Polynomials orthogonal in the Sobolev sense,generated by Chebyshev polynomials orthogonal on a mesh

We consider the problem of constructing polynomials, orthogonal in the Sobolev sense on the finite uniform mesh and associated with classical Chebyshev polynomials of discrete variable. We have found an explicit expression of these polynomials by classicalChebyshev polynomials. Also we have obtained an expansion of new polynomials by generalized powers ofNewton type. We obtain expressions for the deviation of a discrete function and its finite differences from respectively partial sums of its Fourier series on the new system of polynomials and their finite differences.     

Polynomials orthogonal with sign-sensitive weight

We introduce the notion of inner product with sign-sensitive weight and construct systems of nonsymmetrically orthonormalized polynomials. We also study some properties of such polynomials (for example, the properties of Fourier coefficients, quadrature formulas of Gauss type, etc.).Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 737–752, May, 1996.     

Polynomials orthogonal with respect to discrete convolution

The concept of “Discrete Convolution Orthogonality” is introduced and investigated. This leads to new orthogonality relations for the Charlier and Meixner polynomials. This in turn leads to bilinear representations for them. We also show that the zeros of a family of convolution orthogonal polynomials are real and simple. This proves that the zeros of the Rice polynomials are real and simple.     

Annihilator Conditions on Polynomials II

In this paper, we continue the study of various annihilator conditions which were used by Rickart and Kaplansky to abstract the algebraic properties of von Neumann algebras. In our main results, we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of polynomials. These results are somewhat surprising since, in contrast to the polynomial ring case, the nearring of polynomials has substitution for its multiplication operation. Moreover they indicate connections between the ring and nearring structures on polynomials. Examples are provided to illustrate and delimit our results.This revised version was published online in October 2004 with a corrected Received date.The second author was partially supported by the National Science Council of the Republic of China, Taiwan under grant number NSC 90-2115-M-143-001.     

Annihilator Conditions on Polynomials II

In this paper, we continue the study of various annihilator conditions which were used by Rickart and Kaplansky to abstract the algebraic properties of von Neumann algebras. In our main results, we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of polynomials. These results are somewhat surprising since, in contrast to the polynomial ring case, the nearring of polynomials has substitution for its multiplication operation. Moreover they indicate connections between the ring and nearring structures on polynomials. Examples are provided to illustrate and delimit our results.     

Jacobi Polynomials, Type II Codes, and Designs

Jacobi polynomials were introduced by Ozeki in analogy with Jacobi forms of lattices. They are useful to compute coset weight enumerators, and weight enumerators of children. We determine them in most interesting cases in length at most 32, and in some cases in length 72. We use them to construct group divisible designs, packing designs, covering designs, and (t,r)-designs in the sense of Calderbank-Delsarte. A major tool is invariant theory of finite groups, in particular simultaneous invariants in the sense of Schur, polarization, and bivariate Molien series. A combinatorial interpretation of the Aronhold polarization operator is given. New rank parameters for spaces of coset weight distributions and Jacobi polynomials are introduced and studied here.     

Symplectic Groups and Permutation Polynomials,Part II

The linear group trinomial provides a mnemonic device for the recently discovered permutation polynomials of Müller, Cohen, and Matthews, whereas the symplectic group equation generalizes them, thereby giving rise to strong genus zero coverings for characteristic two.     

Polynomials orthogonal with respect to cardinal B-spline weight functions

A stable and efficient discretization procedure is developed to compute the recurrence coefficients for orthogonal polynomials whose weight function is a polynomial cardinal spline of order m ≥ 1. The procedure is compared with a symbolic moment-based method developed recently by G. V. Milovanovi?. Numerical examples are provided for illustration.     

Almost orthogonal operators on the bitorus II

We prove that the operator ${Tf(x,y)=\int^\pi_{-\pi}\int_{|x^{\prime}|<|y^{\prime}|} \frac{e^{iN(x,y) x^{\prime}}}{x^{\prime}}\frac{e^{iN(x,y) y^{\prime}}}{y^{\prime}}f(x-x^{\prime}, y-y^{\prime}) dx^{\prime} dy^{\prime}}$ , with ${x,y \in[0,2\pi]}$ and where the cut off ${|x^{\prime}|<|y^{\prime}|}$ is performed in a smooth and dyadic way, is bounded from L ² to weak- ${L^{2-\epsilon}}$ , any ${\epsilon > 0 }$ , under the basic assumption that the real-valued measurable function N(x, y) is “mainly” a function of y and the additional assumption that N(x, y) is non-decreasing in x, for every y fixed. This is an extension to 2D of C. Fefferman’s proof of a.e. convergence of Fourier series of L ² functions.     

On Brown Polynomials. II

The results of the author’s previous article are improved. We also establish that each irreducible polynomial over a Henselian valued field is a Brown polynomial.

     

Irreducibility of generalized Hermite-Laguerre Polynomials II

In this paper, we show that for each n ≥ 1, the generalised Hermite-Laguerre Polynomials G_¼ and G_¾are either irreducible or linear polynomial times an irreducible polynomial of degree n−1.     

Algebras and orthogonal groups II

The first part of this paper appeared in this Journal (vol.66 (1996), pp. 11-54) and the results of the present paper were described in the introduction to the first part. The theorems and definitions are numbered in continuation of the numbering used in the first part where the bibliography appeared as well; references made in the present paper refer to this joint numbering system and the common bibliography.     

Structural Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations,II

We develop structural formulas satisfied by some families of orthogonal matrix polynomials of size 2 × 2 satisfying second-order differential equations with polynomial coefficients. We consider here three one-parametric families of weight matrices, namely,

Computing the Invariant Polynomials of a Polynomial Matrix. II

The paper considers the problem of computing the invariant polynomials of a general (regular or singular) one-parameter polynomial matrix. Two new direct methods for computing invariant polynomials, based on the W and V rank-factorization methods, are suggested. Each of the methods may be regarded as a method for successively exhausting roots of invariant polynomials from the matrix spectrum. Application of the methods to computing adjoint matrices for regular polynomial matrices, to finding the canonical decomposition into a product of regular matrices such that the characteristic polynomial of each of them coincides with the corresponding invariant polynomial, and to computing matrix eigenvectors associated with roots of its invariant polynomials are considered. Bibliography: 5 titles.     

Resonant Equations with Classical Orthogonal Polynomials. II

Ukrainian Mathematical Journal - We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite and the Laguerre orthogonal polynomials,...