DOCSLIB.ORG

Interpolation Between Hilbert Spaces 3

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Interpolation Between Hilbert Spaces 3 INTERPOLATION BETWEEN HILBERT SPACES YACIN AMEUR Abstract. This note comprises a synthesis of certain results in the theory of exact interpolation between Hilbert spaces. In particular, we examine various characterizations of interpolation spaces and their relations to a number of results in operator-theory and in function-theory. 1. Interpolation theoretic notions 1.1. Interpolation norms. When X, Y are normed spaces, we use the symbol (X; Y ) to denote the totality of bounded linear maps T : X Y with the operator normL → T = sup T x ; x 1 . k k L(X;Y ) {k kY k kX ≤ } When X = Y we simply write (X). L Consider a pair of Hilbert spaces = ( 0, 1) which is regular in the sense that is dense in as well as in H . WeH assumeH that the pair is compatible, i.e., H0 ∩H1 H0 H1 both i are embedded in some common Hausdorff topological vector space . WeH define the K-functional (1) for the couple by M H 2 2 K (t, x)= K t, x ; = inf x0 0 + t x1 1 , t> 0, x . H x=x0+x1{k k k k } ∈M The sum of the spaces 0 and 1 is defined to be the space consisting of all x H 2 H ∈M such that the quantity x Σ := K (1, x) is finite; we denote this space by the symbols k k Σ=Σ( )= + . H H0 H1 We shall soon see that Σ is a Hilbert space (see Lemma 1.1). The intersection ∆ = ∆( )= H H0 ∩ H1 arXiv:1401.6090v5 [math.FA] 29 Jun 2019 is a Hilbert space under the norm x 2 := x 2 + x 2. k k∆ k k0 k k1 A map T : Σ( ) Σ( ) is called a couple map from to if the restriction of H → K H K T to maps boundedly into for i =0, 1. We use the notations T ( ; ) Hi Hi Ki ∈ L H K or T : to denote that T is a couple map. It is easy to check that ( ; ) is a BanachH → K space, when equipped with the norm L H K (1.1) T := max T . k k L( H ; K ) j=0,1{k k L( Hj ;Kj ) } If T 1 we speak of a contraction from to . k k L( H ; K ) ≤ H K A Banach space X such that ∆ X Σ (continuous inclusions) is called ⊂ ⊂ intermediate with respect to the pair . H 1 More precisely, this is the quadratic version of the classical Peetre K-functional. 1 2 YACIN AMEUR Let X, Y be intermediate spaces with respect to couples , , respectively. We say that X, Y are (relative) interpolation spaces if there is aH constantK C such that T : implies that T : X Y and H → K → (1.2) T C T . k k L(X;Y ) ≤ k k L(H;K) In the case when C = 1 we speak about exact interpolation. When = and X = Y we simply say that X is an (exact) interpolation space with respectH K to . H Let H be a suitable function of two positive variables and X, Y spaces interme- diate to the couples , , respectively. We say that the spaces X, Y are of type H (relative to , ) ifH forK any positive numbers M , M we have H K 0 1 (1.3) T Mi, i =0, 1 implies T H(M0,M1). k k L(Hi;Ki) ≤ k k L(X;Y ) ≤ The case H(x, y) = max x, y corresponds to exact interpolation, while H(x, y)= x 1−θy θ corresponds to the{ convexity} estimate (1.4) T T 1−θ T θ . k k L(X;Y ) ≤k k L(H0;K0) k k L(H1;K1) In the situation of (1.4), one says that the interpolation spaces X, Y are of exponent θ with respect to the pairs , . H K 1.2. K-spaces. Given a regular Hilbert couple and a positive Radon measure ̺ on the compactified half-line [0, ] we define anH intermediate quadratic norm by ∞ (1.5) x 2 = x 2 = 1+ t−1 K t, x; d̺(t). k k∗ k k̺ H Z[0,∞] Here the integrand k(t) = 1+ t−1 K(t, x) is defined at the points 0 and by ∞ k(0) = x 2 and k( ) = x 2; we shall write or for the Hilbert space 1 0 ∗ ̺ definedk byk the norm (1.5).