共查询到20条相似文献,搜索用时 15 毫秒
1.
Yuan Xu 《Numerische Mathematik》1994,69(2):233-241
Summary.
The existence of Gaussian cubature for a given measure
depends on whether the corresponding multivariate orthogonal polynomials have
enough common zeros. We examine a class of orthogonal
polynomials of two variables generated from that of one variable.
Received February 9, 1993 / Revised version received
January 18, 1994 相似文献
2.
We prove that in a Banach space admitting a separating polynomial, each weakly null normalized sequence has a subsequence
which is equivalent to the usual basis of some , p an even integer. We show that for each even integer p, the Schatten class admits a separating polynomial. For a space with a basis admitting a 4-homogeneous separating polynomial, we relate the unconditionality
of the basis with the existence of certain type of polynomials defined in terms of infinite symmetric matrices. We find relations
between the properties of the entries of these matrices and the geometrical structure of the space. Finally we study the existence
of convex 4-homogeneous separating polynomials.
Received 3 January 2001 相似文献
3.
Yilmaz Simsek 《Mathematical Methods in the Applied Sciences》2014,37(5):676-685
The purpose of this paper is to define a new class polynomials. Special cases of these polynomials give many famous family of the Bernstein type polynomials and beta polynomials. We also construct generating functions for these polynomials. We investigate some fundamental properties of these functions and polynomials. Using functional equations and generating functions, we derive various identities related to theses polynomials. We also construct interpolation function that interpolates these polynomials at negative integers. Finally, we give a matrix representations of these polynomials. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
4.
《复变函数与椭圆型方程》2012,57(3):271-276
Let $ \Pi_{n,M} $ be the class of all polynomials $ p(z) = \sum _{0}^{n} a_{k}z^{k} $ of degree n which have all their zeros on the unit circle $ |z| = 1$ , and satisfy $ M = \max _{|z| = 1}|\,p(z)| $ . Let $ \mu _{k,n} = \sup _{p\in \Pi _{n,M}} |a_{k}| $ . Saff and Sheil-Small asked for the value of $\overline {\lim }_{n\rightarrow \infty }\mu _{k,n} $ . We find an equivalence between this problem and the Krzyz problem on the coefficients of bounded non-vanishing functions. As a result we compute $$ \overline {\lim }_{n\rightarrow \infty }\mu _{k,n} = {{M} \over {e}}\quad {\rm for}\ k = 1,2,3,4,5.$$ We also obtain some bounds for polynomials with zeros on the unit circle. These are related to a problem of Hayman. 相似文献
5.
Michel Lassalle 《Annals of Combinatorics》1998,2(2):145-163
We present several combinatorial conjectures related to Jack generalized binomial coefficients, or equivalently to shifted Jack polynomials. We prove these conjectures when the degree of these polynomials is 5. 相似文献
6.
This paper is a continuation of [AFK]. The notations used there will be preserved. We will consider completion problems forj
pq-inner polynomials which have a prescribed structure, namely for such matrix polynomials which have a prescribed structure, namely for such matrix polynomials which can be considered as suitably normalized resolvent matrix of an appropriate non-degenerate matricial Schur problem. 相似文献
8.
J.M. Melenk 《Numerische Mathematik》1999,84(1):35-69
Summary. The paper presents results on the approximation of functions which solve an elliptic differential equation by operator adapted
systems of functions. Compared with standard polynomials, these operator adapted systems have superior local approximation
properties. First, the case of Laplace's equation and harmonic polynomials as operator adapted functions is analyzed and rates
of convergence in a Sobolev space setting are given for the approximation with harmonic polynomials. Special attention is
paid to the approximation of singular functions that arise typically in corners. These results for harmonic polynomials are
extended to general elliptic equations with analytic coefficients by means of the theory of Bergman and Vekua; the approximation
results for Laplace's equation hold true verbatim, if harmonic polynomials are replaced with generalized harmonic polynomials.
The Partition of Unity Method is used in a numerical example to construct an operator adapted spectral method for Laplace's
equation that is based on approximating with harmonic polynomials locally.
Received May 26, 1997 / Revised version received September 21, 1998 / Published online September 7, 1999 相似文献
9.
In this paper we develop a theory of slice regular functions on a real alternative algebra A. Our approach is based on a well-known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in A and we prove a Cauchy integral formula for slice functions of class C1. 相似文献
10.
