Refinement equations and spline functions

In this paper, we exploit the relation between the regularity of refinable functions with non-integer dilations and the distribution of powers of a fixed number modulo 1, and show the nonexistence of a non-trivial C ^∞ solution of the refinement equation with non-integer dilations. Using this, we extend the results on the refinable splines with non-integer dilations and construct a counterexample to some conjecture concerning the refinable splines with non-integer dilations. Finally, we study the box splines satisfying the refinement equation with non-integer dilation and translations. Our study involves techniques from number theory and harmonic analysis.     

Addition to V. P. Zastavnyi’s paper “Estimates for sums of moduli of blocks in trigonometric Fourier series”

Analogs of scaling relations are constructed for basis exponential splines with uniform knots corresponding to a linear differential operator of arbitrary order with constant coefficients and real pairwise distinct roots of the characteristic polynomial; the construction does not employ techniques from harmonic analysis.     

A spline interpolation method for solving boundary value problems of potential theory from discretely given data

An interpolation procedure using harmonic splines is described and analyzed for solving (exterior) boundary value problems of Laplace's equation in three dimensions (from discretely given data). The theoretical and computational aspects of the method are discussed. Some numerical examples are given.     

A spline interpolation method for solving boundary value problems of potential theory from discretely given data

An interpolation procedure using harmonic splines is described and analyzed for solving (exterior) boundary value problems of Laplace's equation in three dimensions (from discretely given data). The theoretical and computational aspects of the method are discussed. Some numerical examples are given.     

The Finite Element Method on the Sierpinski Gasket

For certain classes of fractal differential equations on the Sierpinski gasket, built using the Kigami Laplacian, we describe how to approximate solutions using the finite element method based on piecewise harmonic or piecewise biharmonic splines. We give theoretical error estimates, and compare these with experimental data obtained using a computer implementation of the method (available at the web site http://mathlab.cit.cornell.edu/\sim gibbons). We also explain some interesting structure concerning the spectrum of the Laplacian that became apparent from the experimental data. March 29, 2000. Date revised: March 6, 2001. Date accepted: March 21, 2001.     

Error Estimates for the Solution of the Radial Schr?dinger Equation by the Rayleigh--Ritz Finite Element Method

Approximate eigenvalues and eigenfunctions are obtained forthe radial Schr?dinger equation by applying the Rayleigh—Ritzmethod to a function space consisting of polynomial splinesof odd degree. Computable a posteriori error estimates for theeigenfunction error estimates are obtained. The sharpness ofthese estimates is illustrated for the harmonic oscillator andWoods—Saxon potentials, using both cubic splines and piecewisecubic Hermite polynomials.     

Approximation of Surfaces by Fairness Bicubic Splines

In this paper we present an approximation method of surfaces by a new type of splines, which we call fairness bicubic splines, from a given Lagrangian data set. An approximating problem of surface is obtained by minimizing a quadratic functional in a parametric space of bicubic splines. The existence and uniqueness of this problem are shown as long as a convergence result of the method is established. We analyze some numerical and graphical examples in order to prove the validity of our method.     

Free multivariate splines

We introduce a definition of free multivariate splines which generalizes the univariate notion of splines with free knots. We then concentrate on the simplest case, piecewise constant functions and characterize some classes of functions which have a prescribed order of approximation inL _p by these splines. These characterizations involve the classical Besov spaces.     

The local integro cubic splines and their approximation properties

We show the integro cubic splines proposed by Behforooz [1] can be constructed locally by using B-representation of splines. The approximation properties of the local splines are also considered.     

Integral representation of slowly growing equidistant splines

In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|^s). We present an integral representation of such splines with a distribution kernel. This representation is related to the Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.     

A sampling theorem on homogeneous manifolds

We consider a generalization of entire functions of spherical exponential type and Lagrangian splines on manifolds. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a manifold is uniquely determined by its values on some discrete sets of points.

The main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using Lagrangian splines.

     

Convergence of local variational spline interpolation

In this paper we first revisit a classical problem of computing variational splines. We propose to compute local variational splines in the sense that they are interpolatory splines which minimize the energy norm over a subinterval. We shall show that the error between local and global variational spline interpolants decays exponentially over a fixed subinterval as the support of the local variational spline increases. By piecing together these locally defined splines, one can obtain a very good C⁰ approximation of the global variational spline. Finally we generalize this idea to approximate global tensor product B-spline interpolatory surfaces.     

The General Problem of Polynomial Spline Interpolation

We study the general problem of interpolation by polynomial splines and consider the construction of such splines using the coefficients of expansion of a certain derivative in B-splines. We analyze the properties of the obtained systems of equations and estimate the interpolation error.     

Generalization of some extremal properties of splines

We generalize well-known inequalities for the norms of the derivatives of periodic splines with minimal defect, perfect splines, and monosplines.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 403–407, March, 1995.     

Nonsymmetric approximations of classes of periodic functions by splines of defect 2 and Jackson-type inequalities

We determine the exact values of the best (α, β)-approximations and the best one-sided approximations of classes of differentiable periodic functions by splines of defect 2. We obtain new sharp Jackson-type inequalities for the best approximations and the best one-sided approximations by splines of defect 2.     

Cubic Splines with Minimal Norm

Natural cubic interpolatory splines are known to have a minimal L ₂-norm of its second derivative on the C ² (or W ₂ ² ) class of interpolants. We consider cubic splines which minimize some other norms (or functionals) on the class of interpolatory cubic splines only. The cases of classical cubic splines with defect one (interpolation of function values) and of Hermite C ¹ splines (interpolation of function values and first derivatives) with spline knots different from the points of interpolation are discussed.     

Statistical application of barycentric rational interpolants: an alternative to splines

Spline curves, originally developed by numerical analysts for interpolation, are widely used in statistical work, mainly as regression splines and smoothing splines. Barycentric rational interpolants have recently been developed by numerical analysts, but have yet seen very few statistical applications. We give the necesssary information to enable the reader to use barycentric rational interpolants, including a suggestion for a Bayesian prior distribution, and explore the possible statistical use of barycentric interpolants as an alternative to splines. We give the all the necessary formulae, compare the numerical accuracy to splines for some Monte-Carlo datasets, and apply both regression splines and barycentric interpolants to two real datasets. We also discuss the application of these interpolants to data smoothing, where smoothing splines would normally be used, and exemplify the use of smoothing interpolants with another real dataset. Our conclusion is that barycentric interpolants are as accurate as splines, and no more difficult to understand and program. They offer a viable alternative methodology.     

Convergence of discrete and penalized least squares spherical splines

We study the convergence of discrete and penalized least squares spherical splines in spaces with stable local bases. We derive a bound for error in the approximation of a sufficiently smooth function by the discrete and penalized least squares splines. The error bound for the discrete least squares splines is explicitly dependent on the mesh size of the underlying triangulation. The error bound for the penalized least squares splines additionally depends on the penalty parameter.     

Some recurrence formulas for box splines and cone splines

A degree elevation formula for multivariate simplex splines was given by Micchelli [6] and extended to hold for multivariate Dirichlet splines in [8]. We report similar formulae for multivariate cone splines and box_splines. To this end, we utilize a relation due to Dahmen and Micchelli [4] that connects box splines and cone splines and a degree reduction formula given by Cohen, Lyche, and Riesenfeld in [2].