Polynomial approximation on the sphere using scattered data

We consider the problem of approximately reconstructing a function f defined on the surface of the unit sphere in the Euclidean space ℝ^{q +1} by using samples of f at scattered sites. A central role is played by the construction of a new operator for polynomial approximation, which is a uniformly bounded quasi‐projection in the de la Vallée Poussin style, i.e. it reproduces spherical polynomials up to a certain degree and has uniformly bounded L^p operator norm for 1 ≤ p ≤ ∞. Using certain positive quadrature rules for scattered sites due to Mhaskar, Narcowich and Ward, we discretize this operator obtaining a polynomial approximation of the target function which can be computed from scattered data and provides the same approximation degree of the best polynomial approximation. To establish the error estimates we use Marcinkiewicz–Zygmund inequalities, which we derive from our continuous approximating operator. We give concrete bounds for all constants in the Marcinkiewicz–Zygmund inequalities as well as in the error estimates. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fast and accurate interpolation of large scattered data sets on the sphere

In this paper a new efficient algorithm for spherical interpolation of large scattered data sets is presented. The solution method is local and involves a modified spherical Shepard’s interpolant, which uses zonal basis functions as local approximants. The associated algorithm is implemented and optimized by applying a nearest neighbour searching procedure on the sphere. Specifically, this technique is mainly based on the partition of the sphere in a suitable number of spherical zones, the construction of spherical caps as local neighbourhoods for each node, and finally the employment of a spherical zone searching procedure. Computational cost and storage requirements of the spherical algorithm are analyzed. Moreover, several numerical results show the good accuracy of the method and the high efficiency of the proposed algorithm.     

Approximation properties of zonal function networks using scattered data on the sphere

A zonal function (ZF) network is a function of the form x↦∑ _k=1 ⁿ c _k (x · y _k), where x and the y _k's are on the unit sphere in q+1 dimensional Euclidean space, and where the y _k's are scattered points. In this paper, we study the degree of approximation by ZF networks. In particular, we compare this degree of approximation with that obtained with the classical spherical harmonics. In many cases of interest, this is the best possible for a given amount of information regarding the target function. We also discuss the construction of ZF networks using scattered data. Our networks require no training in the traditional sense, and provide theoretically predictable rates of approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

An analysis of wavelet frame based scattered data reconstruction

In real world applications many signals contain singularities, like edges in images. Recent wavelet frame based approaches were successfully applied to reconstruct scattered data from such functions while preserving these features. In this paper we present a recent approach which determines the approximant from shift invariant subspaces by minimizing an ${?}_1$-regularized least squares problem which makes additional use of the wavelet frame transform in order to preserve sharp edges. We give a detailed analysis of this approach, i.e., how the approximation error behaves dependent on data density and noise level. Moreover, a link to wavelet frame based image restoration models is established and the convergence of these models is analyzed. In the end, we present some numerical examples, for instance how to apply this approach to handle coarse-grained models in molecular dynamics.     

Hermite-Birkhoff interpolation of scattered data by radial basis functions

For Hermite-Birkhoff interpolation of scattered multidimensional data by radial basis function φ, existence and characterization theorems and a variational principle are proved. Examples include φ(r)=r^b, Duchon’s thin-plate splines, Hardy’s multiquadrics, and inverse multiquadrics.     

Approximate approximations from scattered data

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe an application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.     

Least squares surface approximation to scattered data using multiquadratic functions

The paper documents an investigation into some methods for fitting surfaces to scattered data. The form of the fitting function is a multiquadratic function with the criteria for the fit being the least mean squared residual for the data points. The principal problem is the selection of knot points (or base points for the multiquadratic basis functions), although the selection of the multiquadric parameter also plays a nontrivial role in the process. We first describe a greedy algorithm for knot selection, and this procedure is used as an initial step in what follows. The minimization including knot locations and the multiquadric parameter is explored, with some unexpected results in terms of “near repeated” knots. This phenomenon is explored, and leads us to consider variable parameter values for the basis functions. Examples and results are given throughout.     

On the least squares fit by radial functions to multidimensional scattered data

Summary This paper investigates some aspects of discrete least squares approximation by translates of certain classes of radial functions. Its specific aims are (i) to provide conditions under which the associated least squares matrix is invertible and (ii) to give upper bounds for the Euclidean norms of the inverses of these matrices (when they exist).The second named author was supported by the National Science Foundation under grant number DMS-8901345     

Improved error bounds for scattered data interpolation by radial basis functions

If additional smoothness requirements and boundary conditions are met, the well-known approximation orders of scattered data interpolants by radial functions can roughly be doubled.