We consider errors of positive quadrature formulas applied to Chebyshev polynomials. These errors play an important role in
the error analysis for many function classes. Hunter conjectured that the supremum of all errors in Gaussian quadrature of
Chebyshev polynomials equals the norm of the quadrature formula. We give examples, for which Hunter's conjecture does not
hold. However, we prove that the conjecture is valid for all positive quadratures if the supremum is replaced by the limit
superior. Considering a fixed positive quadrature formula and the sequence of all Chebyshev polynomials, we show that large
errors are rare. 相似文献
11.
For an ideal I in a regular local ring or a graded ideal I in the polynomial ring we study the limiting behavior of as k goes to infinity. By Kodiyalam’s result it is known that βi(S/Ik) is a polynomial for large k. We call these polynomials the Kodiyalam polynomials and encode the limiting behavior in their generating polynomial. It is shown that the limiting behavior depends only on the coefficients on the Kodiyalam polynomials in the highest possible degree. For these we exhibit lower bounds in special cases and conjecture that the bounds are valid in general. We also show that the Kodiyalam polynomials have weakly descending degrees and identify a situation where the polynomials all have the highest possible degree. 相似文献
12.
F. Marcellán A. Martínez-Finkelshtein P. Martínez-González 《Journal of Computational and Applied Mathematics》2007
We give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity. The leading role is played by the differential equation satisfied by these polynomials. Some new developments, applications and open problems are presented. 相似文献
13.
Stef Graillat 《Mathematics in Computer Science》2007,1(2):337-352
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.
相似文献
14.
J. Liesen 《Constructive Approximation》2001,17(2):267-274
Faber polynomials corresponding to rational exterior mapping functions of degree (m, m − 1) are studied. It is shown that these polynomials always satisfy an (m + 1)-term recurrence. For the special case m = 2, it is shown that the Faber polynomials can be expressed in terms of the classical Chebyshev polynomials of the first
kind. In this case, explicit formulas for the Faber polynomials are derived. 相似文献
15.
We study the moment space corresponding to matrix measures on the unit circle. Moment points are characterized by non-negative definiteness of block Toeplitz matrices. This characterization is used to derive an explicit representation of orthogonal polynomials with respect to matrix measures on the unit circle and to present a geometric definition of canonical moments. It is demonstrated that these geometrically defined quantities coincide with the Verblunsky coefficients, which appear in the Szegö recursions for the matrix orthogonal polynomials. Finally, we provide an alternative proof of the Geronimus relations which is based on a simple relation between canonical moments of matrix measures on the interval [−1, 1] and the Verblunsky coefficients corresponding to matrix measures on the unit circle. 相似文献
16.
We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity and log-concavity, as well as identities involving derivatives. We also prove upper and lower bounds on the moduli of the zeros of these polynomials. 相似文献
17.
We consider sequences of polynomials of hypergeometric type satisfying a three-term recurrence relation with constant coefficients
and general initial conditions. We characterize among these the Chebyshev and Fibonacci polynomials. Furthermore, we show
some necessary conditions and links between the coefficients of the recurrence relation and initial conditions, and the coefficients
of the hypergeometric type differential equation in order that this is satisfied by a sequence of polynomials. 相似文献
18.
We use two nonlinear recurrence relations to define the same sequence of polynomials, a sequence resembling the Chebyshev polynomials of the first kind. Among other properties, we obtain results on their irreducibility and zero distribution. We then study the \(2\times 2\) Hankel determinants of these polynomials, which have interesting zero distributions. Furthermore, if these polynomials are split into two halves, then the zeros of one half lie in the interval \((-1,1)\), while those of the other half lie on the unit circle. Some further extensions and generalizations of these results are indicated. 相似文献
19.
《Quaestiones Mathematicae》2013,36(1):23-33
Abstract Andrews, Knopfmacher and Knopfmacher have used the Schur polynomials to consider the celebrated Rogers-Ramanujan identities in the context of q-Engel expansions. We extend this view using similar polynomials, provided by Sills, in the context of Slater's list of 130 Rogers-Ramanujan type identities. 相似文献
20.
Bruno Iannazzo 《Linear algebra and its applications》2011,434(1):174-184
We study the properties of palindromic quadratic matrix polynomials φ(z)=P+Qz+Pz2, i.e., quadratic polynomials where the coefficients P and Q are square matrices, and where the constant and the leading coefficients are equal. We show that, for suitable choices of the matrix coefficients P and Q, it is possible to characterize by means of φ(z) well known matrix functions, namely the matrix square root, the matrix polar factor, the matrix sign and the geometric mean of two matrices. Finally we provide some integral representations of these matrix functions. 相似文献