Polynomial Interpolation to Data on Flats in

Stimulated by recent work of Hakopian and Sahakian, polynomial interpolation to data at all the s-dimensional intersections of an arbitrary sequence of hyperplanes in ^d is considered, and reduced, by the adjunction of an additional s hyperplanes in general position with respect to the given sequence, to the case s=0 solved much earlier by two of the present authors. In particular, interpolation is from the very same polynomial spaces already used earlier. The difficult question of multiplicity and corresponding matching of derivative information is completely solved, with the number of independent derivative conditions at an intersection exactly equal to that intersection's multiplicity. Also, the consistency requirements placed on the data are minimal in the sense that they need to be checked only at the finitely many 0-dimensional intersections of the hyperplanes involved. The arguments used provide, incidentally, further insights into the two polynomial spaces, (Ξ) and (Ξ), of basic interest in box spline theory.     

Coupled Vandermonde matrices and the superfast computation of Toeplitz determinants

Let n be a positive integer, let be complex numbers and let be a nonsingular n × n complex Toeplitz matrix. We present a superfast algorithm for computing the determinant of T. Superfast means that the arithmetic complexity of our algorithm is , where N denotes the smallest power of 2 that is larger than or equal to n. We show that det T can be computed from the determinant of a certain coupled Vandermonde matrix. The latter matrix is related to a linearized rational interpolation problem at roots of unity and we show how its determinant can be calculated by multiplying the pivots that appear in the superfast interpolation algorithm that we presented in a previous publication. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonlinear multiscale decompositions: The approach of A. Harten

Datadependent interpolatory techniques can be used in the reconstruction step of a multiresolution scheme designed à la Harten. In this paper we carefully analyze the class of Essentially NonOscillatory (ENO) interpolatory techniques described in [11] and their potential to improve the compression capabilities of multiresolution schemes. When dealing with nonlinear multiresolution schemes the issue of stability also needs to be carefully considered.     

Birkhoff interpolation on non-uniformly distributed roots of unity

This paper studies some cases of (0,m)-interpolation on non-uniformly distributed roots of unity that were not covered before. The interpolation problem uses as nodes the zeros of (z ^k +1)(z ³–1) with k=3n+1, 3n+2. Proof of the regularity is more intricate than when k is divisible by 3, the case included in a previous paper by the authors. The interpolation problem appears to be regular for mk+3, a result that is in tune with the case k=3n mentioned before. However, it is necessary to treat the full general 18×18 linear system. For small values of m the determinant is calculated explicitly using MAPLE V, Release 5.     

Convex cubic HERMITE-spline interpolation

We present a method to construct convex cubic C¹-splines which interpolate a given convex data set. The problem is reduced to the solution of a system of linear inequalities. The existence of such convex interpolation splines is assured if the data fulfill slight additional conditions. For stronger conditions some easier methods are developed. Finally, error estimations are given.     

Function cones and interpolation

We consider interpolation of operators acting on functions that belong to a given cone Q with the so‐called decomposition property. The set of all positive functions whose level sets are the level sets of a given function is the main example, and the cone of all decreasing functions is a particular case. As applications, we obtain conditions for the identity (E₀ ∩ Q,E₁ ∩ Q)_θ,p = (E₀,E₁)_θ,p ∩ Q and interpolation results for operators which are bounded when restricted to a given family of characteristic funcions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

New sequent calculi for Visser's Formal Propositional Logic

Two cut‐free sequent calculi which are conservative extensions of Visser's Formal Propositional Logic (FPL) are introduced. These satisfy a kind of subformula property and by this property the interpolation theorem for FPL are proved. These are analogies to Aghaei‐Ardeshir's calculi for Visser's Basic Propositional Logic.     

减少直接法计算量的探讨

§ 1.　TheoreticalBasis　　Manydirectmethodsonlyusetargetfunction(x)andconfinedfunctionGk(x) (k=1 ,2 ,… ,l;listhenumberofconfinedfunction)oftheirfunctionalvaluesateveryknownfeasiblepoints,withoutusingtheconnectionofthesefunctionalvaluesandthefunctionalvaluesatotherfeasiblepointswithintheadjacentdomainsoftheknownfeasiblepoints.Utilizingtheseconnections,weareabletousethefunctionvalueataknownpointtodeterminethefunctionvalueatanotherbetterfeasiblepoint.Bymaintainingcertainfeasiblepointsanddedu…     

Wold-Cramér concordance theorems for interpolation of q-variate stationary processes over locally compact Abelian groups

Salehi and Scheidt [6] have derived several Wold-Cramér concordance theorems for q-variate stationary processes over discrete groups. In this paper we characterize the concordance of the Wold decomposition with respect to families arising in the interpolation problem and the Cramér decomposition for non-full-rank q-variate stationary processes over certain nondiscrete locally compact Abelian (LCA) groups. Moreover, we give an answer to a question of Salehi and Scheidt [6, p. 319] on a characterization of the Wold-Cramér concordance with respect to J₀. As corollary we then deduce a characterization of J₀-regularity.     

On the use of trigonometric approximations in extrapolation methods

Standard trigonometric interpolation formulae are modified for use in extrapolation methods. Some applications are indicated.