     

Interpolation of scattered data: Distance matrices and conditionally positive definite functions

Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R. Franke.     

On approximation of functions on sphere

Let f be an integrable function on the unit sphere Σ _n−1 of R ⁿ (n⩾3) and let σ _N ^δ be the Cesàro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that and where ω(f,t)_p is the 1st-order modulus of continuity of f in L^p-metric which is defined in a way different than in the classical case of n=2. In Memory of Professor M. T. Cheng Project supported by the NSF of China under the grans # 19771009.     

On approximation of functions on sphere

Let f be an integrable function on the unit sphere Σ _n−1 of R ⁿ (n⩾3) and let σ _N ^δ be the Cesàro means of order σ of the Fourier-Laplace series of f. The special value λ:=n−2/2 of σ is known as the critical index. This paper proves that      

Some constructive properties of functions defined on the sphere

The norm of difference operators for functions defined on the sphere is investigated. A mistake in Rustamov's results is pointed out by a counterexample. And correct result is given. Results of convergence in norm of difference operators acting on the differentiable functions on the sphere are obtained. The highest possible degree tending to zero for the moduli of continuity of fractional order for non-constant valued functions are discussed. The work is supported by NSF of China, Grant No. 19771009.     

Some constructive properties of functions defined on the sphere

The norm of difference operators for functions defined on the sphere is investigated. A mistake in Rustamov's results is pointed out by a counterexample. And correct result is given. Results of convergence in norm of difference operators acting on the differentiable functions on the sphere are obtained. The highest possible degree tending to zero for the moduli of continuity of fractional order for non-constant valued functions are discussed.     

Ultradistributional boundary values of harmonic functions on the sphere

We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderón and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.     

Detecting and approximating fault lines from randomly scattered data

Discretely defined surfaces that exhibit vertical displacements across unknown fault lines can be difficult to approximate accurately unless a representation of the faults is known. Accurate representations of these faults enable the construction of constrained approximation models that can successfully overcome common problems such as over-smoothing. In this paper we review an existing method for detecting fault lines and present a new detection approach based on data triangulations and discrete Gaussian curvature (DGC). Furthermore, we show that if the fault line can be described non-parametrically, then accurate support vector machine (SVM) models can be constructed that are independent of the type of triangulation used in the detection algorithms. We shall also see that SVM models are particularly effective when the data produced by the detection algorithms are noisy. We compare the performances of the various new and established models.     

Continuous valuations on the space of Lipschitz functions on the sphere

We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere $S^{n?1}$. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere.     

Constructing prolate spheroidal quaternion wave functions on the sphere

Over the last years, considerable attention has been paid to the role of the prolate spheroidal wave functions (PSWFs) introduced in the early sixties by D. Slepian and H.O. Pollak to many practical signal and image processing problems. The PSWFs and their applications to wave phenomena modeling, fluid dynamics, and filter design played a key role in this development. In this paper, we introduce the prolate spheroidal quaternion wave functions (PSQWFs), which refine and extend the PSWFs. The PSQWFs are ideally suited to study certain questions regarding the relationship between quaternionic functions and their Fourier transforms. We show that the PSQWFs are orthogonal and complete over two different intervals: the space of square integrable functions over a finite interval and the three‐dimensional Paley–Wiener space of bandlimited functions. No other system of classical generalized orthogonal functions is known to possess this unique property. We illustrate how to apply the PSQWFs for the quaternionic Fourier transform to analyze Slepian's energy concentration problem. We address all of the aforementioned and explore some basic facts of the arising quaternionic function theory. We conclude the paper by computing the PSQWFs restricted in frequency to the unit sphere. The representation of these functions in terms of generalized spherical harmonics is explicitly given, from which several fundamental properties can be derived. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Generalization bounds for function approximation from scattered noisy data

We consider the problem of approximating functions from scattered data using linear superpositions of non-linearly parameterized functions. We show how the total error (generalization error) can be decomposed into two parts: an approximation part that is due to the finite number of parameters of the approximation scheme used; and an estimation part that is due to the finite number of data available. We bound each of these two parts under certain assumptions and prove a general bound for a class of approximation schemes that include radial basis functions and multilayer perceptrons. This revised version was published online in June 2006 with corrections to the Cover Date.