Duchon, J. (1978): Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D -splines, R.A.I.R.O. An. Num. 12, no. 4, 325--334.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....simply by integrating the L1 error. In the light of the IR theory, this seems to be too naive. In the IR case it is, in a similar situation, possible to gain an additional d=2 in the order by using a localization trick, which dates back to Duchon s initial work on thin plate splines (cf. [2, 3]) This trick should also work in the sphere setting, but so far nobody has ever tried it. Note that in the just described situation the native space is actually the Sobolev space H (S ) with s = d 2 1. For Euclidean space IR and bounded domains therein, we usually do not know the ....

J. Duchon. Sur l'erreur d'interpolation des fonctions de plusieurs variables pas les D {splines. Rev. Francaise Automat. Informat. Rech. Oper. Anal. Numer., 12(4):325-334, 1978.

....Theorem 1.8. m; k) admissible with bd=2c 1 and 0 m . There exists h 0 0 (depending only on k; of satisfying h : h( h 0 , then for all f 2 W const( k; p where T ; k f = fsg and p : maxfd=2 d=p; 0g. This result was rst proved by Duchon [5] for the particular choice of associated with surface splines. When taken with the construction of Light and Wayne, this amounts to the case when m = are integers and k = m 1. The case p = 1 has been settled by Wu and Schaback [13] while the case when is an integer, m = 0, and k = 1 has ....

....2 (see [14] 3] and [11] for error estimates when f 62 W 2 ) These theorems are actually special cases of the more general result Theorem 5.4. There it is not assumed that has the cone property, but rather that satis es a certain condition related to polynomials (see De nition 4. 1) Duchon [5] has shown that this condition is satis ed if has the cone property, while Golitschek and Light [6] have established this in case : jxj = 1g. An outline of the paper is as follows. In section 2 we de ne and examine certain spaces Y m; k . One useful observation made in Theorem 2.9 is ....

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J. Duchon, Sur l'erreur d'interpolation des fonctions de plusieur variables par les D -splines, RAIRO Analyse Numerique 12 (1978), 325-334.

....f . However, the assumptions are suciently weak to disallow special cases where, for example, f is assumed to satisfy certain boundary conditions, or f is restricted to a certain class of entire functions, or the interpolation points are assumed to satisfy some quasi uniform condition. Duchon [9] has shown that the approximation order of surface spline interpolation is at least p : minfm; m d=2 d=pg for 1 p 1. He actually showed that = o(h p ) for all f 2 H whenever the domain has the cone property (see the following section for the details) Duchon s error analysis was ....

.... , it follows that T f = T T f and hence we obtain f T f = f T 00 f) T 00 (f T f) I II: To express the error in the above form was rst suggested by Mike Powell and can be found in Bejancu [3] That I decays like O(h ) can be obtained by interpolating between results of Duchon [9] and Matveev [16] For II, we imitate Bejancu s approach, and express II in terms of the Lagrange basis as (1.4) II = T (f T f) L : That the right side converges meaningfully and that equality indeed holds in (1.4) will be shown in section 3. Since f T vanishes on , the sum in (1.4) ....

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J. Duchon, Sur l'erreur d'interpolation des fonctions de plusieur variables par les D -splines, RAIRO Analyse Numerique 12 (1978), 325-334.

....on a compact subset Omega of R satisfying a uniform interior cone condition. There is a vast of literature studying this kind of approximation problem by introducing the right space, often called native space, and then giving approximation orders depending on h. We cite for example [4, 8, 9, 11, 19]. Here, we follow [19] because it serves our purposes best and it will come out that the native spaces for our functions are norm equivalent to Sobolev spaces (see theorem 2.1) We start with a positive definite and integrable function Phi and define its Fourier transform by b Phi( ....

J. Duchon, Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D - splines, R.A.I.R.O. Analyse num'erique, vol. 12 no. 4 (1978), pp 325 - 334.

