An asymptotic expansion for the error in a linear map that reproduces polynomials of a certain order

Han's ‘multinode higher-order expansion’ in [H] is shown to be a special case of an asymptotic error expansion available for any bounded linear map on C([a..b]) that reproduces polynomials of a certain order. The key is the formula for the divided difference at a sequence containing just two distinct points.     

On the h-vector of a cohen-macaulay domain in positive characteristic

Here we study the Hilbert function of a Cohen-Macaulay homogeneous domain over an algebraically closed field of positive characteristic. The main tool (and an essential part of the main geometrical results) is the study of the Hilbert function of a general hyperplane section X?P ^r of an integral curve C?P ^r+1 , which is pathological in some sense. In §1, we study the case when Cis a strange curve, i.e., all tangent lines to Cat its simple points pass through a fixed point υ∈P ^r+1 . In §2, we give more refined results under the assumption that the Trisecant Lemma fails for C, i.e., any line spanned by two points of Ccontains one more point of C.     

Pseudozero Set of Real Multivariate Polynomials

The pseudozero set of a system P of polynomials in n variables is the subset of C ⁿ consisting of the union of the zeros of all polynomial systems Q that are near to P in a suitable sense. This concept arises naturally in Scientific Computing where data often have a limited accuracy. When the polynomials of the system are polynomials with complex coefficients, the pseudozero set has already been studied. In this paper, we focus on the case where the polynomials of the system have real coefficients and such that all the polynomials in all the perturbed polynomial systems have real coefficients as well. We provide an explicit definition to compute this pseudozero set. At last, we analyze different methods to visualize this set.      

The role of the endpoint in weighted polynomial approximation with varying weights

For a weight functionw: [a, b]→(0, ∞), we consider weighted polynomials of the formw ⁿ P_n where the degree ofP _n is at mostn. The class of functions that can be approximated with such polynomials depends on the behavior of the densityv(t) of the extremal measure associated withw. We show that every approximable function must vanish at the endpointa ifv(t) behaves like (t?a) ^β ast→a with β>?1/2. We also present an analogous result for internal points. Our results solve some open problems posed by V. Totik and disprove a conjecture of G.G. Lorentz on incomplete polynomials.     

Simultaneous best approximation by three polynomials

Let [a, b] be any interval and let p₀, p₁, p_k be any three polynomials of degrees 0, 1, k, respectively, where k 2. A set of necessary and sufficient conditions for the existence of an f in C[a, b] such that p_i is the best approximation to f from the space of all polynomials of degree less than or equal to i, for all i = 0, 1, k, is given.     

Peano's kernel theorem for vector-valued functions and some applications

We extend the well-known Peano Kernel Theorem to a class of linear operators L : Cⁿ⁺¹([a,b];X}→ X, X being a Branch space, which vanish on abstract polynomials of degree ≤ n. We then recover, in the abstract setting, classical estimates of remainders in polynomials interpolation and quadrature formulas. Finally, we present an application to the error analysis of the trapezoidal time discretization scheme for parabolic evolution equations.     

The convergence of padé approximants and the size of the power series coefficients

Let f be a power series ∑a_izⁱ with complex coefficients. The (n. n) Pade approximant to f is a rational function P/Q where P and Q are polynomials, Q(z) ? 0, of degree ≦ n such that f(z)Q(z)-P(z) = Az²ⁿ⁺¹ + higher degree terms. It is proved that if the coefficients a_i satisfy a certain growth condition, then a corresponding subsequence of the sequence of (n, n) Pade approximants converges to f in the region where the power series f converges, except on an exceptional set E having a certain Hausdorff measure 0. It is also proved that the result is best possible in the sense that we may have divergence on E. In particular,there exists an entire function f such that the sequence of (ny n) Pade approximants diverges everywhere (except at 0)     

On polynomial approximations to f a (Z)(z-a)−1 with complex a and some applications to certain non-hermitian matrices

We are concerned with the problem 1 $$\mathop {min}\limits_{p \in P_n } \mathop {max}\limits_{z \in [ - 1,1]} |w(z)(f_a (z) - p(z))|,a \in C/[ - 1,1],n = 0 \cdots $$ of best polynomial approximation of degree n to f_a(z)=(z?a)^?1 on the unit interval. Here P_n denotes the class of complex polynomials of degree at most n, and ω belongs to a certain classical family of weight functions. For real a the solution of this approximation problem is known. In this paper, we obtain the best approximations for purely imaginary a. For general a, close approximations to the optimal polynomials are derived by solving the approximation problem expli citly for a certain subclass of P_n. We then use these polynomials to devise an iterative method for the solution of linear systems Ax=b with coefficient matrices of the form A=cI+dT where T=T^H and c, d ∈C. Finally, as a further appication of our results, we derive bounds for the decay rates of the inverses of banded matrices A=cI+dT.     

