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2.
M. D. Buhmann 《Advances in Computational Mathematics》2006,24(1-4):57-80
This paper concerns the interpolation with radial basis functions on half-spaces, where the centres are multi-integers restricted
to half-spaces as well. The existence of suitable Lagrange functions is shown for multiquadrics and inverse multiquadrics
radial basis functions, as well as the decay rate and summability of its coefficients. The main technique is a so-called Wiener–Hopf
factorisation of the symbol of the radial basis function and the careful study of the smoothness of its 2π-periodic factors.
Dedicated to Charles A. Micchelli on his 60th Birthday
Mathematics subject classifications (2000) 41A05, 41A15, 41A63, 47A68, 65D05, 65D07. 相似文献
3.
In this paper, we study approximation by radial basis functions including Gaussian, multiquadric, and thin plate spline functions, and derive order of approximation under certain conditions. Moreover, neural networks are also constructed by wavelet recovery formula and wavelet frames. 相似文献
4.
Fornberg Bengt; Flyer Natasha; Hovde Susan; Piret Cecile 《IMA Journal of Numerical Analysis》2008,28(1):121-142
5.
R. A. Brownlee 《Numerical Algorithms》2005,39(1-3):57-68
The error between appropriately smooth functions and their radial basis function interpolants, as the interpolation points fill out a bounded domain in Rd, is a well studied artifact. In all of these cases, the analysis takes place in a natural function space dictated by the choice of radial basis function – the native space. The native space contains functions possessing a certain amount of smoothness. This paper establishes error estimates when the function being interpolated is conspicuously rough.
AMS subject classification 41A05, 41A25, 41A30, 41A63R.A. Brownlee: Supported by a studentship from the Engineering and Physical Sciences Research Council. 相似文献
6.
研究了球面径向基插值对球面函数的逼近问题,给出了一致逼近的上界估计式.文中结果说明,球面径向基插值的逼近阶会随函数光滑性的提高而增加. 相似文献
7.
D. Costarelli & R. Spigler 《分析论及其应用》2013,29(2):169-196
In this paper, a constructive theory is developed for approximating functions of one or more variables by superposition of sigmoidal functions. This is done in the uniform norm as well as in the $L^p$ norm. Results for the simultaneous approximation, with the same order of accuracy, of a function and its derivatives (whenever these exist), are obtained. The relation with neural networks and radial basis functions approximations is discussed. Numerical examples are given for the purpose of illustration. 相似文献
8.
Edward J. Fuselier 《Advances in Computational Mathematics》2008,29(3):311-313
In this paper we derive several new results involving matrix-valued radial basis functions (RBFs). We begin by introducing
a class of matrix-valued RBFs which can be used to construct interpolants that are curl-free. Next, we offer a characterization
of the native space for divergence-free and curl-free kernels based on the Fourier transform. Finally, we investigate the
stability of the interpolation matrix for both the divergence-free and curl-free cases, and when the kernel has finite smoothness
we obtain sharp estimates.
An erratum to this article can be found at 相似文献
9.
In this paper we investigate the approximation of a class of parabolic partial differential equations on the unit spheres SnRn+1 using spherical basis functions. Error estimates in the Sobolev norm are derived.
The results presented in this paper are taken from the authors Ph.D. dissertation under supervision of Professor J.D. Ward and Professor F.J. Narcowich at Texas A&M University.AMS subject classification 35K05, 65M70, 46E22 相似文献
10.
Gregory E. Fasshauer 《Advances in Computational Mathematics》1999,10(1):81-96
We show how conditionally negative definite functions on spheres coupled with strictly completely monotone functions (or functions whose derivative is strictly completely monotone) can be used for Hermite interpolation. The classes of functions thus obtained have the advantage over the strictly positive definite functions studied in [17] that closed form representations (as opposed to series expansions) are readily available. Furthermore, our functions include the historically significant spherical multiquadrics. Numerical results are also presented. This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
11.
Pointwise error estimates for approximation on compact homogeneous manifolds using radial kernels are presented. For a positive definite kernel κ the pointwise error at x for interpolation by translates of κ goes to 0 like ρ
r
, where ρ is the density of the interpolating set on a fixed neighbourhood of x. Tangent space techniques are used to lift the problem from the manifold to Euclidean space, where methods for proving such
error estimates are well established.
Partially supported by NSF Grant DMS-9972004. 相似文献
12.