....fl and h. A basic problem from the point of view of approximation theory is to study the accuracy to which s h approximates f over Omega when h 0, under various smoothness assumptions on f . This problem and its version for scattered interpolation points have been investigated by Duchon [7], Arcang eli and Rabut [1] Madych and Nelson [14] Wu and Schaback [27] Powell [22] Matveev [16] Light and Wayne [13] Schaback [24, 25] and Johnson [9] 12] who estimated the dependence on h of the error (or of some of its derivatives) in the uniform or L p norm (1 p 1) over the ....

....d1 (f; S h ) is the least distance from f to an element of S h , in the uniform norm over Omega Gamma The following result will provide an upper estimate on kT h k1 . Theorem 2 Let Omega ae R d be the closure of a connected, open and bounded set, which satisfies a cone property (see Duchon [7] for a suitable definition of the latter condition) Then, for any parameter fl 0, there exists a constant h 0 0 such that the surface spline functions 1 ; 2 ; n , which satisfy the Lagrange equations (1.5) on the grid Omega hZ d , have the property max x2 Omega n X j=1 ....

Duchon, J. (1978) Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D m -splines. R.A.I.R.O. An. Num. 12, No. 4, 325--334 13

....attributes of an interpolation method is via the notion of L p approximation orders. Surface spline interpolation in Omega is said to provide L p approximation of order fl if kf Gamma T Xi fk Lp ( Omega Gamma = O(ffi fl ) as ffi 0 for all sufficiently smooth functions f . Duchon [8] has shown that if Omega is connected, has the cone property, and has a Lipschitz boundary, then surface spline interpolation in Omega provides L p approximation of order at least fl p : minfm;m d=p Gamma d=2g for p 2 [1 : 1] More precisely, it was shown that for all f 2 H and p 2 [1 : ....

Duchon, J. (1978), Sur l'erreur d'interpolation des fonctions de plusieur variables par les D m - splines, RAIRO Analyse Numerique 12, 325--334.

....at least on Xi, one seeks a nice function s which interpolates the data f j Xi ; that is, which satisfies s( f( 8 2 Xi. The reader is referred to the surveys [3] and [5] for descriptions of a variety of interpolation methods. One such method is that of surface spline interpolation (see [4]) which we now describe. Let m 2 N : f1; 2; 3; g be such that m d=2, and let H m denote the set of all tempered distributions f for which D ff f 2 L 2 : L 2 (R d ) for all jffj = m. For measurable A ae R d and f 2 H m , we define the seminorm jjjf jjj H m (A) 2) d=2 s ....

....to discuss the error between f and s, let us assume that Omega is open, bounded, and has the cone property, and assume also that Xi ae Omega : closure( Omega Gammas The fill distance from Xi to Omega is the quantity ffi : ffi( Xi; Omega Gamma : sup x2 Omega inf 2 Xi jx Gamma j. Duchon [4] has shown that if s is the surface spline interpolant to f at Xi, then (1.1) kf Gamma sk Lp constffi m Gammad=2 d=p jjjf jjj H m 8f 2 H m for 2 p 1 and ffi sufficiently small. What is interesting about the proof of (1.1) is that it hinges not on the fact that s minimizes jjjsjjj H m , ....

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J. Duchon, Sur l'erreur d'interpolation des fonctions de plusieur variables par les D m -splines, RAIRO Analyse Numerique 12 (1978), 325--334.

....n with in (1) Then, however, one get representations of the form (2) in which the corresponding function E is the solution of a boundary value problem on and therefore depends on the geometry of cf. Atteia [2] This is an essential di erence to the one dimensional setting. It is well known, [6], that for functions f 2 BL m( the minimal norm interpolant s f satis es kf s f k L 1 Ch m n=2 kfk BL m( h = sup x2 inf x i 2X kx x i k 2 ; 3) where the constant C is independent of h and f . The purpose of the present paper is to improve this approximation order by m for ....