∞ k k H H Let T ; and suppose that T Mi; then ∈ L H K k k L(Hi;Ki) ≤ (1.6) K t,Tx ; M 2 K M 2t/M 2, x; , x Σ. K ≤ 0 1 0 H ∈ In particular, Mi 1 for i= 0, 1 implies T x x for all x ̺. It ≤ k k K̺ ≤ k k H̺ ∈ H follows that the spaces , are exact interpolation spaces with respect to , . H̺ K̺ H K Geometric interpolation. When the measure ̺ is given by t−θ π d̺(t)= c dt, c = , 0 <θ< 1, θ 1+ t θ sin θπ we denote the norm (1.5) by ∞ dt (1.7) x 2 := c t−θK (t, x) . k kθ θ t Z0 The corresponding space is easily seen to be of exponent θ with respect to . Hθ H In §3.1, we will recognize θ as the geometric interpolation space which has been studied independently by severalH authors, see [27, 40, 25]. INTERPOLATION BETWEEN HILBERT SPACES 3 2 1.3. Pick functions. Let be a regular Hilbert couple. The squared norm x 1 is a densely defined quadraticH form in , which we represent as k k H0 x 2 = Ax , x = A 1/2x 2 k k1 h i0 k k0 where A is a densely defined, positive, injective (perhaps unbounded) operator in . The domain of the positive square-root A1/2 is ∆. H0 Lemma 1.1. We have in terms of the functional calculus in H0 tA (1.8) K (t, x )= x , x , t> 0. 1+ tA 0 tA In the formula (1.8), we have identified the bounded operator 1+tA with its extension to . H0 Proof. Fix x ∆. By a straightforward convexity argument, there is a unique ∈ decomposition x = x0,t + x1,t which is optimal in the sense that (1.9) K(t, x)= x 2 + t x 2 . k 0,t k0 k 1,t k1 It follows that x ∆ for i =0, 1. Moreover, for all y ∆ we have i,t ∈ ∈ d x + ǫy 2 + t x ǫy 2 =0, dǫ{k 0,t k0 k 1,t − k1 }|ǫ=0 i.e., A−1/2x tA1/2x , A1/2y =0, y ∆. h 0,t − 1,t i0 ∈ −1/2 1/2 By regularity, we conclude that A x0,t = tA x1,t, whence tA 1 (1.10) x = x and x = x. 0,t 1+ tA 1,t 1+ tA (Note that the operators in (1.10) extend to bounded operators on 0.) Inserting the relations (1.10) into (1.9), one finishes the proof of the lemma. H Now fix a positive Radon measure ̺ on [0, ]. The norm in the space ̺ (see (1.5)) can be written ∞ H (1.11) x 2 = h(A)x , x , k k̺ h i0 where (1 + t)λ (1.12) h(λ)= d̺(t). 1+ tλ Z[0,∞] The class of functions representable in this form for some positive Radon measure ′ ̺ is the class P of Pick functions, positive and regular on R+. Notice that for the definition (1.11) to make sense, we just need h to be defined on σ(A) 0 , where σ(A) is the spectrum of A. (The value h(0) is irrelevant since A is injective).\{ } A calculus exercise shows that for the space (see (1.7)) we have Hθ (1.13) x 2 = Aθx , x . k kθ h i0 4 YACIN AMEUR 1.4. Quadratic interpolation norms. Let ∗ be any quadratic intermediate space relative to . We write H H x 2 = Bx, x k k∗ h i0 where B is a positive injective operator in (the domain of B1/2 is ∆). H0 For a map T ( ) we shall often use the simplified notations ∈ L H T = T , T = T , T = T . k k k k L(H0) k kA k k L(H1) k kB k k L(H∗) The reader can check the identities T = A1/2T A−1/2 and T = B1/2TB−1/2 . k kA k k k kB k k We shall refer to the following lemma as Donoghue’s lemma, cf. [14, Lemma 1]. Lemma 1.2. If ∗ is exact interpolation with respect to , then B commutes with every projectionH which commutes with A and B = h(A)Hwhere h is some positive Borel function on σ(A). Proof. For an orthogonal projection E on , the condition E 1 is equivalent H0 k kA ≤ to that EAE A, i.e., that E commutes with A. The hypothesis that ∗ be exact interpolation thus≤ implies that every spectral projection of A commutesH with B. It now follows from von Neumann’s bicommutator theorem that B = h(A) for some positive Borel function h on σ(A). In view of the lemma, the characterization of the exact quadratic interpolation norms of a given type H reduces to the characterization of functions h : σ(A) R → + such that for all T ∈ L H (1.14) T M and T M T H(M ,M ), k k≤ 0 k kA ≤ 1 ⇒ k kh(A) ≤ 0 1 or alternatively, (1.15) T ∗T M 2 and T ∗AT M 2 A T ∗h(A)T H(M ,M ) 2 h(A).
