 24. If K1yxand K2yxare two positive definite kernels on X, then the following statements are mutually equivalent:

    1. HK1XHK2X.

    2. There exists 0λ<such that

    K1yxλ2K2yx.

    Note. For any map φfrom a set Xto a Hilbert space H, with the notation xφx, a kernel Kcan be defined by

    Kyx=φxφyforx,yX.E40

    Theorem 25. Let φ:XHbe an arbitrary map and for x,yXlet Kbe defined as

    Kyx=φxφy.

    Then Kis a positive definite kernel.

    Theorem 26. Let Tbe the linear operator from Hto the space of functions on X, defined by

    Tfx=fφxforxX,fH.

    Then RanTcoincides with HKXand

    TfK=PMfforfH,

    where Mis the orthogonal complement of KerT,PMis the orthogonal projection onto M, and Kdenotes the norm in HKX.

    Kolmogorov decomposition. Let Kyxbe a positive definite kernel on an abstract set X. Then there exists a Hilbert space Hand a function φ:XHsuch that

    Kyx=φxφyforx,yX.

    3.2 Operations with RKHSs

    Theorem 27. Let K0be the restriction of the positive definite kernel Kto a nonempty subset X0of Xand let HK0Xand HKXbe the RKHS corresponding to K0and K, respectively. Then

    HK0X0={fX0:fHKX}E41

    and

    hK0=minfK:fX0=hforallhHK0X0.E42

    Remark. If K1yxand K2yxare two positive definite kernels, then

    Kyx=K1yx+K2yx

    is also a positive definite kernel.

    Remark. Let HK1,HK2, and HKbe RKHSs with reproducing kernels K1yx, K2yx, and Kyx, respectively, and let K=K1+K2.Then

    HKX=HK1X+HK2X,

    and for fHK1Xand gHK2X,it follows that

    f+gK2=minf+hK12+ghK22:hHK1XHK2X.E43

    Theorem 28. The intersection HK1XHK2Xof Hilbert spaces HK1Xand HK2Xis again a Hilbert space of functions on Xwith respect to the norm

    f2fK12+fK22.

    In addition the intersection Hilbert space is a RKHS.

    Theorem 29. The reproducing kernel of the space

    HKX=HK1XHK2X

    is determined, as a quadratic form, by

    x,yεy¯εxKyx=inf{x,yηy¯ηxK1yx+x,yζy¯ζxK2yx:εx=ηx+ζx},

    where ϵx,ηx,ζxare an arbitrary complex valued function on Xwith finite support.

    Theorem 30. The tensor product Hilbert space

    HK1XHK2X

    is a RKHS on X×X.

    Theorem 31. The RKHS HKXof the kernel Kyx=K1yxK2yxconsists of all functions fon Xfor which there are sequences gnn0of functions in HK1Xand hnn0of functions in HK2Xso that

    1gnK12hnK22<,1gnxhnx=fx,xX,E44

    and the norm is given by

    fK2=min1gnK12hnK22,

    where the minimum is taken over the set of all sequences gnn0and hn0satisfying (44).

    3.3 Examples of RKHS. Bergman and Hardy spaces

    Bergman space. The space of all analytic functions fon Ωfor which

    Ωfz2dxdy<,z=x+iy

    is satisfied is called the Bergman space on Ωand denoted by A2Ω.

    Remark.A2Ωis a RKHS with respect to the inner product

    fgfgΩΩfzgz¯dxdy,

    and its kernel is called the Bergman kernel on Ωand denoted by BΩwz.

    Bergman kernel for the unit disc. The Bergman kernel for the open unit disc Dis given by

    BDwz=1π11wz¯2forw,zD.E45

    Bergman kernel of a simply connected domain. The Bergman kernel of a simply connected domain ΩCis given by

    BΩwz=1πφwφz¯1φwφz¯2forw,zΩ,E46

    where φis any conformal mapping function from Ωonto D.

    Theorem 32. A conformal mapping from Ωto Dcan be recovered from the Bergman kernel of Ω.

    Jordan curve. A Jordan curve is a continuous 11image of ξ=1in C.

    Green function. A Green functionGwzof Ωis a function harmonic in Ωexcept at z,where it has logarithmic singularity, and continuous in the closure Ω¯,with boundary values Gwz=0for all w∂Ω,where Ωis a finitely connected domain of the complex plane.

    Theorem 33. Let Ωbe a finitely connected domain bounded by analytic Jordan curves, and let Gwzbe the Green’s function of Ω.Then the Bergman kernel function is

    BΩwz=2π2Gwz¯wz,wz.E47

    Hardy space. The closed linear span of φn:n=01in L2(Tis called the (Hilbert type) Hardy space on Tand is denoted by H2T.Here φnξ=ξn.

    Remark.fL2Tbelongs to the Hardy space H2Tif and only if it is orthonormal to all φn(n<0), that is, all Fourier coefficients of fwith negative indices vanish. Then we have

    fgL2=n=0anb¯nforf,gH2T,E48

    where

    an=fφnL2andbn=gφnL2n=01.

    Szegö kernel. The kernel Sξz11ξz¯forξT,zD, or its analytic extension S˜wz11wz¯forw,zDis called the Szegö kernel.

    Notes

    This chapter intends to offer a sample survey for the fundamental concepts of Hilbert spaces and provide an introductory theory of reproducing kernels. We present the basic properties with important theorems and sometimes with punctual notes and remarks to support the subject. However, due to the limit of content and pages, we skipped the proofs of the theorems. The proofs of the first part can be found in [1, 2] and in most of the basic functional analysis books. Besides, the proofs of the second part (related with the reproducing kernels) can easily be found in [3]. The Hilbert space and functional analysis parts of this chapter are based on the books by J.B. Conway [1] and R.G. Douglas [2]. On the other hand, the reproducing kernel part is based on the lecture notes of T. Ando [4] and N. Aronszajn [5], the book of S. Saitoh and Y. Sawano [6], and the book of B. Okutmustur and A. Gheondea [3]. Moreover, the details of Bergman and Hardy spaces are widely explained in the books [7, 8, 9].

    © 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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    Baver Okutmuştur (February 28th 2020). A Survey on Hilbert Spaces and Reproducing Kernels, Functional Calculus, Kamal Shah and Baver Okutmuştur, IntechOpen, DOI: 10.5772/intechopen.91479. Available from:

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