Mean square spline approximation

The approximation to a specified function on the real line by fitting a cubic in a piecewise fashion is achieved by minimizing the deviations in the mean square sense. The coefficients of the cubic are determined sequentially employing the method of dynamic programming. Employing this method a known function is approximated and the results of the computation are tabulated.     

A note on spline approximation for Hadamard finite-part integrals

In this paper we construct product quadrature rules, based on spline interpolation, for the numerical evaluation of singular integrals in the sense of Hadamard. We give a convergence result and examine the behaviour of the stability factor. We also present some numerical tests.     

A multinomial spline approximation scheme using spline quasi-interpolants

This paper presents a multinomial spline approximation scheme based on spline quasi-interpolants. The scheme can be considered as an extension of the usual Bernstein approximation for complex exponentials. Error estimates and numerical examples are given to show that this new scheme could produce highly accurate results.     

Three-monotone spline approximation

For r≥3, n∈N and each 3-monotone continuous function f on [a,b] (i.e., f is such that its third divided differences [x₀,x₁,x₂,x₃]f are nonnegative for all choices of distinct points x₀,…,x₃ in [a,b]), we construct a spline s of degree r and of minimal defect (i.e., s∈C^r−1[a,b]) with n−1 equidistant knots in (a,b), which is also 3-monotone and satisfies ‖f−s‖_L∞[a,b]≤cω₄(f,n⁻¹,[a,b])_∞, where ω₄(f,t,[a,b])_∞ is the (usual) fourth modulus of smoothness of f in the uniform norm. This answers in the affirmative the question raised in [8, Remark 3], which was the only remaining unproved Jackson-type estimate for uniform 3-monotone approximation by piecewise polynomial functions (ppfs) with uniformly spaced fixed knots.Moreover, we also prove a similar estimate in terms of the Ditzian–Totik fourth modulus of smoothness for splines with Chebyshev knots, and show that these estimates are no longer valid in the case of 3-monotone spline approximation in the L_p norm with p<∞. At the same time, positive results in the L_p case with p<∞ are still valid if one allows the knots of the approximating ppf to depend on f while still being controlled.These results confirm that 3-monotone approximation is the transition case between monotone and convex approximation (where most of the results are “positive”) and k-monotone approximation with k≥4 (where just about everything is “negative”).     

Perfect spline approximation

Our study of perfect spline approximation reveals: (i) it is closely related to ΣΔ modulation used in one-bit quantization of bandlimited signals. In fact, they share the same recursive formulae, although in different contexts; (ii) the best rate of approximation by perfect splines of order r with equidistant knots of mesh size h is h^r−1. This rate is optimal in the sense that a function can be approximated with a better rate if and only if it is a polynomial of degree <r.The uniqueness of best approximation is studied, too. Along the way, we also give a result on an extremal problem, that is, among all perfect splines with integer knots on , (multiples of) Euler splines have the smallest possible norms.     

Tensor spline approximation

The cyclic-shift tensor-factorization interpolation method recently described by de Boor can be used in particular for least-squares fitting of multivariate data on a rectangular grid and for evaluation of the resulting tensor-product splines, taking advantage of existing linear algebra and univariate spline software. We discuss the computational details of this method, pointing out variants and suggesting techniques for dealing with ill-conditioned least-squares problems.     

A note on square divisible designs

We consider square divisible designs with parameters n, m, k=r, 0 and . We show that being disjoint induces an equivalence relation on the block set of such a design and that any two disjoint blocks meet precisely the same point classes. Also, the intersection number of two blocks depends only on their equivalence classes. The number of blocks disjoint with a given block is at most n–1; equality holds for all blocks iff the dual of the given design is also divisible with the same parameters. We then give a few applications.The author gratefully acknowledges the support of the Deutsche Forschungsgemeinschaft via a Heisenberg grant during the time of this research.     

A note on simultaneous diophantine approximation

Refining earlier investigations due to J.M.MACK [7] by a method of MORDELL it is proved that for any two irrational numbers α, β there exist infinitely many pairs of fractions p/r, q/r satisfying the inequalities $$|\alpha - \frac{p}{r}|< \frac{8}{{13}}r^{ - 3/2} ,|\beta - \frac{q}{r}|< \frac{8}{{13}}r^{ - 3/2} .$$ .     

Moment-preserving spline approximation on finite intervals

Summary Continuing previous wotk, we discuss the problem of approximating a functionf on the interval [0, 1] by a spline function of degreem, withn (variable) knots, matching, as many of the initial moments off as possible. Additional constraints on the derivatives of the approximation at one endpoint of [0, 1] may also be imposed. We show that, if the approximations exist, they can be represented in terms of generalized Gauss-Lobatto and Gauss-Radau quadrature rules relative to appropriate moment functionals or measures (depending off). Pointwise convergence asn, for fixedm>0, is shown for functionsf that are completely monotonic on [0, 1], among others. Numerical examples conclude the paper.The work of the first author was supported by theMinistero della Pubblica Istruzione and by theConsiglio Nazionale delle Ricerche. The work of the second author was supported, in part, by the National Science Foundation under grant DCR-8320561     

A linear approach to shape preserving spline approximation

This paper deals with the approximation of a given large scattered univariate or bivariate data set that possesses certain shape properties, such as convexity, monotonicity, and/or range restrictions. The data are approximated for instance by tensor-product B-splines preserving the shape characteristics present in the data.Shape preservation of the spline approximant is obtained by additional linear constraints. Constraints are constructed which are local linear sufficient conditions in the unknowns for convexity or monotonicity. In addition, it is attractive if the objective function of the resulting minimisation problem is also linear, as the problem can then be written as a linear programming problem. A special linear approach based on constrained least squares is presented, which in the case of large data reduces the complexity of the problem sets in contrast with that obtained for the usual ₂-norm as well as the -norm.An algorithm based on iterative knot insertion which generates a sequence of shape preserving approximants is given. It is investigated which linear objective functions are suited to obtain an efficient knot insertion method.     

A short note on weighted mean matrices

In the present paper we have established a relation between (N, p _n ) and (N, q _n ) weighted mean matrices, when considered as bounded operators on 1^p, 1 < p < ∞.     

A note on testing two-dimensional normal mean

Summary For the problem of testing a composite hypothesis with one-sided alternatives of the mean vector of a two-dimensional normal distribution, a characterization of similar tests is presented and an unbiased test dominating the likelihood ratio test is proposed. A sufficient condition for admissibility is given, which implies the result given by Cohen et al. (1983,Studies in Econometrics, Time Series and Multivariate Statistics, Academic Press): the admissibility of the likelihood ratio test.     

Limit theorems in spline approximation

In this paper we establish new asymptotic relations of the form