Load more

Recommended publications
  • Symmetry Preserving Interpolation Erick Rodriguez Bazan, Evelyne Hubert
    Symmetry Preserving Interpolation Erick Rodriguez Bazan, Evelyne Hubert To cite this version: Erick Rodriguez Bazan, Evelyne Hubert. Symmetry Preserving Interpolation. ISSAC 2019 - International Symposium on Symbolic and Algebraic Computation, Jul 2019, Beijing, China. 10.1145/3326229.3326247. hal-01994016 HAL Id: hal-01994016 https://hal.archives-ouvertes.fr/hal-01994016 Submitted on 25 Jan 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symmetry Preserving Interpolation Erick Rodriguez Bazan Evelyne Hubert Université Côte d’Azur, France Université Côte d’Azur, France Inria Méditerranée, France Inria Méditerranée, France [email protected] [email protected] ABSTRACT general concept. An interpolation space for a set of linear forms is The article addresses multivariate interpolation in the presence of a subspace of the polynomial ring that has a unique interpolant for symmetry. Interpolation is a prime tool in algebraic computation each instantiated interpolation problem. We show that the unique while symmetry is a qualitative feature that can be more relevant interpolants automatically inherit the symmetry of the problem to a mathematical model than the numerical accuracy of the pa- when the interpolation space is invariant (Section 3).
    [Show full text]
  • Real Interpolation of Sobolev Spaces
    MATH. SCAND. 105 (2009), 235–264 REAL INTERPOLATION OF SOBOLEV SPACES NADINE BADR Abstract We prove that W 1 is a real interpolation space between W 1 and W 1 for p>q and 1 ≤ p < p p1 p2 0 1 p<p2 ≤∞on some classes of manifolds and general metric spaces, where q0 depends on our hypotheses. 1. Introduction 1 ∞ Do the Sobolev spaces Wp form a real interpolation scale for 1 <p< ? The aim of the present work is to provide a positive answer for Sobolev spaces on some metric spaces. Let us state here our main theorems for non-homogeneous Sobolev spaces (resp. homogeneous Sobolev spaces) on Riemannian mani- folds. Theorem 1.1. Let M be a complete non-compact Riemannian manifold satisfying the local doubling property (Dloc) and a local Poincaré inequality ≤ ∞ ≤ ≤ ∞ 1 (Pqloc), for some 1 q< . Then for 1 r q<p< , Wp is a real 1 1 interpolation space between Wr and W∞. To prove Theorem 1.1, we characterize the K-functional of real interpola- tion for non-homogeneous Sobolev spaces: Theorem 1.2. Let M be as in Theorem 1.1. Then ∈ 1 + 1 1. there exists C1 > 0 such that for all f Wr W∞ and t>0 1 1 ∗∗ 1 ∗∗ 1 r 1 1 ≥ r | |r r +|∇ |r r ; K f, t ,Wr ,W∞ C1t f (t) f (t) ≤ ≤ ∞ ∈ 1 2. for r q p< , there is C2 > 0 such that for all f Wp and t>0 1 1 q∗∗ 1 q∗∗ 1 r 1 1 ≤ r | | q +|∇ | q K f, t ,Wr ,W∞ C2t f (t) f (t) .