.... f 0 ; f 0 s f BL m (R n ) Z 0 X j j=m m 2 f(x) 1 A (f(x) s f (x) dx An application of the Cauchy Schwarz inequality then gives f 0 ; f 0 s f BL m (R n ) Ckfk BL 2m( kf s f k L 2 ( 8) where C depends only on n and m: Next, from Proposition 3 of [6] with k = 0 and p = 2, we have kf s f k L 2 Ch m kf 0 s f k BL m (R n ) for some C 0 independent of h and f . Inserting this in (8) we obtain with the orthogonality condition f 0 s f ; s f BL m (R n ) 0 kf 0 s f k 2 BL m (R n ) f 0 ; f 0 s f BL m (R n ....

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Duchon J. 1978 Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D m - splines. R.A.I.R.O. Analyse numerique 12, 325-334.

....interpolant to f using the seminorm j Delta j. That is, f(a i ) If) a i ) i = 1; m, and amongst all such interpolants, jIf j is minimal. Looking at Equation (3) it is not immediately obvious whether the direct or indirect form of the seminorm is important. However, work by Duchon [2] explains why the direct form of the seminorm is to be preferred in general. Roughly speaking, Duchon showed that if Omega is a open set in IR n with Lipschitz continuous boundary, then one can improve the error estimate to jf(x) Gamma (If) x)j P(x)jf Gamma If j Omega for all x 2 Omega . ....

....us back to familiar cases. For example, if all but one of the a j are zero, then our seminorm becomes one of the usual Sobolev seminorms of the appropriate (integer) order. The error estimates mentioned in the introduction were investigated very thoroughly for this case in the papers of Duchon [2, 3]. If a j = 1, j = 0; 1; m and is zero otherwise, and we take k = 0, then the usual Sobolev norm on IR n is obtained. The reader can consult Adams [1] or H#rmander [4] for details of the Sobolev theory) Of course, if k = 0 and a 0 6= 0, then the seminorm is always in fact a norm. 3 ....

Duchon J. Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D m splines, RAIRO Analyse Numerique 12 (4), 1978, 325334.

....of order k. Here and in the sequel we use the standard multi index notation with j j : P i i for 2 IN n . Convergence of interpolation on regular grids has been studied extensively by Buhmann (see e.g: 1] for a comprehensive treatment) and others. For the case of scattered data Duchon [3] treated the thin plate spline case OE(r) r 2 log r, while Jackson [6] proved a general, but non quantitative convergence result. The dissertation of Wu [13] of 1986 contained a rather general Hilbert space theory for Kriging and related radial basis function methods in the case of a single ....

....may be completed to form a Hilbert space with inner product (f 1 ; f 2 ) OE = Z IR n f 1 (t) f 2 (t) t) Gamma1 dt which was thoroughly studied by Madych and Nelson in [9] 10] This is why the rather restrictive condition (4.2) occurs there, too, while (4. 2) does not appear in [3] and [6] We do not want to make direct use of Hilbert space properties here, but proceed directly to an estimate of ( q (x) in the next section. The application of the Cauchy Schwarz inequality to (4.1) allows the error bound to be factored into a term c f depending on f , but not on the ....

Duchon,J., Sur L'erreur d'interpolation des fonctions de plusiers variables par les D m \Gamma splines, RAIRO Analyse Numerique 12 (1978) 325-334.

....EB in such a way that the constant C in kEBuk W k 2 (R d ) Ckuk W k 2 (B) is independent of the radius and the position of the ball B (cf [14] Thus (4. 3) leads to kD ff u Gamma D ff s u k L2 (B) C vol(B) 1 2 kP (ff) X; Phi k L1 (B) ku B k W k 2 (R d ) According to [5] there exist M , M 1 , h 2 0 and for h h 2 a finite subset T h Omega such that the balls B(t; h) and B(t; Mh) with radius h and Mh respectively, centered at t 2 T h satisfy B(t; h) Omega [ t2Th B(t; Mh) 6 HOLGER WENDLAND and such that P t2Th B(t;Mh) M 1 . Here A denotes the ....

J. Duchon, Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D m -splines, R.A.I.R.O. Analyse num'erique, Vol. 12, No. 4 (1978), pp 325 - 334.

