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1.
It is often observed that interpolation based on translates of radial basis functions or non-radial kernels is numerically
unstable due to exceedingly large condition of the kernel matrix. But if stability is assessed in function space without considering
special bases, this paper proves that kernel-based interpolation is stable. Provided that the data are not too wildly scattered,
the L
2 or L
∞ norms of interpolants can be bounded above by discrete ℓ 2 and ℓ ∞ norms of the data. Furthermore, Lagrange basis functions are uniformly bounded and Lebesgue constants grow at most like the
square root of the number of data points. However, this analysis applies only to kernels of limited smoothness. Numerical
examples support our bounds, but also show that the case of infinitely smooth kernels must lead to worse bounds in future
work, while the observed Lebesgue constants for kernels with limited smoothness even seem to be independent of the sample
size and the fill distance. 相似文献
2.
We establish extension theorems for functions in spaces which arise naturally in studying interpolation by radial basic functions. These spaces are akin in some way to the non-integer-valued Sobolev spaces, although they are considerably more general. Such extensions allow us to establish local error estimates in a way which we make precise in the introductory section of our paper. There are many other applications of these fundamental results, including improved Lp error estimates for interpolation by shifts of a single basic function, but these applications have been left to a later paper. 相似文献
3.
In this paper, we obtain the Lebesgue constants for interpolatory ?-splines of third order with uniform nodes, i.e., the norms of interpolation operators from C to C describing the process of interpolation of continuous bounded and continuous periodic functions by ?-splines of third order with uniform nodes on the real line. As a corollary, we obtain exact Lebesgue constants for interpolatory polynomial parabolic splines with uniform nodes. 相似文献
4.
Mixed modulus of smoothness in weighted Lebesgue spaces with Muckenhoupt weights are investigated. Using mixed modulus of smoothness we obtain Potapov type direct and inverse estimates of angular trigonometric approximation of functions in these spaces. Also we obtain equivalences between mixed modulus of smoothness and K-functional and realization functional. Fractional order modulus of smoothness is considered as well. 相似文献
5.
The paper obtains error estimates for approximation by radial basis functions on the sphere. The approximations are generated
by interpolation at scattered points on the sphere. The estimate is given in terms of the appropriate power of the fill distance
for the interpolation points, in a similar manner to the estimates for interpolation in Euclidean space. A fundamental ingredient
of our work is an estimate for the Lebesgue constant associated with certain interpolation processes by spherical harmonics.
These interpolation processes take place in ``spherical caps' whose size is controlled by the fill distance, and the important
aim is to keep the relevant Lebesgue constant bounded. This result seems to us to be of independent interest.
March 27, 1997. Dates revised: March 19, 1998; August 5, 1999. Date accepted: December 15, 1999. 相似文献
6.
Considering the measurable and nonnegative functions ? on the half-axis [0, ∞) such that ?(0) = 0 and ?( t) → ∞ as t → ∞, we study the operators of weak type (?, ?) that map the classes of ?-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ? n. We prove interpolation theorems for the subadditive operators of weak type (? 0, ? 0) bounded in L ∞(? n) and subadditive operators of weak types (? 0, ? 0) and (? 1, ? 1) in L ?(? n ) under some assumptions on the nonnegative and increasing functions ?( x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (? 0, ? 0) bounded from L ∞(? n) to BMO(? n). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces. 相似文献
7.
In the paper [Y. Xu, Lagrange interpolation on Chebyshev points of two variables, J. Approx. Theory 87 (1996) 220–238], the author introduced a set of Chebyshev-like points for polynomial interpolation (by a certain subspace of polynomials) in the square [-1,1] 2, and derived a compact form of the corresponding Lagrange interpolation formula. In [L. Bos, M. Caliari, S. De Marchi, M. Vianello, A numerical study of the Xu polynomial interpolation formula in two variables, Computing 76(3–4) (2005) 311–324], we gave an efficient implementation of the Xu interpolation formula and we studied numerically its Lebesgue constant, giving evidence that it grows like , n being the degree. The aim of the present paper is to provide an analytic proof to show that the Lebesgue constant does have this order of growth. 相似文献
8.
A discrete Fourier analysis on the fundamental domain Ω
d
of the d-dimensional lattice of type A
d
is studied, where Ω 2 is the regular hexagon and Ω 3 is the rhombic dodecahedron, and analogous results on d-dimensional simplex are derived by considering invariant and anti-invariant elements. Our main results include Fourier analysis
in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange
interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation
is shown to be in the order of (log n)
d
. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already
appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature
formulas, which provides only the second known example of such formulas. 相似文献
9.
This paper explores the quality of polynomial interpolation approximations over the sphere S
r−1⊂R
r in the uniform norm, principally for r=3. Reimer [17] has shown there exist fundamental systems for which the norm ‖Λ
n
‖ of the interpolation operator Λ
n
, considered as a map from C(S
r−1) to C(S
r−1), is bounded by d
n
, where d
n
is the dimension of the space of all spherical polynomials of degree at most n. Another bound is d
n
1/2(λavg/λmin )1/2, where λavg and λmin are the average and minimum eigenvalues of a matrix G determined by the fundamental system of interpolation points. For r=3 these bounds are (n+1)2 and (n+1)(λavg/λmin )1/2, respectively. In a different direction, recent work by Sloan and Womersley [24] has shown that for r=3 and under a mild regularity assumption, the norm of the hyperinterpolation operator (which needs more point values than interpolation) is bounded by O(n
1/2), which is optimal among all linear projections. How much can the gap between interpolation and hyperinterpolation be closed? For interpolation the quality of the polynomial interpolant is critically dependent on the choice of interpolation points. Empirical evidence in this paper suggests that for points obtained by maximizing λmin , the growth in ‖Λ
n
‖ is approximately n+1 for n<30. This choice of points also has the effect of reducing the condition number of the linear system to be solved for the interpolation weights. Choosing the points to minimize the norm directly produces fundamental systems for which the norm grows approximately as 0.7n+1.8 for n<30. On the other hand, ‘minimum energy points’, obtained by minimizing the potential energy of a set of (n+1)2 points on S
2, turn out empirically to be very bad as interpolation points. This paper also presents numerical results on uniform errors for approximating a set of test functions, by both interpolation and hyperinterpolation, as well as by non-polynomial interpolation with certain global basis functions. 相似文献
10.
