To what extent is a large space of matrices not closed under product?

Let (K) be a field. Given an arbitrary linear subspace V of M_n(K) of codimension less than n-1, a classical result states that V generates the (K)-algebra M_n(K). Here, we strengthen this statement in three ways: we show that M_n(K) is spanned by the products of the form AB with (A,B)∈V²; we prove that every matrix in M_n(K) can be decomposed into a product of matrices of V; finally, when V is a linear perplane of M_n(K) and n>2, we show that every matrix in M_n(K) is a product of two elements of V.     

On the Durrmeyer-type modification of some discrete approximation operators

In [10], for continuous functionsf from the domain of certain discrete operatorsL _n the inequalities are proved concerning the modulus of continuity ofL _nf. Here we present analogues of the results obtained for the Durrmeyer-type modification $\tilde L_n $ ofL _n. Moreover, we give the estimates of the rate of convergence of $\tilde L_n f$ in Hölder-type norms     

Twisted inner products and contraction inequalities on spaces of contravariant and covariant tensors

Given positive integers n and p, and a complex finite dimensional vector space V, we let S_n,p(V) denote the set of all functions from V×V×?×V-(n+p copies) to C that are linear and symmetric in the first n positions, and conjugate linear symmetric in the last p positions. Letting κ=min{n,p} we introduce twisted inner products, [·,·]_s,t,1?s,t?κ, on S_n,p(V), and prove the monotonicity condition [F,F]_s,t?[F,F]_u,v is satisfied when s?u?κ,t?v?κ, and F∈S_n,p(V). Using the monotonicity condition, and the Cauchy-Schwartz inequality, we obtain as corollaries many known inequalities involving norms of symmetric multilinear functions, which in turn imply known inequalities involving permanents of positive semidefinite Hermitian matrices. New tensor and permanental inequalities are also presented. Applications to partial differential equations are indicated.     

On the matrices of given rank in a large subspace

Let V be a linear subspace of M_n,p(K) with codimension lesser than n, where K is an arbitrary field and n?p. In a recent work of the author, it was proven that V is always spanned by its rank p matrices unless n=p=2 and K?F₂. Here, we give a sufficient condition on codim V for V to be spanned by its rank r matrices for a given r∈?1,p-1?. This involves a generalization of the Gerstenhaber theorem on linear subspaces of nilpotent matrices.     

Rational interpolation to solutions of Riccati difference equations on elliptic lattices

It is shown how to define difference equations on particular lattices {x_n}, n∈Z, where the x_ns are values of an elliptic function at a sequence of arguments in arithmetic progression (elliptic lattice). Solutions to special difference equations (elliptic Riccati equations) have remarkable simple (!) interpolatory continued fraction expansions.     

Idempotent elements determined matrix algebras

Let M_n(R) be the algebra of all n×n matrices over a unital commutative ring R with 2 invertible, V be an R-module. It is shown in this article that, if a symmetric bilinear map {·,·} from M_n(R)×M_n(R) to V satisfies the condition that {u,u}={e,u} whenever u²=u, then there exists a linear map f from M_n(R) to V such that . Applying the main result we prove that an invertible linear transformation θ on M_n(R) preserves idempotent matrices if and only if it is a Jordan automorphism, and a linear transformation δ on M_n(R) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points.     

On polyvectors of vector spaces and hyperplanes of projective Grassmannians

We investigate relationships between polyvectors of a vector space V, alternating multilinear forms on V, hyperplanes of projective Grassmannians and regular spreads of projective spaces. Suppose V is an n-dimensional vector space over a field F and that A_n-1,k(F) is the Grassmannian of the (k − 1)-dimensional subspaces of PG(V) (1  ? k ? n − 1). With each hyperplane H of A_n-1,k(F), we associate an (n − k)-vector of V (i.e., a vector of ∧^n−kV) which we will call a representative vector of H. One of the problems which we consider is the isomorphism problem of hyperplanes of A_n-1,k(F), i.e., how isomorphism of hyperplanes can be recognized in terms of their representative vectors. Special attention is paid here to the case n = 2k and to those isomorphisms which arise from dualities of PG(V). We also prove that with each regular spread of the projective space PG(2k-1,F), there is associated some class of isomorphic hyperplanes of the Grassmannian A_2k-1,k(F), and we study some properties of these hyperplanes. The above investigations allow us to obtain a new proof for the classification, up to equivalence, of the trivectors of a 6-dimensional vector space over an arbitrary field F, and to obtain a classification, up to isomorphism, of all hyperplanes of A_5,3(F).     

The Lebesgue constant for Lagrange interpolation on equidistant nodes

Summary Forn=1, 2, 3, ..., let _n denote the Lebesgue constant for Lagrange interpolation based on the equidistant nodesx _{k, n} =k, k=0, 1, 2, ...,n. In this paper an asymptotic expansion for log _n is obtained, thereby improving a result of A. Schönhage.     

The asymptotic accuracy of rational best approximations to ez on a disk

The method described by D. Braess (J. Approx. Theory40 (1984), 375–379) is applied to study approximation of e^z on a disk rather than an interval. Let E_mn be the distance in the supremum norm on ¦z¦ ? ? from e^z to the set of rational functions of type (m, n). The analog of Braess' result turns out to be $\text{E}{}_{\mathrm{mn}}\text{~}\text{m! n! ?}{}^{\mathrm{m\; +\; n\; +1}}\text{(m + n)! (m + n +1)!}$ as m + n → ∞ This formula was obtained originally for a special case by E. Saff (J. Approx. Theory9 (1973), 97–101).     

