Wendland, H. (1997): Sobolev-type error estimates for interpolation by radial basis functions. In: A. Le Mehaute, C. Rabut, L.L. Schumaker, eds., Surface Fitting and Multiresolution Methods, Vanderbilt Univ. Press, pp. 337--344.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 been handled by Wendland [12]. We expect that the theorem remains true in case d=2 bd=2c 1, but our techniques are unable to cope with this case. Without the restriction m , our error analysis breaks down. However, the case m can be salvaged if one employs T ; h ;k f instead of T ; k f , where h : h) ....

H. Wendland, Sobolev-type error estimates for interpolation by radial basis functions, Surface Fitting and Multiresolution Methods (A. LeMehaute, C. Rabut, and L.L. Schumaker, eds.), Vanderbilt University Press, Nashville, Tenessee USA, 1997, pp. 337-344.

....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 eventually generalized by Wu and Schaback [21] and Wendland [20] to apply to a large family of radial basis function interpolation methods. At the same time, there were e orts to understand the special case when and = hZ . Although this special case is quite di erent from the desired setup, it was a tempting case because it falls in line with the very ....

H. Wendland, Sobolev-type error estimates for interpolation by radial basis functions, Surface Fitting and Multiresolution Methods (A. LeMehaute, C. Rabut, and L.L. Schumaker, eds.), Vanderbilt University Press, Nashville, Tenessee USA, 1997, pp. 337-344.

.... technique works for norms that can be localized properly, and this was clearly pointed out in the recent paper [4] In particular, it works for spaces that are norm equivalent to Sobolev spaces, and this covers the case of Wendland s compactly supported unconditionally positive definite functions [10]. Using s f Gammas f = 0; one can replace kf 00 k 2 by kf 00 Gamma s 00 f k 2 in the right hand side and combine the above inequalities into kf 00 Gamma s 00 f k 2 2 h 2 p 90 kf 00 Gamma s 00 f k 2 kf (4) k 2 kf 00 Gamma s 00 f k 2 h 2 p 90 kf (4) k 2 kf ....

Wendland, H., Sobolev--type error estimates for interpolation by radial basis functions, in: Curves and Surfaces in Geometric Design, A. Le M'ehaut'e and C. Rabut and L.L. Schumaker (eds.), Vanderbilt University Press, Nashville, TN, 1997

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H. Wendland, Sobolev-type error estimates for interpolation by radial basis functions, pp. 337--344. In: A. LeM'ehaut'e, C. Rabut and L. L. Schumaker (eds.), Surface Fitting and Multiresolution Methods, Vanderbilt University Press, Nashville, TN, 1997.

....is of quite some importance. Error bounds normally are rst derived for functions in the native space, and are then moved over to other spaces [48] The generalized RBF interpolation setting can be proven to be optimal in at least three dioeerent aspects, and therefore the attained error bounds ([57] for Wendland s functions in Sobolev spaces) are optimal with respect to all other linear recovery processes based on the same data and working in the same space of functions. We cannot say much more here, and we refer the reader to survey articles on RBF theory [9, 43, 44, 47, 50, 52] 4 ....

H. Wendland, Sobolev-type error estimates for interpolation by radial basis functions, in: Surface Fitting and Multiresolution Methods, A. LeM#haut#, C. Rabut, L.L. Schumaker, eds., Vanderbilt University Press, Nashville, pp. 337-344, 1997.

....(x) CF (h) For the basis functions we especially investigate, the order of conditional positive definiteness and F (h) are given in table 1. As a reference for the Sobolev splines we give [4] The results for the compactly supported radial basis functions and explicit formulas can be found in [13, 14]. The degree of the polynomial is minimal under the following conditions: 1) Phi(x) OE(kxk 2 ) is a compactly supported function which consists of a univariate polynomial within its support. 2) The function Phi is positive definite on R d and the even extension of OE is in C 2 . 4. ....

....of x and X. Now, we form u B : EB (E Omega ujB) 2 W k 2 (R d ) for a ball B R d . It is possible to choose the extension mapping 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 ....

H. Wendland, Sobolev-type error estimates for interpolation by radial basis functions, in: Surface Fitting and Multiresolution Methods, A. LeM'ehaut'e, C. Rabut, L. L. Schumaker, eds., Vanderbilt University Press, Nashville, 1997, pp 337-344.

....(4) where u j are the Lagrange functions from spanf Phi( Delta Gamma x 1 ) Phi( Delta Gamma xN )g P d m . They satisfy not only u j (x k ) ffi jk , they also reproduce polynomials up to a degree less than m, i.e. N X j=1 p(x j )u j (x) p(x) p 2 P d m : 5) Several papers [3, 4, 5, 7, 10] deal with the accuracy of this interpolation process. They provide error bounds on the interpolation error in terms of the fill distance, whenever the function f comes from the native space. In this paper we want to show that these bounds cannot be improved without further conditions on the ....

....L 1 (R d ) then there exist bounds on the L p error of the form kf Gamma s f;X k Lp ch s1=2 d=p X jf j Nm; Phi ( Omega Gamma ; 2 p 1; 13) and kf Gamma s f;X k Lp ( Omega Gamma ch s1=2 d=2 X jf j Nm; Phi ( Omega Gamma ; 1 p 2; 14) c. f. 3] for the results on thin plate splines and [10] for Phi 2 L 1 (R d ) We shall show that the orders given in (13) cannot be improved for functions from the native space. To do this, we need a stability result on the interpolation process, which we cite from [6] Lemma 4 .1 For X = fx 1 ; xN g Omega let fl X be the largest number ....

Wendland, H., Sobolev-type error estimates for interpolation by radial basis functions, in: A. LeM'ehaut'e, C. Rabut, and L.L Schumaker (eds.), Surface Fitting and Multiresolution Methods, 337-344, Vanderbilt University Press, Nashville, TN, 1997.

....s f;X k L1 ( Omega ) Ch s1=2 kfk Phi (3.7) for f 2 G Omega ; Phi . Inverse Theorems for RBF 5 Actually, in [14] the theorem is stated in a more localized version, but the proof holds true in this situation. There are several other papers giving error bounds of this form, some of them are [1, 3, 5, 12]. Next, we need a stability result on the interpolation process. Therefore, we define the separation distance q X : 1 2 min j 6=k kx j Gamma x k k 2 and cite from [9] Theorem 3.3. Let Phi 2 cpd(m) satisfy the decay condition (2.5) For X = fx 1 ; xN g Omega denote by AX; Phi the ....

Wendland, H., Sobolev-type error estimates for interpolation by radial basis functions, in: A. LeM'ehaut'e, C. Rabut, and L.L Schumaker (eds.), Surface Fitting and Multiresolution Methods , pp 337-344, Vanderbilt University Press, Nashville, TN, 1997.

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Wendland, H. (1997): Sobolev-type error estimates for interpolation by radial basis functions. In: A. Le Mehaute, C. Rabut, L.L. Schumaker, eds., Surface Fitting and Multiresolution Methods, Vanderbilt Univ. Press, pp. 337--344.

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