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1.
Let Δ be a triangulation of some polygonal domain Ω ⊂ R2 and let Sqr(Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S
q
r
(Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh
q
+1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices.
Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for
a superspline subspace of S
q
r
(Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof
is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines
developed in [7] and [18]. 相似文献
2.
M. J. Johnson 《Constructive Approximation》1998,14(3):429-438
The functions φ
m
:=|.|
2m-d
if d is odd, and φ
m
:=|.|
2m-d
\log|.| if d is even, are known as surface splines, and are commonly used in the interpolation or approximation of smooth functions. We show that if one's domain is the unit
ball in R
d
, then the approximation order of the translates of φ
m
is at most m . This is in contrast to the case when the domain is all of R
d
where it is known that the approximation order is exactly 2m .
April 23, 1996. Date revised: May 5, 1997. 相似文献
3.
Integration and approximation in arbitrary dimensions 总被引:13,自引:0,他引:13
We study multivariate integration and approximation for various classes of functions of d variables with arbitrary d. We consider algorithms that use function evaluations as the information about the function. We are mainly interested in
verifying when integration and approximation are tractable and strongly tractable. Tractability means that the minimal number
of function evaluations needed to reduce the initial error by a factor of ɛ is bounded by C(d)ɛ−p
for some exponent p independent of d and some function C(d). Strong tractability means that C(d) can be made independent of d. The ‐exponents of tractability and strong tractability are defined as the smallest powers of ɛ{-1} in these bounds.
We prove that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space
whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure. We obtain bounds on the ‐exponents,
and for some cases we find their exact values. For some weighted Korobov and Sobolev spaces, the strong ‐exponent is the same
as the ‐exponent for d=1, whereas for the third space it is 2.
For approximation we also consider algorithms that use general evaluations given by arbitrary continuous linear functionals
as the information about the function. Our main result is that the ‐exponents are the same for general and function evaluations.
This holds under the assumption that the orthonormal eigenfunctions of the covariance operator have uniformly bounded L∞ norms. This assumption holds for spaces with shift-invariant kernels. Examples of such spaces include weighted Korobov spaces.
For a space with non‐shift‐invariant kernel, we construct the corresponding space with shift-invariant kernel and show that
integration and approximation for the non-shift-invariant kernel are no harder than the corresponding problems with the shift-invariant
kernel. If we apply this construction to a weighted Sobolev space, whose kernel is non-shift-invariant, then we obtain the
corresponding Korobov space. This enables us to derive the results for weighted Sobolev spaces.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
4.
Recently, A. Cohen, R. A. DeVore, P. Petrushev, and H. Xu investigated nonlinear approximation in the space BV (R
2
). They modified the classical adaptive algorithm to solve related extremal problems. In this paper, we further study the
modified adaptive approximation and obtain results on some extremal problems related to the spaces V
σ,p
r
(R
d
) of functions of ``Bounded Variation" and Besov spaces B
α
(R
d
).
November 23, 1998. Date revised: June 25, 1999. Date accepted: September 13, 1999. 相似文献
5.
In this paper we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere S
r−1
⊂ Rr. The hyperinterpolation approximation L
n
ƒ, where ƒ ∈ C(S
r
−1), is derived from the exact L
2 orthogonal projection Π ƒ onto the space P
n
r
(S
r
−1) of spherical polynomials of degree n or less, with the Fourier coefficients approximated by a positive weight quadrature rule that integrates exactly all polynomials
of degree ≤ 2n. We extend to arbitrary r the recent r = 3 result of Sloan and Womersley [9], by proving that under an additional “quadrature regularity” assumption on the quadrature
rule, the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere is O(n
r
/2−1), which is the same as that of the orthogonal projection Πn, and best possible among all linear projections onto P
n
r
(S
r
−1). 相似文献
6.
V. N. Temlyakov 《Constructive Approximation》2000,16(3):399-426
The question of finding an optimal dictionary for nonlinear m -term approximation is studied in this paper. We consider this problem in the periodic multivariate (d variables) case for classes of functions with mixed smoothness. We prove that the well-known dictionary U
d
which consists of trigonometric polynomials (shifts of the Dirichlet kernels) is nearly optimal among orthonormal dictionaries.
Next, it is established that for these classes near-best m -term approximation, with regard to U
d
, can be achieved by simple greedy-type (thresholding-type) algorithms.
The univariate dictionary U is used to construct a dictionary which is optimal among dictionaries with the tensor product structure.
June 22, 1998. Date revised: March 26, 1999. Date accepted: March 22, 1999. 相似文献
7.
Stable locally supported bases are constructed for the spaces \cal S
d
r
(\triangle) of polynomial splines of degree d≥ 3r+2 and smoothness r defined on triangulations \triangle , as well as for various superspline subspaces. In addition, we show that for r≥ 1 , in general, it is impossible to construct bases which are simultaneously stable and locally linearly independent.
February 2, 2000. Date revised: November 27, 2000. Date accepted: March 7, 2001. 相似文献
8.
J. Yoon 《Constructive Approximation》2001,17(2):227-247
A new multivariate approximation scheme on R
d
using scattered translates of the “shifted” surface spline function is developed. The scheme is shown to provide spectral
L
p
-approximation orders with 1 ≤ p ≤ ∞, i.e., approximation orders that depend on the smoothness of the approximands. In addition, it applies to noisy data
as well as noiseless data. A numerical example is presented with a comparison between the new scheme and the surface spline
interpolation method. 相似文献
9.
We prove that a convex functionf ∈ L
p[−1, 1], 0<p<∞, can be approximated by convex polynomials with an error not exceeding Cω
3
ϕ
(f,1/n)p where ω
3
ϕ
(f,·) is the Ditzian-Totik modulus of smoothness of order three off. We are thus filling the gap between previously known estimates involving ω
3
ϕ
(f,1/n)p, and the impossibility of having such estimates involving ω4. We also give similar estimates for the approximation off by convexC
0 andC
1 piecewise quadratics as well as convexC
2 piecewise cubic polynomials.
Communicated by Dietrich Braess 相似文献
10.
YANG Shouzhi & PENG Lizhong Department of Mathematics Shantou University Shantou China LMAM School of Mathematical Sciences Peking University Beijing China 《中国科学A辑(英文版)》2006,49(1):86-97
An algorithm is presented for raising an approximation order of any given orthogonal multiscaling function with the dilation factor a. Let φ(x) = [φ1(x),φ2(x),…,φr(x)]T be an orthogonal multiscaling function with the dilation factor a and the approximation order m. We can construct a new orthogonal multiscaling function φnew(x) = [ φT(x). f3r 1(x),φr 2(x),…,φr s(x)}T with the approximation order m L(L ∈ Z ). In other words, we raise the approximation order of multiscaling function φ(x) by increasing its multiplicity. In addition, we discuss an especial setting. That is, if given an orthogonal multiscaling function φ(x) = [φ1 (x), φ2(x), …, φr(x)]T is symmetric, then the new orthogonal multiscaling function φnew(x) not only raise the approximation order but also preserve symmetry. Finally, some examples are given. 相似文献