The approximation by q-Bernstein polynomials in the case q ↓ 1

Let B_n (f, q; x), n=1, 2, ... , 0 < q < ∞, be the q-Bernstein polynomials of a function f, B_n (f, 1; x) being the classical Bernstein polynomials. It is proved that, in general, {B_n (f, q_n; x)} with q_n ↓ 1 is not an approximating sequence for f ∈C[0, 1], in contrast to the standard case q_n ↓ 1. At the same time, there exists a sequence 0 < δ_n ↓ 0 such that the condition implies the approximation of f by {B_n (f, q_n; x)} for all f ∈C[0, 1]. Received: 15 March 2005     

WEIGHTED Lp-APPROXIMATION OF DERIVATIVES BY THE METHOD OF BETA OPERATORS

The author consides Beta operators β_nf on suitable Sobolev type subspace of L_p[0, ∞) and characterizes the global rate of approximation of derivatives f^(τ) through corresponding derivatives (β_nf)^(τ) in an appropriate weighted L_p-metric by the rate of Ditzian and Totik's τ-th order weighted modulus of Smoothness.     

Spaces of matrices without non-zero eigenvalues in their field of definition, and a question of Szechtman

Let V be a vector space of dimension n over any field F. Extreme values for the possible dimension of a linear subspace of End_F(V) with a particular property are considered in two specific cases. It is shown that if E₁ is a subspace of End_F(V) and there exists an endomorphism g of V, not in E₁, such that for every hyperplane H of V some element of E₁ agrees with g on H, then E₁ has dimension at least . This answers a question that was posed by Szechtman in 2003. It is also shown that a linear subspace of M_n(F) in which no element possesses a non-zero eigenvalue in F may have dimension at most . The connection between these two properties, which arises from duality considerations, is discussed.     

Good distance graphs and the geometry of matrices

Denote by G=(V,∼) a graph which V is the vertex set and ∼ is an adjacency relation on a subset of V×V. In this paper, the good distance graph is defined. Let (V,∼) and (V^′,∼^′) be two good distance graphs, and φ:V→V^′ be a map. The following theorem is proved: φ is a graph isomorphism ⇔φ is a bounded distance preserving surjective map in both directions ⇔φ is a distance k preserving surjective map in both directions (where k<diam(G)/2 is a positive integer), etc. Let D be a division ring with an involution such that both |F∩Z_D|?3 and D is not a field of characteristic 2 with D=F, where and Z_D is the center of D. Let H_n(n?2) be the set of n×n Hermitian matrices over D. It is proved that (H_n,∼) is a good distance graph, where A∼B⇔rank(A-B)=1 for all A,B∈H_n.     

Note on the Dočev-Grosswald asymptotic series for generalized Bessel polynomials

The Do?ev-Grosswald asymptotic series for the generalized Bessel polynomials y_n(z; a, b) is extended to O(1/n⁴) relative accuracy. The differential equation of the asymptotic factor, derived from the differential equation for y_n(z; a, b), is the basis of a different and easier method that employs simple recurrence relations and much less algebra for obtaining the same series. This is applied to the important special case of a = 1 to obtain the asymptotic series to O(1/n¹¹) relative accuracy.     

On Genuine Bernstein–Durrmeyer Operators

We continue the studies on the so–called genuine Bernstein–Durrmeyer operators U _n by establishing a recurrence formula for the moments and by investigating the semigroup T(t) approximated by U _n . Moreover, for sufficiently smooth functions the degree of this convergence is estimated. We also determine the eigenstructure of U _n , compute the moments of T(t) and establish asymptotic formulas. Received: January 26, 2007.     

A uniform asymptotic expansion for weighted sums of exponentials

We consider the random variable Z_n,α=Y₁+2^αY₂+?+n^αY_n, with α∈R and Y₁,Y₂,… independent and exponentially distributed random variables with mean one. The distribution function of Z_n,α is in terms of a series with alternating signs, causing great numerical difficulties. Using an extended version of the saddle point method, we derive a uniform asymptotic expansion for P(Z_n,α<x) that remains valid inside (α≥−1/2) and outside (α<−1/2) the domain of attraction of the central limit theorem. We discuss several special cases, including α=1, for which we sharpen some of the results in Kingman and Volkov (2003).     

Convergence rate of a new Bezier variant of Chlodowsky operators to bounded variation functions

In the present paper, we estimate the rate of pointwise convergence of the Bézier Variant of Chlodowsky operators _Cn,α$C_{n,\alpha }$ for functions, defined on the interval extending infinity, of bounded variation. To prove our main result, we have used some methods and techniques of probability theory.     

Ranks derived from multilinear maps

Let V₁, V₂ and V₃ be vector spaces over any field k. An element T∈V₁⊗V₂⊗V₃ induces for each i=1,2,3 a k-linear map where is the dual vector space of V_i. We characterize all integer triplets (r₁,r₂,r₃) such that there exists a tensor T with r_i=rankT_i, and we explain how these ranks are related to the higher secant varieties of various Segre varieties. We also study the case T∈V₁⊗?⊗V_n with n>3, giving necessary conditions on the ranks of all induced linear maps.     

Approximation of monomials by lower degree polynomials

Our topic is the uniform approximation ofx ^k by polynomials of degreen (n on the interval [–1, 1]. Our major result indicates that good approximation is possible whenk is much smaller thann ² and not possible otherwise. Indeed, we show that the approximation error is of the exact order of magnitude of a quantity,p _k,n , which can be identified with a certain probability. The numberp _k,n is in fact the probability that when a (fair) coin is tossedk times the magnitude of the difference between the number of heads and the number of tails exceedsn.