The Embedding Theory of Native Spaces

In the theory of radial basis functions, mathematicians use linear combinations of the translates of the radial basis functions as interpolants. The set of these linear combinations is a normed vector space. This space can be completed and become a Hilbert space, called native space, which is of great importance in the last decade. The native space then contains some abstract elements which are not linear combinations of radial basis functions. The meaning of these abstract elements is not fully known. This paper presents some interpretations for the these elements. The native spaces are embedded into some well-known spaces. For example, the Sobolev-space is shown to be a native space. Since many differential equations have solutions in the Sobolev-space, we can therefore approximate the solutions by linear combinations of radial basis functions. Moreover, the famous question of the embedding of the native space into L²() is also solved by the author.     

The Equivalence Theory of Native Spaces

In this paper some equivalence definitions are given for native spaces which were introduced by Madych and Nelson and have become influential in the theory of radial basis functions. The abstract elements in native spaces are interpreted. Moreover, Weinrich and Iske's theories are unified.     

The Equivalence Theory of Native Spaces

In this paper some equivalence definitions are given for native spaces which were introduced by Madych and Nelson and have become influential in the theory of radial basis functions. The abstract elements in native spaces are interpreted. Moreover, Weinrich and Iske's theories are unified.     

Spherical Scattered Data Quasi-interpolation by Gaussian Radial Basis Function

Since the spherical Gaussian radial function is strictly positive definite, the authors use the linear combinations of translations of the Gaussian kernel to interpolate the scattered data on spheres in this article. Seeing that target functions are usually outside the native spaces, and that one has to solve a large scaled system of linear equations to obtain combinatorial coefficients of interpolant functions, the authors first probe into some problems about interpolation with Gaussian radial functions. Then they construct quasiinterpolation operators by Gaussian radial function, and get the degrees of approximation. Moreover, they show the error relations between quasi-interpolation and interpolation when they have the same basis functions. Finally, the authors discuss the construction and approximation of the quasi-interpolant with a local support function.     

Approximation of functions with zero integrals over balls by linear combinations of the Laplace operator eigenfunctions

It is proved that the special linear combinations of Bessel functions are dense in the C ^∞-topology in the space of functions with zero integrals over balls of fixed radii on an arbitrary open domain U ì \mathbbRⁿ U \subset {\mathbb{R}^n} . Some generalizations of this result for solutions of some convolution equations of the form f * T = 0, where T is radial, are obtained. Analogous results for rank-one symmetric spaces of the noncompact type are considered.     

Inclusions in general spaces: Hoelder stability, solution schemes and Ekeland's principle

We present two basic lemmas on exact and approximate solutions of inclusions and equations in general spaces. Its applications involve Ekeland's principle, characterize calmness, lower semicontinuity and the Aubin property of solution sets in some Hoelder-type setting and connect these properties with certain iteration schemes of descent type. In this way, the mentioned stability properties can be directly characterized by convergence of more or less abstract solution procedures. New stability conditions will be derived, too. Our basic models are (multi-) functions on a complete metric space with images in a linear normed space.     

On the structure of function spaces in optimal recovery of point functionals for ENO-schemes by radial basis functions

Summary. Radial basis functions are used in the recovery step of finite volume methods for the numerical solution of conservation laws. Being conditionally positive definite such functions generate optimal recovery splines in the sense of Micchelli and Rivlin in associated native spaces. We analyse the solvability to the recovery problem of point functionals from cell average values with radial basis functions. Furthermore, we characterise the corresponding native function spaces and provide error estimates of the recovery scheme. Finally, we explicitly list the native spaces to a selection of radial basis functions, thin plate splines included, before we provide some numerical examples of our method. Received March 14, 1995     

Error estimates for interpolation of rough data using the scattered shifts of a radial basis function

The error between appropriately smooth functions and their radial basis function interpolants, as the interpolation points fill out a bounded domain in R^d, 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.     

Approximation in rough native spaces by shifts of smooth kernels on spheres

Within the conventional framework of a native space structure, a smooth kernel generates a small native space, and “radial basis functions” stemming from the smooth kernel are intended to approximate only functions from this small native space. Therefore their approximation power is quite limited. Recently, Narcowich et al. (J. Approx. Theory 114 (2002) 70), and Narcowich and Ward (SIAM J. Math. Anal., to appear), respectively, have studied two approaches that have led to the empowerment of smooth radial basis functions in a larger native space. In the approach of [NW], the radial basis function interpolates the target function at some scattered (prescribed) points. In both approaches, approximation power of the smooth radial basis functions is achieved by utilizing spherical polynomials of a (possibly) large degree to form an intermediate approximation between the radial basis approximation and the target function. In this paper, we take a new approach. We embed the smooth radial basis functions in a larger native space generated by a less smooth kernel, and use them to approximate functions from the larger native space. Among other results, we characterize the best approximant with respect to the metric of the larger native space to be the radial basis function that interpolates the target function on a set of finite scattered points after the action of a certain multiplier operator. We also establish the error bounds between the best approximant and the target function.     

