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1.
Predrag M. Rajković Sladjana D. Marinković Miomir S. Stanković 《The Ramanujan Journal》2006,12(2):245-255
We construct the sequence of orthogonal polynomials with respect to an inner product which is defined by q-integrals over a collection of intervals in the complex plane. We prove that they are connected with little q-Jacobi polynomials. For such polynomials we discuss a few representations, a recurrence relation, a difference equation,
a Rodrigues-type formula and a generating function.
2000 Mathematics Subject Classification Primary—33D45, 42C05 相似文献
2.
《Mathematical Methods in the Applied Sciences》2018,41(13):5308-5326
The aim of this paper is to study complete polynomial systems in the kernel space of conformally invariant differential operators in higher spin theory. We investigate the kernel space of a generalized Maxwell operator in 3‐dimensional space. With the already known decomposition of its homogeneous kernel space into 2 subspaces, we investigate first projections from the homogeneous kernel space to each subspace. Then, we provide complete polynomial systems depending on the given inner product for each subspace in the decomposition. More specifically, the complete polynomial system for the homogenous kernel space is an orthogonal system wrt a given Fischer inner product. In the case of the standard inner product in L2 on the unit ball, the provided complete polynomial system for the homogeneous kernel space is a partially orthogonal system. Further, if the degree of homogeneity for the respective subspaces in the decomposed kernel spaces approaches infinity, then the angle between the 2 subspaces approaches π/2. 相似文献
3.
An affine de Casteljau type algorithm to compute q-Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define
a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix
representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bézier
patch and a tensor product q-Bernstein Bézier patch. Finally, q-Bernstein polynomials B
n,m
(f;x,y) for a function f(x,y), (x,y)∈[0,1]×[0,1] are defined and fundamental properties are discussed.
AMS subject classification (2000) 65D17 相似文献
4.
5.
Iserles et al. (J. Approx. Theory 65:151–175, 1991) introduced the concepts of coherent pairs and symmetrically coherent pairs of measures with the aim of obtaining Sobolev
inner products with their respective orthogonal polynomials satisfying a particular type of recurrence relation. Groenevelt
(J. Approx. Theory 114:115–140, 2002) considered the special Gegenbauer-Sobolev inner products, covering all possible types of coherent pairs, and proves certain
interlacing properties of the zeros of the associated orthogonal polynomials. In this paper we extend the results of Groenevelt,
when the pair of measures in the Gegenbauer-Sobolev inner product no longer form a coherent pair.
This research is supported by grants from CNPq and FAPESP. 相似文献
6.
7.
María Álvarez de Morales Juan J. Moreno–Balcázar Teresa E. Pérez Miguel A. Piñar 《Acta Appl Math》2000,61(1-3):257-266
In this work, we study algebraic and analytic properties for the polynomials { Q
n
}
n 0, which are orthogonal with respect to the inner product
where , R such that – 2 > 0. 相似文献
8.
Antônio BrandãoJr. 《Rendiconti del Circolo Matematico di Palermo》2008,57(2):265-278
Let M
n
(K) be the algebra of all n × n matrices over an infinite field K. This algebra has a natural ℤ
n
-grading and a natural ℤ-grading. Finite bases for its ℤ
n
-graded identities and for its ℤ-graded identities are known. In this paper we describe finite generating sets for the ℤ
n
-graded and for the ℤ-graded central polynomials for M
n
(K)
Partially supported by CNPq 620025/2006-9 相似文献
9.
C. Brezinski 《Journal of Approximation Theory》2010,162(12):2290-2302
Let c be a linear functional defined by its moments c(xi)=ci for i=0,1,…. We proved that the nonlinear functional equations P(t)=c(P(x)P(αx+t)) and P(t)=c(P(x)P(xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. This equation relates the polynomials of the family with those of the scaled and shifted family. Other types of nonlinear functional equations whose solutions are formal orthogonal polynomials are also presented. Applications to Legendre and Chebyshev polynomials are given. Then, orthogonality with respect to a definite inner product is studied. When c is an integral functional with respect to a weight function, the preceding functional equations are nonlinear integral equations, and these results lead to new characterizations of orthogonal polynomials on the real line, on the unit circle, and, more generally, on an algebraic curve. 相似文献
10.
Starting from the addition formula for q-disk polynomials, which is an identity in noncommuting variables, we establish a basic analogue in commuting variables of
the addition and product formula for disk polynomials. These contain, as limiting cases, the addition and product formula
for little q-Legendre polynomials. As q tends to 1 the addition and product formula for disk polynomials are recovered.
Date received: September 29, 1995. Date revised: May 20, 1996. 相似文献
11.
Let μ be a finite positive Borel measure supported in [−1,1] and introduce the discrete Sobolev-type inner productwhere the mass points ak belong to [−1,1], Mk,i0, i=0,…,Nk−1, and Mk,Nk>0. In this paper, we study the asymptotics of the Sobolev orthogonal polynomials by comparison with the orthogonal polynomials with respect to the measure μ and we prove that they have the same asymptotic behaviour. We also study the pointwise convergence of the Fourier series associated to this inner product provided that μ is the Jacobi measure. We generalize the work done by F. Marcellán and W. Van Assche where they studied the asymptotics for only one mass point in [−1,1]. The same problem with a finite number of mass points off [−1,1] was solved by G. López, F. Marcellán and W. Van Assche in a more general setting: they consider the constants Mk,i to be complex numbers. As regards the Fourier series, we continue the results achieved by F. Marcellán, B. Osilenker and I.A. Rocha for the Jacobi measure and mass points in
. 相似文献
12.
