17 research outputs found
Strictly positive definite kernels on the -sphere: beyond radial symmetry
The paper introduces a new characterisation of strictly positive definiteness
for kernels on the 2-sphere without assuming the kernel to be radially
(isotropic) or axially symmetric. The results use the series expansion of the
kernel in spherical harmonics. Then additional sufficient conditions are proven
for kernels with a block structure of expansion coefficients. These generalise
the result derived by Chen et al. 2003 for radial kernels to non-radial
kernels
Advances in radial and spherical basis function interpolation
The radial basis function method is a widely used technique for interpolation of scattered data. The method is meshfree, easy to implement independently of the number of dimensions, and for certain types of basis functions it provides spectral accuracy. All these properties also apply to the spherical basis function method, but the class of applicable basis functions, positive definite functions on the sphere, is not as well studied and understood as the radial basis functions for the Euclidean space. The aim of this thesis is mainly to introduce new techniques for construction of Euclidean basis functions and to establish new criteria for positive definiteness of functions on spheres.
We study multiply and completely monotone functions, which are important for radial basis function interpolation because their monotonicity properties are in some cases necessary and in some cases sufficient for the positive definiteness of a function. We enhance many results which were originally stated for completely monotone functions to the bigger class of multiply monotone functions and use those to derive new radial basis functions. Further, we study the connection of monotonicity properties and positive definiteness of spherical basis functions. In the processes several new sufficient and some new necessary conditions for positive definiteness of spherical radial functions are proven. We also describe different techniques of constructing new radial and spherical basis functions, for example shifts. For the shifted versions in the Euclidean space we prove conditions for positive definiteness, compute their Fourier transform and give integral representations. Furthermore, we prove that the cosine transforms of multiply monotone functions are positive definite under some mild extra conditions. Additionally, a new class of radial basis functions which is derived as the Fourier transforms of the generalised
Gaussian φ(t) = e−tβ is investigated.
We conclude with a comparison of the spherical basis functions, which we derived in this thesis and those spherical basis functions well known. For this numerical test a set of test functions as well as recordings of electroencephalographic data are used to evaluate the performance of the different basis functions
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On radial basis functions
Many sciences and other areas of research and applications from engineering to economics require the approximation of functions that depend on many variables. This can be for a variety of reasons. Sometimes we have a discrete set of data points and we want to find an approximating function that completes this data; another possibility is that precise functions are either not known or it would take too long to compute them explicitly. In this snapshot we want to introduce a particular method of approximation which uses functions called radial basis functions. This method is particularly useful when approximating functions that depend on very many variables. We describe the basic approach to approximation with radial basis functions, including their computation, give several examples of such functions and show some applications
-summability and Fourier series of B-splines with respect to their knots
We study the -summability of functions in the -dimensional torus
and so-called -invariant functions. Those are functions
on the torus whose Fourier coefficients depend only on the -norm of
their indices. Such functions are characterized as divided differences that
have as knots for . It leads us to consider the -dimensional
Fourier series of univariate B-splines with respect to its knots, which turns
out to enjoy a simple bi-orthogonality that can be used to obtain an orthogonal
series of the B-spline function
New methods for quasi-interpolation approximations: resolution of odd-degree singularities
In this paper, we study functional approximations where we choose the
so-called radial basis function method and more specifically,
quasi-interpolation. From the various available approaches to the latter, we
form new quasi-Lagrange functions when the orders of the singularities of the
radial function's Fourier transforms at zero do not match the parity of the
dimension of the space, and therefore new expansions and coefficients are
needed to overcome this problem. We develop explicit constructions of infinite
Fourier expansions that provide these coefficients and make an extensive
comparison of the approximation qualities and - with a particular focus -
polynomial precision and uniform approximation order of the various formulae.
One of the interesting observations concerns the link between algebraic
conditions of expansion coefficients and analytic properties of localness and
convergence
Strictly positive definite non-isotropic kernels on two-point homogeneous manifolds : the asymptotic approach
We present sufficient conditions for a family of positive definite kernels on a compact two-point homogeneous space to be strictly positive definite based on their expansion in eigenfunctions of the Laplace–Beltrami operator. We also present a characterisation of this kernel class. The family analyzed is a generalization of the isotropic kernels and the case of a real sphere is analyzed in details