Matrix Extension with Symmetry and Applications to Symmetric Orthonormal Complex M-wavelets

Matrix extension with symmetry is to find a unitary square matrix P of 2π-periodic trigonometric polynomials with symmetry such that the first row of P is a given row vector p of 2π-periodic trigonometric polynomials with symmetry satisfying p[`(p)]^T=1\mathbf {p}\overline{\mathbf{p}}^{T}=1 . Matrix extension plays a fundamental role in many areas such as electronic engineering, system sciences, wavelet analysis, and applied mathematics. In this paper, we shall solve matrix extension with symmetry by developing a step-by-step simple algorithm to derive a desired square matrix P from a given row vector p of 2π-periodic trigonometric polynomials with complex coefficients and symmetry. As an application of our algorithm for matrix extension with symmetry, for any dilation factor M, we shall present two families of compactly supported symmetric orthonormal complex M-wavelets with arbitrarily high vanishing moments. Wavelets in the first family have the shortest possible supports with respect to their orders of vanishing moments; their existence relies on the establishment of nonnegativity on the real line of certain associated polynomials. Wavelets in the second family have increasing orders of linear-phase moments and vanishing moments, which are desirable properties in numerical algorithms.     

Parameter choices and a better bound on the list size in the Guruswami–Sudan algorithm for algebraic geometry codes

Given an algebraic geometry code C_L(D, aP){C_{\mathcal L}(D, \alpha P)}, the Guruswami–Sudan algorithm produces a list of all codewords in C_L(D, aP){C_{\mathcal L}(D, \alpha P)} within a specified distance of a received word. The initialization step in the algorithm involves parameter choices that bound the degree of the interpolating polynomial and hence the length of the list of codewords generated. In this paper, we use simple properties of discriminants of polynomials over finite fields to provide improved parameter choices for the Guruswami–Sudan list decoding algorithm for algebraic geometry codes. As a consequence, we obtain a better bound on the list size as well as a lower degree interpolating polynomial.     

Chamber behavior of double Hurwitz numbers in genus 0

We study double Hurwitz numbers in genus zero counting the number of covers CP¹→CP¹ with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.     

A case of the dynamical Mordell–Lang conjecture

We prove a special case of a dynamical analogue of the classical Mordell–Lang conjecture. Specifically, let φ be a rational function with no periodic critical points other than those that are totally invariant, and consider the diagonal action of φ on (\mathbb P¹)^g{(\mathbb P^1)^g}. If the coefficients of φ are algebraic, we show that the orbit of a point outside the union of the proper preperiodic subvarieties of (\mathbb P¹)^g{(\mathbb P^1)^g} has only finite intersection with any curve contained in (\mathbb P¹)^g{(\mathbb P^1)^g}. We also show that our result holds for indecomposable polynomials φ with coefficients in \mathbb C{\mathbb C}. Our proof uses results from p-adic dynamics together with an integrality argument. The extension to polynomials defined over \mathbb C{\mathbb C} uses the method of specialization coupled with some new results of Medvedev and Scanlon for describing the periodic plane curves under the action of (φ, φ) on \mathbb A²{\mathbb A^2}.     

On discrete q-ultraspherical polynomials and their duals

A special case of the big q-Jacobi polynomials Pn(x;a,b,c;q), which corresponds to a=b=−c, is shown to satisfy a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<q⁻¹). Since Pn(x;qα,qα,−qα;q) tend to Gegenbauer (or ultraspherical) polynomials in the limit as q→1, this family represents another q-extension of these classical polynomials, different from the continuous q-ultraspherical polynomials of Rogers. For a dual family with respect to the polynomials Pn(x;a,a,−a;q) (i.e., for dual discrete q-ultraspherical polynomials) we also find new orthogonality relations with extremal measures.     

On genus and embeddings of torsion-free nilpotent groups of class two

Summary We study embeddings between torsion-free nilpotent groups having isomorphic localizations. Firstly, we show that for finitely generated torsion-free nilpotent groups of nilpotency class 2, the property of having isomorphicP-localizations (whereP denotes any set of primes) is equivalent to the existence of mutual embeddings of finite index not divisible by any prime inP. We then focus on a certain family Γ of nilpotent groups whose Mislin genera can be identified with quotient sets of ideal class groups in quadratic fields. We show that the multiplication of equivalence classes of groups in Γ induced by the ideal class group structure can be described by means of certain pull-back diagrams reflecting the existence of enough embeddings between members of each Mislin genus. In this sense, the family Γ resembles the family N₀ of infinite, finitely generated nilpotent groups with finite commutator subgroup. We also show that, in further analogy with N₀, two groups in Γ with isomorphic localizations at every prime have isomorphic localizations at every finite set of primes. We supply counterexamples showing that this is not true in general, neither for finitely generated torsion-free nilpotent groups of class 2 nor for torsion-free abelian groups of finite rank. Supported by DGICYT grant PB94-0725 This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.     

