The volume of restricted moment spaces

In this paper we compute the volume of restricted moment spaces under very general conditions on the functions generating the moment space.     

Piecewise polynomial spaces and geometric continuity of curves

Summary We say that a curve has geometric continuity if its curvatures and Frenet frame are continuous. In this paper we introduce spaces of piecewise polynomials which can be used to model space curves which have geometric continuity. We show that the basic theoretical properties of ordinary spline functions also hold for these spaces. These results extend and unify recent work on Beta-splines and Nu-splines which are used as a design tool in computer-aided geometric design of free form curves and surfaces.This work was initiated when the first author was on Sabbatical at Thomas J. Watson IBM Research Center, and was partially supported by the U.S.-Israel Binational Foundation, grant no. 86-00243/1.     

On weight functions which admit explicit Gauss-Turán quadrature formulas

The main purpose of this paper is the construction of explicit Gauss-Turán quadrature formulas: they are relative to some classes of weight functions, which have the peculiarity that the corresponding -orthogonal polynomials, of the same degree, are independent of . These weights too are introduced and discussed here. Moreover, highest-precision quadratures for evaluating Fourier-Chebyshev coefficients are given.

     

On translation invariant operators which preserve the B-spline recurrence

It was observed in [4] that the Hilbert transform of the univariate B-spline preserves the B-spline recurrence. Motivated by this observation, we characterize translation invariant operators that preserve the multivariate B-spline recurrence and analogous results are also provided for the multivariate cube spline. Charles A. Micchelli was supported in part by the US National Science of Foundation under grant CCR-0407476. Yuesheng Xu was supported in part by the US National Science Foundation under grant CCR-0407476, by the Natural Science Foundation of China under grant 10371122, by the Chinese Academy of Sciences under the program “One Hundred Distinguished Chinese Scientists” and by Ministry of Education, People’s Republic of China, under the Changjian Scholarship through Zhongshan University.     

Divided differences,hyperbolic equations,and lifting distributions

We discuss the relationship between divided differences, fundamental functions of hyperbolic equations, multivariate interpolation, and polyhedral splines.     

On the Cholesky factorization of the Gram matrix of locally supported functions

Cholesky factorization of bi-infinite and semi-infinite matrices is studied and in particular the following is proved. If a bi-infinite matrixA has a Cholesky factorization whose lower triangular factorL and its lower triangular inverse decay exponentially away from the diagonal, then the semi-infinite truncation ofA has a lower triangular Cholesky factor whose elements approach those ofL exponentially. This result is then applied to studying the asymptotic behavior of splines obtained by orthogonalizing a large finite set of B-splines, in particular identifying the limiting profile when the knots are equally spaced.The first and second authors were partially supported by Nato Grant #920209, the second author also by the Alexander von Humboldt Foundation, and the last two authors by the Italian Ministry of University and Scientific and Technological Research.     

Convexity and Bernstein polynomials onk-simploids

This paper is concerned with Bernstein polynomials onk-simploids by which we mean a cross product ofk lower dimensional simplices. Specifically, we show that if the Bernstein polynomials of a given functionf on ak-simploid form a decreasing sequence thenf +l, wherel is any corresponding tensor product of affine functions, achieves its maximum on the boundary of thek-simploid. This extends recent results in [1] for bivariate Bernstein polynomials on triangles. Unlike the approach used in [1] our approach is based on semigroup techniques and the maximum principle for second order elliptic operators. Furthermore, we derive analogous results for cube spline surfaces.This work was partially supported by NATO Grant No. DJ RG 639/84.