....on a compact subset Omega of R d satisfying a uniform interior cone condition. There is a vast of literature studying this kind of approximation problem by introducing the right space, often called native space, and then giving approximation orders depending on h. We cite for example [4, 8, 9, 11, 19]. Here, we follow [19] because it serves our purposes best and it will come out that the native spaces for our functions are norm equivalent to Sobolev spaces (see theorem 2.1) We start with a positive definite and integrable function Phi and define its Fourier transform by b Phi( 2 ) ....

J. Duchon, Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D m - splines, R.A.I.R.O. Analyse num'erique, vol. 12 no. 4 (1978), pp 325 - 334.

....h = sup t2 Omega inf a2A jt Gammaaj. If f 2 C( Omega Gamma say, and Uf is its interpolant, then one might hope to get kf Gamma Ufk = O(h ) as h 0, where is some measure of the smoothness of f . We will establish such error bounds in this paper. Early work in this area was due to Duchon [3], who developed the theory of surface splines. His error estimates rest crucially on the property that his interpolant preserves polynomials of degree k Gamma 1. Since his interpolants are all special cases of the ones just described, the polynomial preservation property of U , that is, Up = p ....

....asymptotic behaviour of the power function OE(0) Gamma 2 X r=1 p r (x)OE(jx Gamma a r j) X r;s=1 p r (x)p s (x)OE(ja r Gamma a s j) in the special case that K = Pi k Gamma1 . We need some straightforward results from Lagrange interpolation theory. These may be found in Duchon [3], and in a wide variety of other places, particularly in the researches of finite element theory. 3.1 Definition Let b = fb 1 ; b g be a set of points in IR n which is unisolvent with respect to Pi k Gamma1 . Then L b : C(IR n ) Gamma Pi k Gamma1 will denote the Lagrange ....

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Duchon J. Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D m --splines, RAIRO Analyse Numerique 12(4), 1978, 325--334.

....function h, is just the h spline Hermite interpolant. The results on error estimation and convergence rate of the h spline interpolant generalise those in [11, 12, 18, 10] to the case of Hermite interpolation. 1 Introduction The classic variational approach of interpolation proposed by Duchon [4, 5], and further discussed by Meinguet [13] is to find a interpolant u 2 D Gammam L 2 (IR d ) ff : D ff f 2 L 2 (IR d ) 8jffj = mg minimising the quadratic functional kuk 2 m : X jffj=m c ff Z IR d jD ff u(x)j 2 dx; 1) under the interpolatory constraints u(x i ) f i ; ....

.... positive constants K; ffl 0 such that for every 0 ffl ffl 0 , Omega ae [fB(t; fflK) t 2 T ffl g; where T ffl = ft 2 IR d : B(t; ffl) ae Omega g; B(t; ffl) fx 2 IR d : jx Gamma tj fflg: If Omega satisfies the cone condition, then the above requirements will naturally be met; see [5]. Let fL 1 ; LN g be the interpolating functionals with supporting set X. Let s be the h spline Hermite interpolant to f . For each fi 2 ZZ d ; 0 jfij r and x 2 Omega Gamma we want to get the convergence rate of jD fi (s Gamma f) x)j on Omega Gamma as the coverage of X to the ....

J. Duchon (1978), Sur l'Erreur d'Interpolation des Fonctions de Plusieurs Variables par les D m -splines, RAIRO Analyse Numerique 12, 325--334.

.... r 0 such that for every x 2 Omega a unit vector (x) exists such that the cone C(x; x) r) fx y : y 2 R d ; kyk 2 = 1; y T (x) cos( 2 [0; r]g is contained in Omega Gamma The choice of centers we want to investigate here is motivated by the following Lemma of Duchon [1], which we include here for the convenience of the reader. Lemma 3 .1 Let Omega be an open subset of R d satisfying an interior cone condition. Then there exist M 0 and h 0 0 such that for all 0 h h 0 the set X h : fx 2 2h p d Z d : B(x; h) Omega g satisfies Omega [ ....

Duchon, J., Sur l'erreur d'interpolation des fonctions de plusiers variables par les D m -splines, R.A.I.R.O. Analyse num'erique 12 (1978), 325-334.