Asymptotics of Differentiated Bernstein Polynomials

It is shown that the m th-order derivative of the n th-order Bernstein polynomial of a function f satisfying a certain Lipschitz condition, can be written for n\rightarrow +∈fty as a singular integral of Gauss—Weierstrass type, m times differentiated (in a certain sense) under the integral sign. The theorem is applied to yield an overdifferentiation formula, involving p times differentiated Bernstein polynomials of functions that are not C ^p . December 1, 1998. Dates revised: July 22, 1999 and January 11, 2000. Date accepted: February 1, 2000.     

Characterization of seminuclear sets in a finite projective plane

LetC be a set ofq + a points in the desarguesian projective plane of orderq, such that each point ofC is on exactly 1 tangent, and onea+ 1-secant (a>1). Then eitherq=a + 2 andC consists of the symmetric difference of two lines, with one further point removed from each line, orq=2a + 3 andC is projectively equivalent to the set of points {(0,1,s),(s, 0, 1),(1,s, 0): -s is not a square inGF(q)}.     

On the triple points of singular maps

The number of triple points (mod 2) of a self-transverse immersion of a closed 2n-manifold M into 3n-space are known to equal one of the Stiefel-Whitney numbers of M. This result is generalized to the case of generic (i.e. stable) maps with singularities. Besides triple points and Stiefel-Whitney numbers, a certain linking number of the manifold of singular values with the rest of the image is involved in the generalized equation which corrects an erroneous formula in [9].? If n is even and the closed manifold is oriented then the equations mentioned above make sense over the integers. Together, the integer- and mod 2 generalized equations imply that a certain Stiefel-Whitney number of closed oriented 4k-manifolds vanishes. This Stiefel-Whitney number is in fact the first in a family which vanish on such manifolds. Received: October 12, 2001     

POLYNOMIALS RELATED TO HALL NUMBERS

Let Λ be a finite dimensional algebra of finite representation type over a finite field k. For any modules A, B and Pin mod Λ with P projective, we prove that there exists a polynomial ? ^B _(P)Λ over Z whose evaluation at |E| for any conservative finite field extension E of Λ is the sum of Hall numbers F ^{B E} _{C ^E A ^E} where C ^E runs through isoclasses in mod Λ ^E and P ^E is the projective cover of C ^E . As a consequence of this result and its dual version, Hall polynomials ? ^E _CA exist when C or A is semisimple. As applications of the main result, we obtain the existence of Hall polynomials for Nakayama algebras and some selfinjective algebras.

     

Existence and uniqueness of an invariant probability for a class of Feller Markov chains

We consider the class of Feller Markov chains on a phase spaceX whose kernels mapC ₀ (X), the space of bounded continuous functions that vanish at infinity, into itself. We provide a necessaryand sufficient condition for the existence of an invariant probability measure using a generalized Farkas Lemma. This condition is a Lyapunov type criterion that can be checked in practice. We also provide a necessaryand sufficient condition for existence of aunique invariant probability measure. When the spaceX is compact, the conditions simplify.     

Approximation of unbounded functions via compactification

Approximants to functions f(s) that are allowed to possess infinite limits on their interval of definition, are constructed.To this end a compactification of Rⁿ is developed which is based on the projection of Rⁿ on a bowl-shaped subset of a parabolic surface. This compactification induces a bijection and a metric with several desirable properties that make it a useful tool for rational approximation of unbounded functions.Roughly speaking this compactification enables us to show that unbounded functions can be approximated by rational functions on a closed interval; thus we also obtain an extension to Weierstrass’ celebrated theorem. An extension to a Fourier-type theorem is also obtained. Roughly speaking, our result states that unbounded periodic functions can be approximated by quotients of certain trigonometric sums. The characteristics of the main results are the following. The approximations do not require the original approximated function to possess a restricted rate of growth. Neither do they require that the approximated function possess any amount of smoothness. Moreover, the numerator and denominator, in an approximating quotient are guaranteed not to vanish simultaneously. Furthermore, some of the proposed approximations are guaranteed to be bounded at every point at which the original approximated function is bounded. Beside the tool of compactification we also employ Bernstein polynomials and Cesaro means of “trigonometric sums”.