Spline Methods In Discrete Mathematics

Spline functions not only play important roles in approximation theory, CAGD and wavelets,but also are closely related with some branches in pure mathematics,such as abstract algebra, algebraic geometry and combinatorics.In this dissertation,some prob-lems about divided differences,spline methods in discrete geometry and combinatorics, biorthogonal systems in asymptotic analysis are discussed.(1)A conjecture about several divided difference formulas with super-convergence proposed by Xinghua Wang,Heyu Wang and Mingjun Lai,is proved in this dissertation. A type of more general super convergent remainders for Lagrange interpolation is also derived which contains Brodskii[2],de Boor [3],Floater [4],Xinghua Wang's[5,6] recent derivative expansions as special cases.(2)Spline methods in discrete geometry are proposed.The volumes of portions of hy-percubes determined by hyperplanes play important roles in discrete geometry and other branches of Mathematics.It seems that Laplace and Polya investigated this problem in the context of a question in probability. They gave the integral expression and asymp-totic formula for the volume of a central slab of a hypercube.Polya's formula and its close relatives are striking important in solving the problems arising in cube slicing.For example,Good conjecture was solved by Hensley with the help of probabilistic methods and Polya's formula. Hensley also gave another conjecture about the upper bound for the volume of the slice.Ball used probabilistic methods,ending it up making ingenious estimates on integrals corresponding to the integral formula for the volume treated by Polya. In this dissertation,we observed that the volume of the slice can be transformed to an equivalent problem in spline theory. Having related it directly to B-splines[14], we can take advantage of many powerful spline techniques to derive various results of these objects,the asymptotic properties of B-splines and their derivatives are investi-gated and the convergence orders are also derived.The results given by Laplace and Polya are reproved with brevity and elegance by using spline notation.The asymptotic properties of B-splines are not only paly a part in cube slicing but also striking important in combinatorial enumeration.(3)Investigate a series problems arising in combinatorial enumeration by spline the- ory. Different from the traditional combinatorial method,splines as functions of a con-tinuous nature provide an analysis method in combinatorial enumeration which usually considered as counting discrete objects.The two areas interact each other in the context of investigating the interrelationship between Eulerian numbers and B splines.For in-stance,Worpitzky's identity and the integral expression for Eulerian numbers are special cases of Marsden's identity and integral formula for B-splines respectively.Spline explanations for some mixed volumes are derived by exploring the relations between mixed volumes and splines.Constructing log-concave sequences by using the explanation,a partial answer to the open problems which was proposed by Schmidt and Simion([13]) is given.Investigate the log-concavity for some combinatorial sequences by B-splines.Specifi-cally, we use B-splines to prove that the sequence of the coefficients of the descent polyno-mials,Ddn(t),are log-concave,which is a much stronger conclusion than unimodal given by E.Steingrimsson.Spline methods for some topics in asymptotic combinatorics is proposed in context of observing the relations between three combinatorial numbers and splines.The asymptotic property of Eulerian numbers was examined by L.Carlitz et.al.[9] in 1972.They gave the convergence order O{d-3/4)with the help of central limit theorem in probability theory. Nevertheless,in this dissertation,we give a more precise convergence order O(d-3/2)with the help of the spline theory.(4)The orthonormal property of the Hermite polynomials can be considered as a biorthogonal relation between the derivatives of the Gaussian and the Hermite polynomi-als.We extend the biorthogonal relation between Hermite polynomials and the derivatives of the Gaussian to a family of functionsφthat approximate the Gaussian function and to construct a family of Appell sequences of "biorthogonal polynomials" that approximate the Hermite polynomials.The Appell polynomials and the distributional derivatives ofφform a biorthognal system.In particular,the Appell polynomials generated by the uni-form B-spline of order N are the classical Bernoulli polynomials of order N and the Appell polynomials generated by the binomial distributions of order N are the classical Euler polynomials of order N.After suitably normalized,they converge to Hermite polynomials as N→∞.A necessary and sufficient condition for two Appell polynomial sequences whose generating functions satisfy the a-scaling equation is derived.A criterion theorem for polynomial sequences that approximate to Hermite polynomi-als is derived and making the asymptotic properties of generalized Buchholz polynomials, Ultraspherical(Gegenbauer)polynomials and Laguerre polynomials as corollaries. A criterion theorem for polynomial sequences that approximate to generalized La-guerre polynomials is derived and the asymptotic formulas for Meixner-Pllaczek polyno-mials and Meixner polynomials are derived by using the theorem.Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme.The theorems mentioned above has verified some of the Askey scheme.