Sobolev spaces of symmetric functions and applications

We prove sharp pointwise estimates for functions in the Sobolev spaces of radial functions defined in a ball. As a consequence, we obtain some imbeddings of such Sobolev spaces in weighted Lq-spaces. We also prove similar imbeddings for Sobolev spaces of functions with partial symmetry. Our techniques lead to new Hardy type inequalities. It is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates. We apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the Hénon type under both Dirichlet and Navier boundary conditions. The delicate question of the regularity of these solutions is also established.     

Embeddings of Beppo–Levi spaces in Hölder–Zygmund spaces, and a new method for radial basis function interpolation error estimates

The Beppo–Levi native spaces which arise when using polyharmonic splines to interpolate in many space dimensions are embedded in Hölder–Zygmund spaces. Convergence rates for radial basis function interpolation are inferred in some special cases.     

Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting

In this paper we discuss Sobolev bounds on functions that vanish at scattered points in a bounded, Lipschitz domain that satisfies a uniform interior cone condition. The Sobolev spaces involved may have fractional as well as integer order. We then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions (RBFs). These estimates include situations in which the target function does not belong to the native space of the RBF.

     

Semigroups and abstract Gevrey spaces

Conditions are found under which a closed linear operator A in a Banach space X generates a continuous semigroup in a linear topological space Y which is dense in X. The space Y is an abstract Gevrey space associated with the operator A. This is an abstract setting for some results for hyperbolic systems with data in spaces of Gevrey functions.     

Nontrivial expansions of zero and representation of analytic functions by series of simple fractions

We propose a modification of the previously-known abstract scheme that reduces the problem of expansion of elements of a locally convex space in series over the system of eigenvectors of some linear operator to the question of existence of a nontrivial expansion of zero in this space. We implement this general scheme for the spaces of analytic functions in domains of the extended complex plane and the systems of simple fractions that are the eigenfunctions of the Pommier operator.     

Perturbations of Banach Frames and Atomic Decompositions

Banach frames and atomic decompositions are sequences that have basis-like properties but which need not be bases. In particular, they allow elements of a Banach space to be written as linear combinations of the frame or atomic decomposition elements in a stable manner. In this paper we prove several functional — analytic properties of these decompositions, and show how these properties apply to Gabor and wavelet systems. We first prove that frames and atomic decompositions are stable under small perturbations. This is inspired by corresponding classical perturbation results for bases, including the Paley — Wiener basis stability criteria and the perturbation theorem el kato. We introduce new and weaker conditions which ensure the desired stability. We then prove quality properties of atomic decompositions and consider some consequences for Hilbert frames. Finally, we demonstrate how our results apply in the practical case of Gabor systems in weighted L² spaces. Such systems can form atomic decompositions for L²_w(IR), but cannot form Hilbert frames but L²_w(IR) unless the weight is trivial.     

On hyperbolic summation and hyperbolic moduli of smoothness

This paper deals with the method of hyperbolic summation of tensor product orthogonal spline functions onI ^d. The spaces, defined in terms of the order of the best approximation by the elements of the space spanned by the tensor product functions with indices from a given hyperbolic set, are described both in terms of the coefficients in some basis and as interpolation spaces. Moreover, the hyperbolic modulus of smoothness is studied, and some relations between hyperbolic summation and hyperbolic modulus of smoothness are established.     

Gradient superconvergence on uniform simplicial partitions of polytopes

Superconvergence of the gradient for the linear simplicial finite-elementmethod applied to elliptic equations is a well known featurein one, two, and three space dimensions. In this paper we showthat, in fact, there exists an elegant proof of this featureindependent of the space dimension. As a result, superconvergencefor dimensions four and up is proved simultaneously. The keyingredient will be that we embed the gradients of the continuouspiecewise linear functions into a larger space for which wedescribe an orthonormal basis having some useful symmetry properties.Since gradients and rotations of standard finite-element functionsare in fact the rotation-free and divergence-free elements ofRaviart–Thomas and Nédélec spaces in threedimensions, we expect our results to have applications alsoin those contexts.     

Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants

Recently, error estimates have been made available for divergence-free radial basis function (RBF) interpolants. However, these results are only valid for functions within the associated reproducing kernel Hilbert space (RKHS) of the matrix-valued RBF. Functions within the associated RKHS, also known as the ``native space' of the RBF, can be characterized as vector fields having a specific smoothness, making the native space quite small. In this paper we develop Sobolev-type error estimates when the target function is less smooth than functions in the native space.

     

On the solution of the non-local parabolic partial differential equations via radial basis functions

In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions is considered. The approximate solution is found using the radial basis functions collocation method. There are some difficulties in computing the solution of the time dependent partial differential equations using radial basis functions. If time and space are discretized using radial basis functions, the resulted coefficient matrix will be very ill-conditioned and so the corresponding linear system cannot be solved easily. As an alternative method for solution, we can use finite-difference methods for discretization of time and radial basis functions for discretization of space. Although this method is easy to use but an accurate solution cannot be provided. In this work an efficient collocation method is proposed for solving non-local parabolic partial differential equations using radial basis functions. Numerical results are presented and are compared with some existing methods.