    [Show full text]
  • Characterizations of Screened Sobolev Spaces Noah Stevenson and Ian Tice DEPARTMENTOF MATHEMATICAL SCIENCES,CARNEGIE MELLON UNIVERSITY
    Characterizations of Screened Sobolev Spaces Noah Stevenson and Ian Tice DEPARTMENT OF MATHEMATICAL SCIENCES,CARNEGIE MELLON UNIVERSITY An approach which is promising in the resolution of this issue is to swap the inhomogeneous Sobolev spaces for their homogeneous counterparts, Does the topology on the screened Sobolev spaces Background W_ k;p (Ω). The upshot is that the latter families allow for a larger variety of Research admit a Fourier space characterization? behaviors at infinity. For this approach to have any chance of being fruitful in the Leoni and Tice gave the following partial frequency & Motivation study of boundary value problems, it is essential to identify and understand the Questions n characterization of these spaces: for f 2 S R it holds Partial differential equations (PDEs) are trace spaces associated to the homogeneous Sobolev spaces. Recall that the trace Z 1 2s 2 ^ 2 2 central to the modeling of natural space associated to a Sobolev space on a domain is a space of functions holding Can the screened Sobolev spaces be [f]W~ s;2 minfjξj ; jξj g f (ξ) dξ ; (2) (σ) n understood through interpolation theory? R phenomena. As a result, the study of these the possible boundary values and, for higher order Sobolev spaces, powers of the where 0 < s < 1 and constant screening function σ = 1. equations and their solutions has held a normal derivative. Interpolation theory is a set of tools which This expression suggests that the high-mode part of a prominent place in mathematics for For functions of finite energy in planar strips these trace spaces were first s;2 take certain pairs of topological vector member of W~ behaves like a member of the centuries.
    [Show full text]
  • Interpolation of Compact Operators by the Methods of Calderon and Gustavsson-Peetre
    Proceedings of the Edinburgh Mathematical Society (1995) 38, 261-276 I INTERPOLATION OF COMPACT OPERATORS BY THE METHODS OF CALDERON AND GUSTAVSSON-PEETRE by M. CWIKEL* and N. J. KALTONf (Received 15th July 1993) Let X = (X0,A'i) and \ = {Y0, V,) be Banach couples and suppose T:X-»Y is a linear operator such that T:X0-+ Yo is compact. We consider the question whether the operator T:[X0, A'1]fl->[T0> y,]e is compact and show a positive answer under a variety of conditions. For example it suffices that Xo be a UMD-space or that Xo is reflexive and there is a Banach space so that X0 = [W,Xl~\I for some 0<ot< 1. 1991 Mathematics subject classification: 46M35. 1. Introduction Let X=(X0,Xi) and \ = (Y0,Y1) be Banach couples and let T be a linear opeator such that T:X->Y (meaning, as usual, that T:XQ + X^ YO+ 7, and T:Xj-*Yj boun- dedly for y=0,1). Interpolation theory supplies us with a variety of interpolation functors F for generating interpolation spaces, i.e. functors F which when applied to the couples X and Y yield spaces F(X) and F(Y) having the property that each T as above maps F(X) into F(Y) with bound for some absolute constant C depending only on the functor F. It will be convenient here to use the customary notation ||T||x_v=max(||7'||Xo,||T||;ri). Further general background about interpolation theory and Banach couples can be found e.g.
    [Show full text]
  • Interpolation and Extrapolation Diplomarbeit Zur Erlangung Des Akademischen Grades Diplom-Mathematiker
    Interpolation and Extrapolation Diplomarbeit zur Erlangung des akademischen Grades Diplom-Mathematiker Friedrich-Schiller-Universität Jena Fakultät für Mathematik und Informatik eingereicht von Peter Arzt geb. am 5.7.1982 in Bernau Betreuer: Prof. Dr. H.-J. Schmeißer Jena, 5.11.2009 Abstract We define Lorentz-Zygmund spaces, generalized Lorentz-Zygmund spaces, slowly varying functions, and Lorentz-Karamata spaces. To get an interpolation charac- terization of Lorentz-Karamata spaces, we examine the K- and the J-method of real interpolation with function parameters in quasi-Banach spaces. In particular, we study the Kalugina class BK and prove the Equivalence Theorem and the Reitera- tion Theorem with function parameters. Finally, we define the ∆ and Σ method of extrapolation and achieve an extrap- olation characterization that allows us, in particular, to characterize generalized Lorentz-Zygmund spaces by Lorentz spaces. Contents Introduction 4 1 Preliminaries 6 1.1 Notations . 6 1.2 Quasi-Banach spaces . 6 1.3 Non-increasing rearrangement and Lorentz spaces . 8 1.4 Slowly varying functions and Lorentz-Karamata spaces . 12 2 Interpolation 17 2.1 Classical Real Interpolation . 17 2.2 The function classes BK and BΨ .................... 23 2.3 Interpolation with function parameters . 30 2.4 Interpolation with slowly varying parameters . 43 2.5 Interpolation with logarithmic parameters . 47 3 Characterization of Extrapolation Spaces 54 3.1 ∆-Extrapolation . 54 3.2 Σ-Extrapolation . 61 3.3 Variations . 70 3.4 Applications to concrete function spaces . 72 Bibliography 77 3 Introduction n For 0 < p, q ≤ ∞, α ∈ R, and Ω ⊆ R with finite Lebesgue measure, the Lorentz- Zygmund space Lp,q(log)α(Ω) is the space of all complex-valued functions f on Ω such that 1 Z ∞ q 1 α ∗ q dt kf|Lp,q(log L)α(Ω)k = t p (1 + |log t|) f (t) < ∞.
