Harvey Diamond

A characterization of multivariate quasi-interpolation formulas and its applications

SummaryLet ϕ be a compactly supported function on ℝs andS (ϕ) the linear space withgenerator ϕ; that is,S (ϕ) is the linear span of the multiinteger translates of ϕ. It is well known that corresponding to a generator ϕ there are infinitely many quasi-interpolation formulas. A characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasi-interpolants. We introduce a generalization of interpolation called μ-interpolation and a notion of higher order quasi-interpolation called μ-approximation. A characterization of μ-approximants similar to that of quasi-interpolants is obtained with similar applications. Among these applications are estimating least-squares approximants without matrix inversion, surface fitting to incomplete or semi-scattered discrete data, and constructing generators with one-point quasi-interpolation formulas. It will be seen that the exact values of the generator ϕ at the multi-integers ℤs facilitates the above study. An algorithm to yield this information for box splines is discussed.

A General framework for local interpolation

SummaryWe present a general framework for the construction of local interpolation methods with a given approximation order. Some applications to multivariate spline spaces are presented.

Interpolation by Multivariate Splines

A general interpolation scheme by multivariate splines at regular sample points is introduced. This scheme guarantees the local optimal order of approximation to sufficiently smooth data functions. A discussion on numerical implementation is included. 1. Introduction. In this paper we introduce a very general interpolation scheme by multivariate splines. Based on the quasi-interpolation formulas devel- oped in (3), we show that the interpolating multivariate splines so obtained give the optimal order of approximation to sufficiently smooth functions. Let q be a nonnegative locally supported piecewise polynomial function sym- metric with respect to the origin, and S the linear span of all the translates q(. -j), j E Z8, of q. Hence, S is a multivariate spline space on a certain grid partition A, with certain smoothness joining conditions, and of certain total degree, induced by q. Denote by q the Fourier transform of q. We assume that q is normalized, that is,

Minimax Policies for Unobservable Inspections

On a finite closed lime interval an inspector wishes lo detect an event as soon as possible after its occurrence. A loss fd is incurred if the event remains undetected for a period d. Generally, f is assumed continuous and monotone increasing. Exactly N inspections are allowed. We seek a randomized inspection policy yielding the minimax expected loss. The problem is reformulated as a two-person zero-sum game between the inspector and an inspectee and a pair of equilibrium strategies is sought. An explicit solution for the linear case, fd = d is obtained and its asymptotic properties N large determined. A computational procedure for calculating the optimal policies in the nonlinear case is presented. Two additional related games are briefly treated. The optimal inspection policies each have the salient property of being a randomization over a one parameter family of pure strategies.

Shape-preserving quasi-interpolation and interpolation by box spline surfaces

This paper is devoted to the study of shape-preserving approximation and interpolation of functions by box spline surfaces on three and four directional meshes. The properties of positivity, monotonicity, and convexity are considered. A characterization of the grid spacing is given which guarantees the preservation of these properties for functions in certain Lipschitz classes.

Gaussian approximation of expected utility

Abstract We apply Gaussian methods to the approximation of expected utility. An explicit formula, in terms of mean, variance and skewness, is developed for the two-point Gaussian method. We derive the following characterization: the Gaussian method on n points approximates E[U(X)] by simulating X with the unique discrete random variable on n (or fewer) points which matches the first 2n moments, E[X0],…,E[X2n−1], of X.

Defining Exponential and Trigonometric Functions Using Differential Equations

Summary This note addresses the question of how to rigorously define the functions exp(x), sin(x), and cos(x), and develop their properties directly from that definition. We take a differential equations approach, defining each function as the solution of an initial value problem. Assuming only the basic existence/uniqueness theorem for solutions of linear differential equations, we derive the standard properties and identities associated with these functions. Our target audience is undergraduates with a calculus background.

On an asymptotic property of expected utility

Suppose Sn = Σi = 1n Xi where the Xi are elements of a sequence of random variables. Letting U be a risk-averse utility function, we show that the asymptotic approximation E[U(Sn)] ∼ U(E[Sn]) holds under generally applicable conditions.

Inclusion theorems for the absolute summability of divergent integrals

Some inclusion theorems are obtained relating the absolute summability of divergent integrals of the form fâf(x)dx under three summability methods: Abelian A(x), Abelian A(lnx) and Stieltjes S(x).

The Rational Approximation of Small Angles

Summary Can you approximate small angles between vectors using only rational operations on the coordinates? In two dimensions, yes, with the right trigonometry function. What about more than two dimensions?