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1.
This paper is concerned with a generalization of Bernstein–Bézier curves. A one parameter family of rational Bernstein–Bézier
curves is introduced based on a de Casteljau type algorithm. A subdivision procedure is discussed, and matrix representation
and degree elevation formulas are obtained. We also represent conic sections using rational q-Bernstein–Bézier curves.
AMS subject classification (2000) 65D17 相似文献
2.
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
3.
M. I. Grigor’ev V. N. Malozemov A. N. Sergeev 《Computational Mathematics and Mathematical Physics》2006,46(11):1872-1881
Analytical principles of the theory of Bézier curves are presented. A new approach to the construction of composite Bézier curves of prescribed smoothness both on a plane and in a multidimensional Euclidean space is proposed. 相似文献
4.
Explicit formulae for the Bézier coefficients of the constrained dual Bernstein basis polynomials are derived in terms of the Hahn orthogonal polynomials. Using difference properties of the latter polynomials, efficient recursive scheme is obtained to compute these coefficients. Applications of this result to some problems of CAGD is discussed. 相似文献
5.
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
6.
A classical result on the expansion of an analytic function in a series of Jacobi polynomials is extended to a class of q-orthogonal polynomials containing the fundamental Askey–Wilson polynomials and their special cases. The function to be expanded
has to be analytic inside an ellipse in the complex plane with foci at ±1. Some examples of explicit expansions are discussed.
相似文献
7.
Wolter Groenevelt 《Constructive Approximation》2009,29(1):85-127
Big q-Jacobi functions are eigenfunctions of a second-order q-difference operator L. We study L as an unbounded self-adjoint operator on an L
2-space of functions on ℝ with a discrete measure. We describe explicitly the spectral decomposition of L using an integral transform ℱ with two different big q-Jacobi functions as a kernel, and we construct the inverse of ℱ.
相似文献
8.
This is an expository paper that aims to give an essentially self-contained survey of a so-called q-analogue of classical polynomials from their characterizations by Hahn's property and a Rodrigues formula which allows us to construct them. Integral representations of corresponding forms are given, except in some cases where the problem remains open. 相似文献
9.
In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases.
Dedicated to Richard Askey on the occasion of his 70th birthday.
2000 Mathematics Subject Classification Primary—33D50; Secondary—33C50
Supported in part by NSERC grant #A6197. 相似文献
10.
For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relations which admit a
dual system of orthogonal polynomials. For the q = 0 orthogonal limit functions we discuss interpretations on p-adic spaces. In the little 0-Jacobi case we also discuss product formulas.
Dedicated to Dick Askey on the occasion of his seventieth birthday.
2000 Mathematics Subject Classification Primary—33D45, 33D80
Work done at KdV Institute, Amsterdam and supported by NWO, project number 613.006.573. 相似文献
11.
We continue studying the generalized coherent states for oscillator-like systems associated with a given family of orthogonal
polynomials. We consider the case of generalized oscillators generated by the Charlier q-polynomials.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 39–46, April, 2008. 相似文献
12.
We construct certain raising operators of row type for Macdonald's symmetric polynomials by an interpolation method. 相似文献
13.
James Haglund 《Advances in Applied Mathematics》1998,20(4):450-487
Connections betweenq-rook polynomials and matrices over finite fields are exploited to derive a new statistic for Garsia and Remmel'sq-hit polynomial. Both this new statisticmatand another statistic for theq-hit polynomial ξ recently introduced by Dworkin are shown to induce different multiset Mahonian permutation statistics for any Ferrers board. In addition, for the triangular boards they are shown to generate different families of Euler–Mahonian statistics. For these boards the ξ family includes Denert's statisticden, and gives a new proof of Foata and Zeilberger's Theorem that (exc, den) is equidistributed with (des, maj). Thematfamily appears to be new. A proof is also given that theq-hit polynomials are symmetric and unimodal. 相似文献
14.
We firstly establish the fourth order difference equation satisfied by the Laguerre-Hahn polynomials orthogonal on special non-uniform lattices in general case, secondly give it explicitly for the cases of polynomials r-associated to the classical polynomials orthogonal on linear, q-linear and q-nonlinear (Askey-Wilson) lattices, and thirdly give it semi-explicitly for the class one Laguerre-Hahn polynomials orthogonal on linear lattice. 相似文献
15.
The tensor product of a positive and a negative discrete series representation of the quantum algebra Uq(su(1,1)) decomposes as a direct integral over the principal unitary series representations. Discrete terms can appear, and these terms are a finite number of discrete series representations, or one complementary series representation. From the interpretation as overlap coefficients of little q-Jacobi functions and Al-Salam and Chihara polynomials in base q and base q–1, two closely related bilinear summation formulas for the Al-Salam and Chihara polynomials are derived. The formulas involve Askey-Wilson polynomials, continuous dual q-Hahn polynomials and little q-Jacobi functions. The realization of the discrete series as q-difference operators on the spaces of holomorphic and anti-holomorphic functions, leads to a bilinear generating function for a certain type of 21-series, which can be considered as a special case of the dual transmutation kernel for little q-Jacobi functions. 相似文献
16.
We give examples of generalized modular forms with the property that their divisors are supported at the cusps and the exponents
in their q-product expansions take infinitely many values.
相似文献
17.
18.
Gert Vegter. 《Mathematics of Computation》2000,69(230):691-720
Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the de Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives.
19.
Irrationality measures are given for the values of the series
, where
and Wn is a rational valued Fibonacci or Lucas form, satisfying a second order linear recurrence. In particular, we prove irrationality
of all the numbers
where fn and ln are the Fibonacci and Lucas numbers, respectively.
2000 Mathematics Subject Classification Primary—11J82, 11B39 相似文献
20.
In a recent paper Ismail et al. (Algebraic Methods and q-Special Functions (J.F. van Diejen and L. Vinet, eds.) CRM Proceding and Lecture Notes, Vol. 22, American Mathematical Society, 1999, pp. 183–200) have established a continuous orthogonality relation and some other properties of a 21-Bessel function on a q-quadratic grid. Dick Askey (private communication) suggested that the Bessel-type orthogonality found in Ismail et al. (1999) at the 21-level has really a general character and can be extended up to the 87-level. Very-well-poised 87-functions are known as a nonterminating version of the classical Askey–Wilson polynomials (SIAM J. Math. Anal. 10 (1979), 1008–1016; Memoirs Amer. Math. Soc. Number
319 (1985)). Askey's conjecture has been proved by the author in J. Phys. A: Math. Gen. 30 (1997), 5877–5885. In the present paper which is an extended version of Suslov (1997) we discuss in detail properties of the orthogonal 87-functions. Another type of the orthogonality relation for a very-well-poised 87-function was recently found by Askey et al. J. Comp. Appl. Math. 68 (1996), 25–55. 相似文献