    [Show full text]
  • Lecture Notes on Function Spaces
    Lecture Notes on Function Spaces space invaders Karoline Götze TU Darmstadt, WS 2010/2011 Contents 1 Basic Notions in Interpolation Theory 5 2 The K-Method 15 3 The Trace Method 19 3.1 Weighted Lp spaces . 19 3.2 The spaces V (p, θ, X0,X1) ........................... 20 3.3 Real interpolation by the trace method and equivalence . 21 4 The Reiteration Theorem 26 5 Complex Interpolation 31 5.1 X-valued holomorphic functions . 32 5.2 The spaces [X, Y ]θ and basic properties . 33 5.3 The complex interpolation functor . 36 5.4 The space [X, Y ]θ is of class Jθ and of class Kθ................ 37 6 Examples 41 6.1 Complex interpolation of Lp -spaces . 41 6.2 Real interpolation of Lp-spaces . 43 6.2.1 Lorentz spaces . 43 6.2.2 Lorentz spaces and the K-functional . 45 6.2.3 The Marcinkiewicz Theorem . 48 6.3 Hölder spaces . 49 6.4 Slobodeckii spaces . 51 6.5 Functions on domains . 54 7 Function Spaces from Fourier Analysis 56 8 A Trace Theorem 68 9 More Spaces 76 9.1 Quasi-norms . 76 9.2 Semi-norms and Homogeneous Spaces . 76 9.3 Orlicz Spaces LA ................................ 78 9.4 Hardy Spaces . 80 9.5 The Space BMO ................................ 81 2 Motivation Examples of function spaces: n 1 α p k,p s,p s C(R ; R),C ([−1, 1]),C (Ω; C),L (X, ν),W (Ω),W ,Bp,q, n where Ω domain in R , α ∈ R+, 1 ≤ p, q ≤ ∞, (X, ν) measure space, k ∈ N, s ∈ R+.
    [Show full text]
  • Ideal Interpolation, H-Bases and Symmetry Erick Rodriguez Bazan, Evelyne Hubert
    Ideal Interpolation, H-Bases and Symmetry Erick Rodriguez Bazan, Evelyne Hubert To cite this version: Erick Rodriguez Bazan, Evelyne Hubert. Ideal Interpolation, H-Bases and Symmetry. ISSAC 2020 - International Symposium on Symbolic and Algebraic Computation, Jul 2020, Kalamata, Greece. 10.1145/3373207.3404057. hal-02482098v2 HAL Id: hal-02482098 https://hal.inria.fr/hal-02482098v2 Submitted on 7 Jul 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Ideal Interpolation, H-Bases and Symmetry Erick Rodriguez Bazan Evelyne Hubert Université Côte d’Azur, France Université Côte d’Azur, France Inria Méditerranée, France Inria Méditerranée, France [email protected] [email protected] ABSTRACT for each instantiated interpolation problem, that is both invariant Multivariate Lagrange and Hermite interpolation are examples of and of minimal degree. An interpolation space for Λ identifies with ideal interpolation. More generally an ideal interpolation problem the quotient space K»x¼/I. Hence a number of operations related is defined by a set of linear forms, on the polynomial ring, whose to I can already be performed with a basis of an interpolation kernels intersect into an ideal. space for Λ: decide of membership to I, determine normal forms For an ideal interpolation problem with symmetry, we address of polynomials modulo I and compute matrices of multiplication the simultaneous computation of a symmetry adapted basis of the maps in K»x¼/I.