Raymond Mortini 《Monatshefte für Mathematik》2009,158(1):81-95
We characterize those sequences (x
n
) in the spectrum of H
∞ whose Nevanlinna–Pick interpolation problems admit thin Blaschke products as solutions. We also study under which conditions
there is a Blaschke product B with prescribed zero-set distribution and solving problems of the form B(x) = f
n
(x) for every x ∈ P(x
n
), where P(x
n
) is the Gleason part associated with the point x
n
and where (f
n
) is an arbitrary sequence of functions in the unit ball of H
∞. As a corollary we get a new characterization of Carleson–Newman Blaschke products in terms of bounded universal functions,
a result first proved by Gallardo and Gorkin.
相似文献
13.
In this paper, we relate several questions about cutting planes to a fundamental problem in the geometry of numbers, namely,
the closest vector problem. Using this connection we show that the dominance, membership and validity problems are NP-complete
for Chvátal and split cuts.
Received: August 28, 2001 / Accepted: March 2002?Published online May 8, 2002 相似文献
14.
Radial basis functions (RBFs) have found important applications in areas such as signal processing, medical imaging, and neural networks since the early 1980s. Several applications require that certain physical properties are satisfied by the interpolant, for example, being divergence-free in case of incompressible data. In this paper we consider a class of customized (e.g., divergence-free) RBFs that are matrix-valued and have compact support; these are matrix-valued analogues of the well-known Wendland functions. We obtain stability estimates for a wide class of interpolants based on matrix-valued RBFs, also taking into account the size of the compact support of the generating RBF. We conclude with an application based on an incompressible Navier–Stokes equation, namely the driven-cavity problem, where we use divergence-free RBFs to solve the underlying partial differential equation numerically. We discuss the impact of the size of the support of the basis function on the stability of the solution.
AMS subject classification 65D05 相似文献
15.
Sonia Berrios 《Journal of Mathematical Analysis and Applications》2008,337(1):556-575
Let E and F be complex Banach spaces, and let U be an open ball in E. We show that if E has a shrinking and unconditional basis, then every holomorphic function that is weakly continuous on U-bounded sets is weakly uniformly continuous on U-bounded sets. 相似文献
16.
The aim of this paper is to define the Besov–Morrey spaces and the Triebel– Lizorkin–Morrey spaces and to present a decomposition
of functions belonging to these spaces. Our results contain an answer to the conjecture proposed by Mazzucato.
The first author is supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
The second author is supported by Fūjyukai foundation and the 21st century COE program at Graduate School of Mathematical
Sciences, the University of Tokyo. 相似文献
17.
The cut polytope of a graph arises in many fields. Although much is known about facets of the cut polytope of the complete
graph, very little is known for general graphs. The study of Bell inequalities in quantum information science requires knowledge
of the facets of the cut polytope of the complete bipartite graph or, more generally, the complete k-partite graph. Lifting is a central tool to prove certain inequalities are facet inducing for the cut polytope. In this paper
we introduce a lifting operation, named triangular elimination, applicable to the cut polytope of a wide range of graphs.
Triangular elimination is a specific combination of zero-lifting and Fourier–Motzkin elimination using the triangle inequality.
We prove sufficient conditions for the triangular elimination of facet inducing inequalities to be facet inducing. The proof
is based on a variation of the lifting lemma adapted to general graphs. The result can be used to derive facet inducing inequalities
of the cut polytope of various graphs from those of the complete graph. We also investigate the symmetry of facet inducing
inequalities of the cut polytope of the complete bipartite graph derived by triangular elimination.
相似文献
18.
S. Norvidas 《Lithuanian Mathematical Journal》2009,49(2):185-189
For a compact set K in ℝ
n
, let B
2
K
be the set of all functions f ∈ L
2(ℝ2) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B
2
K
by finite exponential sums
in the space , as τ → ∞. 相似文献
19.
Richard K. Gordon W. Elliott Hutchcraft 《Numerical Methods for Partial Differential Equations》2010,26(4):945-956
This article discusses the use of radial basis functions in the solution of partial differential equations. In particular, an investigation of how to implement boundary conditions at material interfaces is presented. A comparison is made of solutions obtained using only radial basis functions with those obtained using either jump functions or elliptical functions in conjunction with standard radial basis functions. The purpose of adding either jump functions or elliptical functions is to alleviate the inaccurate handling of the boundary conditions at material interfaces. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010 相似文献
20.
Radial Basis Functions (RBF) have found a wide area of applications. We consider the case of polyharmonic RBF (called sometimes polyharmonic splines) where the data are on special grids of the form having practical importance. The main purpose of the paper is to consider the behavior of the polyharmonic interpolation splines on such grids for the limiting process 0.$"> For a large class of data functions defined on it turns out that there exists a limit function This limit function is shown to be a polyspline of order on strips. By the theory of polysplines we know that the function is smooth up to order everywhere (in particular, they are smooth on the hyperplanes , which includes existence of the normal derivatives up to order while the RBF interpolants are smooth only up to the order The last fact has important consequences for the data smoothing practice.