....of OE, and this technique allows to prove conditional positive definiteness for a variety of radial basis functions. We list a few examples: Multiquadrics OE(r) c 2 r 2 ) fi=2 for fi 2 IR Gammad n 2ZZ and 2q fi [8] Thin plate splines OE(r) r fi for fi 2 IR 0 n 2ZZ and 2q fi [3, 4, 5], Thin plate splines OE(r) Gamma1) fi=2 1 r fi log r for fi 2 2IN, 2q fi [3, 4, 5] Gaussians OE(r) e Gammaffr 2 for ff 0 and q 0. Our main purpose here is to study the error f(x) Gamma s f (x) of different radial basis function interpolants on different spaces. Curiously ....

....radial basis functions. We list a few examples: Multiquadrics OE(r) c 2 r 2 ) fi=2 for fi 2 IR Gammad n 2ZZ and 2q fi [8] Thin plate splines OE(r) r fi for fi 2 IR 0 n 2ZZ and 2q fi [3, 4, 5] Thin plate splines OE(r) Gamma1) fi=2 1 r fi log r for fi 2 2IN, 2q fi [3, 4, 5], Gaussians OE(r) e Gammaffr 2 for ff 0 and q 0. Our main purpose here is to study the error f(x) Gamma s f (x) of different radial basis function interpolants on different spaces. Curiously enough, each conditionally positive definite function Phi does not only define an interpolation ....

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Duchon, J., Sur L'erreur d'interpolation des fonctions de plusieurs variables par les D m \Gamma splines, RAIRO Anal. Num. 12 (1978), 325-334.

....singularity is certainly not integrable in general. One way of resolving this problem is to consider pseudo functions as defined in Schwartz [20] This in turn would involve us in the notion of the finite part of a divergent integral as defined by Hadamard [6] This is the approach taken by Duchon [2, 3]. Other approaches via analytic continuation or the theory of homogeneous distributions are also possible, see [8] However, it is possible to avoid these technicalities by a rather elegant trick. This trick will actually yield improvements in the theory, as we shall eventually see. The following ....

....which parallels that of Madych and Nelson [11] We adhere pretty closely to their development and notation, so as to make for an easy comparison. However, we would emphasize that better results (in the sense that the hypotheses are less restrictive) can be obtained by following arguments of Duchon [2], Madych and Nelson [12] or Light and Wayne [10] Theorem 4.5 Let w be a weight function satisfying Assumptions (A3.1) A3.4) Suppose a 1 ; am 2 IR n are so ordered that a 1 ; a is a unisolvent set with respect to k Gamma1 . Let p 1 ; p 2 k Gamma1 be such ....

Duchon J. Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D m -- splines, RAIRO Analyse Numerique 12 (4), 1978, 325--334.

No context found.

Duchon, J. (1978): Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D -splines, R.A.I.R.O. An. Num. 12, no. 4, 325--334.

No context found.

Duchon, J. (1978): Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D m -splines, R.A.I.R.O. An. Num. 12, no. 4, 325--334.

No context found.

J. Duchon, Sur l'erreur d'interpolation des fonctions de plusiers variables par les D -splines, RAIRO. Anal. Num'er. 12 (1978), 325--334.

No context found.

Duchon, J., Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D -splines, R.A.I.R.O. Analyse num'erique 12 no. 4 (1978), 325--334.

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D2. Duchon, J. (1978), Sur l'erreur d'interpolation des fonctions de plusieur variables par les D m - splines, RAIRO Analyse Numerique 12, 325--334.

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Du2. Duchon, J., Sur l'erreur d'interpolation des fonctions de plusieur variables par les D m -splines, RAIRO Analyse Numerique 12 (1978), 325--334.

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Duchon, J. (1978): Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D m -splines. RAIRO Analyse Numerique 12, 325--334

No context found.

Duchon J., Sur l'erreur d'interpolation des fonctions de plusieurs variables par les D m --splines, RAIRO Analyse Numerique 12 (4), 1978, 325--334.

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