Dual generalized Bernstein basis 总被引:1,自引:0,他引:1
The generalized Bernstein basis in the space Πn of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced by Philips [Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511–518], is given by the formula [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT Numer. Math. 44 (2004) 63–78], We give explicitly the dual basis functions for the polynomials , in terms of big q-Jacobi polynomials Pk(x;a,b,ω/q;q), a and b being parameters; the connection coefficients are evaluations of the q-Hahn polynomials. An inverse formula—relating big q-Jacobi, dual generalized Bernstein, and dual q-Hahn polynomials—is also given. Further, an alternative formula is given, representing the dual polynomial (0jn) as a linear combination of min(j,n-j)+1 big q-Jacobi polynomials with shifted parameters and argument. Finally, we give a recurrence relation satisfied by , as well as an identity which may be seen as an analogue of the extended Marsden's identity [R.N. Goldman, Dual polynomial bases, J. Approx. Theory 79 (1994) 311–346]. 相似文献
13.
A. Martinez-Finkelshtein 《Constructive Approximation》2000,16(1):73-84
In this paper we obtain the strong asymptotics for the sequence of orthogonal polynomials with respect to the inner product
where ρ
0
and ρ
1
are weights which satisfy Szegő's condition, supported on a smooth Jordan closed curve or arc.
December 14, 1997. Date revised: September 21, 1998. Date accepted: November 16, 1998. 相似文献
14.
Boyd Geoff Micchelli Charles A. Strang Gilbert Zhou Ding-Xuan 《Advances in Computational Mathematics》2001,14(4):379-391
Every s×s matrix A yields a composition map acting on polynomials on R
s
. Specifically, for every polynomial p we define the mapping C
A
by the formula (C
A
p)(x):=p(Ax), xR
s
. For each nonnegative integer n, homogeneous polynomials of degree n form an invariant subspace for C
A
. We let A
(n) be the matrix representation of C
A
relative to the monomial basis and call A
(n) a binomial matrix. This paper studies the asymptotic behavior of A
(n) as n. The special case of 2×2 matrices A with the property that A
2=I corresponds to discrete Taylor series and motivated our original interest in binomial matrices. 相似文献
15.
Francisco Luquin 《Mathematische Nachrichten》1996,181(1):261-275
Some results of A.O. Gelfond concerning monic polynomials of least deviation (uniform) from zero together with their derivatives are extended to certain Lp - norms and to several variables. As an application we extend a recent result of Xiaoming Huang to functions f∈C(n)n)([?1, 1]) which satisfy f(n)(x)≥1. 相似文献
16.
The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm
over a compact set in ℂ
d
. We study this problem on general sets but devote special attention to product sets such as cube and polydisk. We also establish
a multivariate analog of the Hilbert–Fekete upper bound for the integer Chebyshev constant, which depends on the dimension
of space. In the case of single-variable polynomials in the complex plane, our estimate coincides with the Hilbert–Fekete
result.
相似文献
17.
André Draux 《Numerical Algorithms》2000,24(1-2):31-58
Some methods of numerical analysis, used for obtaining estimations of zeros of polynomials, are studied again, more especially
in the case where the zeros of these polynomials are all strictly positive, distinct and real. They give, in particular, formal
lower and upper bounds for the smallest zero. Thanks to them, we produce new formal lower and upper bounds of the constant
in Markov-Bernstein inequalities in L
2 for the norm corresponding to the Laguerre and Gegenbauer inner products. In fact, since this constant is the inverse of
the square root of the smallest zero of a polynomial, we give formal lower and upper bounds of this zero. Moreover, a new
sufficient condition is given in order that a polynomial has some complex zeros.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
18.
A generating function for nonstandard orthogonal polynomials involving differences: the Meixner case
In this paper we deal with a family of nonstandard polynomials orthogonal with respect to an inner product involving differences.
This type of inner product is the so-called Δ-Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond
to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials).
The aim of this work is to obtain a generating function for the Δ-Meixner–Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for Laguerre–Sobolev
orthogonal polynomials. 相似文献
19.
The paper describes a new space of variable degree polynomials. This space is isomorphic to ℙ6, possesses a Bernstein like basis and has generalized tension properties in the sense that, for limit values of the degrees,
its functions approximate quadratic polynomials. The corresponding space of C
3, variable degree splines is also studied. This spline space can be profitably used in the construction of shape preserving
curves or surfaces.
AMS subject classification (2000) 65D07, 65D17, 65D10 相似文献
20.
The Al–Salam & Carlitz polynomials are q–generalizations of the classical Hermite polynomials. Multivariable generalizations of these polynomials are introduced via a generating function involving a multivariable hypergeometric function which is the q–analogue of the type–A Dunkl integral kernel. An eigenoperator is established for these polynomials and this is used to prove orthogonality with respect to a certain Jackson integral inner product. This inner product is normalized by deriving a q–analogue of the Mehta integral, and the corresponding normalization of the multivariable Al–Salam & Carlitz polynomials is derived from a Pieri–type formula. Various other special properties of the polynomials are also presented, including their relationship to the shifted Macdonald polynomials and the big–q Jacobi polynomials. 相似文献