Louise A. Raphael

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,

Convergence Rates of Multiscale and Wavelet Expansions

We prove several results that characterize the rate at which wavelet and multiresolution expansions converge to functions in a given Sobolev space in the supremum error norm. Some of the results are proved without assuming the existence of a scaling function in the multiresolution analysis. Necessary and sufficient conditions are given for convergence at given rates in terms of behavior of Fourier transforms of the wavelet or scaling function near the origin. Such conditions turn out in special cases to be equivalent to moment and other known determining convergence rates.

Equisummability of eigenfunction expansions under analytic multipliers

In this paper we present some abstract criteria for equisummability of expansions in eienfunctions of certain pairs of elliptic operators on general domains of [w”. The criteria to be developed originate in some cases from general Banach space arguments, and in others from the more specific spatial nature of the differential operators involved. The importance of our analysis of L”-equisummability of two operators is that the question of convergence of the two summability means is reduced essentially to showing that the difference of the modified resolvent operators is uniformly bounded. We apply these criteria to give a simple proof of an equisummability result of Gurarie and Kon [4] for a certain class of elliptic operators whose leading terms are positive and lower order terms have coefficients which are singular on a nowhere dense set. The key technique is to analyze kernels of the resolvents and use ,&‘-radial bounds of the resolvents developed in [3]. The prototypical case for equisummability is found in the study of equiconvergence for differential operators. Classically Haar [ 51, Walsh [lo], Birkhoff [2], and Tamarkin [lo] have shown that the difference between expansions with respect to eigenfunctions of a Sturm-Liouville operator (or Birkhoff series) and the ordinary Fourier series tends to zero uniformly in every finite interval. In this case the multiplier or summator function is the characteristic function. In some one-dimensional cases where such equiconvergence fails, Stone [9], Levitan and Sargsjan [6], and Benzinger [1] have showed equisummability for Riesz typical means of eigenfunction expansions of differential operators, where the summator function is

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).