W. Light and H. Wayne, On power functions and error estimates for radial basis function interpolation, J. Approx. Theory 92 (1998), 245--266. Home/Search Document Details and Download
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On the Accuracy of Surface Spline Approximation and Interpolation .. - Bejancu (2000) (Correct)
....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 domain Omega Gamma Further, Matveev [17] and Bejancu [2, 3] proved that the decay of the error as h 0 is significantly ....
....Using the change of variables v = ht in (3.6) we find P 2 h (x) h fl Z R d j Theta x (h Gamma1 v)j 2 kvk Gammad Gammafl dv : 3. 11) Thus the cone condition on Omega implies the existence of h 0 0 such that the following estimate holds (cf. Wu and Schaback [27] Light and Wayne [13]) max x2 Omega P 2 h (x) const(fl; Omega Gamma h fl ; 8 h h 0 : 3.12) Note that, if fl 2 (0; 2) then (3.12) can be established without assuming the cone condition for Omega Gamma as demonstrated by the author in [3, Section 5.3] The last two displays imply max x2 Omega Z R d ....
Light, W., Wayne, H. (1998) On power functions and error estimates for radial basis function interpolation. J. Approx. Theory 92, 245--266
Native Hilbert Spaces for Radial Basis Functions I - Schaback (1997) (8 citations) (Correct)
.... Gamma Pi P ) f) M X j=1 j f(x j ) Gamma q X k=1 p k (x j )f( k ) M X j=1 j f(x j ) Gamma q X k=1 f( k ) M X j=1 j p k (x j ) and since the sets X = fx 1 ; xM g and Xi are disjoint, the coefficients must vanish. There is an easy possibility used in [9] to go over from here to a fully positive definite case. Using an early idea from Golomb and Weinberger [4] we form a new function K : Omega Theta Omega IR by K(x; y) Psi(x; y) q X j=1 p j (x)p k (y) 25) and a new inner product (f; g) Phi : q X j=1 f( j )g( j ) f Gamma ....
Light, W., and H. Wayne, On power functions and error estimates for radial basis function approximation, J. Approx. Theory 92 (1998) 245-- 266
Optimal Approximation Orders in L p for Radial Basis Functions - Wendland (1998) (3 citations) (Correct)
....(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 ....
....spline or Phi is in addition to (12) in 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 ; ....
Light, W. and Wayne, H., On power functions and error estimates for radial basis function interpolation, J. of Approx. Theory 92 (1998), 245-266.
Inverse and Saturation Theorems for Radial Basis Function.. - Schaback, Wendland (1998) (4 citations) (Correct)
....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 ....
Light, W. and Wayne, H., On power functions and error estimates for radial basis function interpolation, J. of Approx. Theory 92 (1998), 245-266.
Spaces of Distributions and Interpolation by Translates of a.. - Light, Wayne (1997) Self-citation (Light Wayne) (Correct)
....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 that p s (a j ) 1 if s = j and is zero ....
Light, W.A. and H.S.J. Wayne, On power functions and error estimates for radial basis function interpolation, J. Approx. Th. (to appear).
Optimal Approximation Orders In L_p For Radial Basis Functions - Wendland (2000) (Correct)
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W. Light and H. Wayne, On power functions and error estimates for radial basis function interpolation, J. Approx. Theory 92 (1998), 245--266.
Native Hilbert Spaces for Radial Basis Functions II - Schaback (1999) (3 citations) (Correct)
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Light, W.L., H. Wayne, On power functions and error estimates for radial basis function interpolation, J. Approx. Theory 92 (1998), 245--267.
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