This paper discusses the problem of choosing the Lagrange interpolation points T = ( t0, t1,…, tn) in the interval −1 t 1 to minimize the norm of the error, considered as an operator from the Hardy space H2( R) of analytic functions to the space C[−1, 1]. It is shown that such optimal choices converge for fixed n, as R → ∞, to the zeros of a Chebyshev polynomial. Asymptotic estimates are given for the norm of the error for these optimal interpolations, as n → ∞ for fixed R. These results are then related to the problem of choosing optimal interpolation points with respect to the Eberlein integral. This integral is based on a probability measure over certain classes of analytic functions, and is used to provide an average interpolation error over these classes. The Chebyshev points are seen to be limits of optimal choices in this case also. 相似文献
11.
Summary In recent years, with the attention to the radial-basis function by mathematicians, more and more research is concentrated
on the Gaussian cardinal interpolation. The main purpose of this paper is to discuss the asymptotic behavior of Lebesgue constants
of the Gaussian cardinal interpolation operator ℒ λ from l (ℤ) into L (ℝ), that is, ∥ℒ λ∥ 1. We obtain the strong asymptotic estimate of the Lebesgue constants which improves the results of Riemenschneider and Sivakumar
in [11]. 相似文献
12.
We establish uniform estimates for the weighted Lebesgue constant of Lagrange interpolation for a large class of exponential weights on [-1, 1]. We deduce theorems on uniform convergence of weighted Lagrange interpolation together with rates of convergence. 相似文献
13.
In this paper, we complete our investigations of mean convergence of Lagrange interpolation for fast decaying even and smooth exponential weights on the line. In doing so, we also present a summary of recent related work on the line and [–1,1] by the authors, Szabados, Vertesi, Lubinsky and Matjila. We also emphasize the important and fundamental ideas, applied in our proofs, that were developed by Erds, Turan, Askey, Freud, Nevai, Szabados, Vértesi and their students and collaborators. These methods include forward quadrature estimates, orthogonal expansions, Hilbert transforms, bounds on Lebesgue functions and the uniform boundedness principle. 相似文献
14.
Summary Padé approximants are a frequently used tool for the solution of mathematical problems. One of the main drawbacks of their use for multivariate functions is the calculation of the derivatives of f(x
1, ..., x
p
). Therefore multivariate Newton-Padé approximants are introduced; their computation will only use the value of f at some points. In Sect. 1 we shall repeat the univariate Newton-Padé approximation problem which is a rational Hermite interpolation problem. In Sect. 2 we sketch some problems that can arise when dealing with multivariate interpolation. In Sect. 3 we define multivariate divided differences and prove some lemmas that will be useful tools for the introduction of multivariate Newton-Padé approximants in Sect. 4. A numerical example is given in Sect. 5, together with the proof that for p=1 the classical Newton-Padé approximants for a univariate function are obtained. 相似文献
15.
In general, Banach space-valued Riemann integrable functions defined on [0, 1] (equipped with the Lebesgue measure) need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X* is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31(2), 697-703 (2001)] that L^1[0, 1] has the weak Lebesgue property. 相似文献
16.
Let X be a linear space, and H a Hilbert space. Let
denote a set of n distinct points in X designated by x1, ..., xn (these points are called nodes). It is desired to interpolate arbitrary data on
by a function in the linear span of the n functions, [formula] where yk are n distinct points in X (called knots), Tv are linear maps from X to H, and Fν are some suitable univariate functions. In this paper, we discuss the solvability of this interpolation scheme. For the case in which the nodes and knots coincide, we give a convenient condition which is equivalent to the nonsingularity of the interpolation matrices. We obtain some sufficient conditions for the case in which the nodes and knots do not necessarily coincide. 相似文献
17.
We consider interpolation by spherical harmonics at points on a ( d−1)-dimensional sphere and show that, in the limit, as the points coalesce under an angular scaling, the Lebesgue function of the process converges to that of an associated algebraic interpolation problem for the original angles considered as points in
d−1. 相似文献
18.
Some estimates for the convolution operators with kernels of type (l, r) on Lebesgue spaces with power weights and Herz-type spaces in the setting of homogeneous groups are established. 相似文献
19.
Summary. I derive a posteriori error estimates for two-point boundary value problems and parabolic equations in one dimension based on interpolation error
estimates. The interpolation error estimates are obtained from an extension of the error formula for the Lagrange interpolating
polynomial in the case of symmetrically-spaced interpolation points. From this formula pointwise and seminorm a priori estimates of the interpolation error are derived. The interpolant in conjunction with the a priori estimates is used to obtain asymptotically exact a posteriori error estimates of the interpolation error. These a posteriori error estimates are extended to linear two-point boundary problems and parabolic equations. Computational results demonstrate
the convergence of a posteriori error estimates and their effectiveness when combined with an hp-adaptive code for solving parabolic systems.
Received April 17, 2000 / Revised version received September 25, 2000 / Published online May 30, 2001 相似文献
20.
We study interpolation methods associated to polygons and establish estimates for the norms of interpolated operators. Our results explain the geometrical base of estimates in the literature. Applications to interpolation of weighted Lp-spaces are also given. 相似文献
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