    [Show full text]
  • Noncommutative Boyd Interpolation Theorems
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 367, Number 6, June 2015, Pages 4079–4110 S 0002-9947(2014)06185-0 Article electronically published on December 10, 2014 NONCOMMUTATIVE BOYD INTERPOLATION THEOREMS SJOERD DIRKSEN Abstract. We present a new, elementary proof of Boyd’s interpolation the- orem. Our approach naturally yields a noncommutative version of this re- sult and even allows for the interpolation of certain operators on 1-valued noncommutative symmetric spaces. By duality we may interpolate several well-known noncommutative maximal inequalities. In particular we obtain a version of Doob’s maximal inequality and the dual Doob inequality for non- commutative symmetric spaces. We apply our results to prove the Burkholder- Davis-Gundy and Burkholder-Rosenthal inequalities for noncommutative mar- tingales in these spaces. 1. Introduction Symmetric Banach function spaces play a pivotal role in many fields of mathe- matical analysis, especially probability theory, interpolation theory and harmonic analysis. A cornerstone result in the interpolation theory of these spaces is the Boyd interpolation theorem, named after D.W. Boyd. Together with the Calder´on- Mitjagin theorem, which characterizes the symmetric Banach function spaces which 1 ∞ are an exact interpolation space for the couple (L (R+),L (R+)), Boyd’s theorem provides an invaluable tool for the analysis of symmetric spaces. The history of Boyd’s interpolation theorem begins with the announcement of Marcinkiewicz [28], shortly before his death, of an extension of the Riesz-Thorin theorem. Let us say that a sublinear operator T is of Marcinkiewicz weak type (p, p) p if for any f ∈ L (R+), 1 −1 p ≤ p d(v; Tf) Cv f L (R+) (v>0), where d(·; Tf) denotes the distribution function of Tf.
    [Show full text]
  • Interpolation of Hilbert and Sobolev Spaces: Quantitative Estimates and Counterexamples
    Interpolation of Hilbert and Sobolev Spaces: Quantitative Estimates and Counterexamples S. N. Chandler-Wilde,∗ D. P. Hewett,y A. Moiola∗ August 20, 2014 Dedicated to Vladimir Maz'ya, on the occasion of his 75th Birthday Abstract This paper provides an overview of interpolation of Banach and Hilbert spaces, with a focus on establishing when equivalence of norms is in fact equality of norms in the key results of the theory. (In brief, our conclusion for the Hilbert space case is that, with the right normalisations, all the key results hold with equality of norms.) In the final section we apply the Hilbert space results to the s s n Sobolev spaces H (Ω) and He (Ω), for s 2 R and an open Ω ⊂ R . We exhibit examples in one and two dimensions of sets Ω for which these scales of Sobolev spaces are not interpolation scales. In the cases when they are interpolation scales (in particular, if Ω is Lipschitz) we exhibit examples that show that, in general, the interpolation norm does not coincide with the intrinsic Sobolev norm and, in fact, the ratio of these two norms can be arbitrarily large. 1 Introduction This paper provides in the first two sections a self-contained overview of the key results of the real method of interpolation for Banach and Hilbert spaces. This is a classical subject of study (see, e.g., [4,5, 23, 24] and the recent review paper [3] for the Hilbert space case), and it might be thought that there is little more to be said on the subject.
    [Show full text]
  • Pivot Duality of Universal Interpolation and Extrapolation Spaces
    PIVOT DUALITY OF UNIVERSAL INTERPOLATION AND EXTRAPOLATION SPACES CHRISTIAN BARGETZ1 AND SVEN-AKE WEGNER2 Abstract. It is a widely used method, for instance in perturbation theory, to associate with a given C0-semigroup its so-called interpolation and extrapolation spaces. In the model case of the 2 shift semigroup acting on L (R), the resulting chain of spaces recovers the classical Sobolev scale. In 2014, the second named author defined the universal interpolation space as the projective limit of the interpolation spaces and the universal extrapolation space as the completion of the inductive limit of the extrapolation spaces, provided that the latter is Hausdorff. In this note we use the notion of the dual with respect to a pivot space in order to show that the aforementioned inductive limit is Hausdorff, already complete, and can be represented as the dual of the projective limit whenever a power of the generator of the initial semigroup is a self-adjoint operator. In the case of the classical Sobolev scale we show that the duality holds, and that the two universal spaces were already studied by Laurent Schwartz in the 1950s. Our results and examples complement the approach of Haase, who in 2006 gave a different definition of universal extrapolation spaces in the context of functional calculi. Haase avoids the inductive limit topology precisely for the reason that it a priori cannot be guaranteed that the latter is always Hausdorff. We show that this is indeed the case provided that we start with a semigroup defined on a reflexive Banach space. 1. The classical Sobolev scale We start by considering the following generic example.
    [Show full text]
  • A Note on Homogeneous Sobolev Spaces of Fractional Order
    A NOTE ON HOMOGENEOUS SOBOLEV SPACES OF FRACTIONAL ORDER LORENZO BRASCO AND ARIEL SALORT Abstract. We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev{Slobodecki˘ınorm. We compare it to the fractional Sobolev space obtained by the K−method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible. Contents 1. Introduction1 1.1. Motivations1 1.2. Aims3 1.3. Results4 1.4. Plan of the paper5 2. Preliminaries5 2.1. Basic notation5 2.2. Sobolev spaces6 2.3. A homogeneous Sobolev{Slobodecki˘ıspace6 2.4. Another space of functions vanishing at the boundary7 3. An interpolation space8 4. Interpolation VS. Sobolev-Slobodecki˘ı 11 4.1. General sets 11 4.2. Convex sets 14 4.3. Lipschitz sets and beyond 20 5. Capacities 23 6. Double-sided estimates for Poincar´econstants 26 Appendix A. An example 27 Appendix B. One-dimensional Hardy inequality 30 Appendix C. A geometric lemma 31 References 32 1. Introduction arXiv:1806.08945v1 [math.FA] 23 Jun 2018 1.1. Motivations. In the recent years there has been a great surge of interest towards Sobolev spaces of fractional order. This is a very classical topic, essentially initiated by the Russian school in the 50s of the last century, with the main contributions given by Besov, Lizorkin, Nikol'ski˘ı, Slobodecki˘ıand their collaborators.
    [Show full text]
  • Shape-Driven Interpolation with Discontinuous Kernels: Error Analysis, Edge Extraction and Applications in Mpi
    SHAPE-DRIVEN INTERPOLATION WITH DISCONTINUOUS KERNELS: ERROR ANALYSIS, EDGE EXTRACTION AND APPLICATIONS IN MPI S. DE MARCHI∗, W. ERBy , F. MARCHETTIz , E. PERRACCHIONEx , AND M. ROSSINI{ Abstract. Accurate interpolation and approximation techniques for functions with discontinuities are key tools in many applications as, for instance, medical imaging. In this paper, we study an RBF type method for scattered data interpolation that incorporates discontinuities via a variable scaling function. For the construction of the discontinuous basis of kernel functions, information on the edges of the interpolated function is necessary. We characterize the native space spanned by these kernel functions and study error bounds in terms of the fill distance of the node set. To extract the location of the discontinuities, we use a segmentation method based on a classification algorithm from machine learning. The conducted numerical experiments confirm the theoretically derived convergence rates in case that the discontinuities are a priori known. Further, an application to interpolation in magnetic particle imaging shows that the presented method is very promising. Key words. Meshless approximation of discontinuous functions; radial basis function (RBF) interpolation; variably scaled discontinuous kernels (VSDKs); Gibbs phenomenon; segmentation and classification with kernel machines, Magnetic Particle Imaging (MPI) AMS subject classifications. 41A05, 41A25, A1A30, 65D05 1. Introduction. Data interpolation is an essential tool in medical imaging. It is required for geometric alignment, registration of images, to enhance the quality on display devices, or to reconstruct the image from a compressed amount of data [7, 26, 40]. Interpolation techniques are needed in the generation of images as well as in post-processing steps.
    [Show full text]