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1.
Shayne Waldron 《Constructive Approximation》2009,30(1):33-52
This paper considers tight frame decompositions of the Hilbert space ℘
n
of orthogonal polynomials of degree n for a radially symmetric weight on ℝ
d
, e.g., the multivariate Gegenbauer and Hermite polynomials. We explicitly construct a single zonal polynomial p∈℘
n
with the property that each f∈℘
n
can be reconstructed as a sum of its projections onto the orbit of p under SO(d) (symmetries of the weight), and hence of its projections onto the zonal polynomials p
ξ
obtained from p by moving its pole to ξ∈S:={ξ∈ℝ
d
:|ξ|=1}. Furthermore, discrete versions of these integral decompositions also hold where SO(d) is replaced by a suitable finite subgroup, and S by a suitable finite subset. One consequence of our decomposition is a simple closed form for the reproducing kernel for
℘
n
.
相似文献
2.
We introduce polynomials B
n
i
(x;ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as
q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating
B
n
i
(x;ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
3.
A. Bodin 《Commentarii Mathematici Helvetici》2003,78(1):134-152
We give a global version of Lê-Ramanujam μ-constant theorem for polynomials. Let , , be a family of polynomials of n complex variables with isolated singularities, whose coefficients are polynomials in t. We consider the case where some numerical invariants are constant (the affine Milnor number μ(t), the Milnor number at infinity λ(t), the number of critical values, the number of affine critical values, the number of critical values at infinity). Let n=2, we also suppose the degree of the is a constant, then the polynomials and are topologically equivalent. For we suppose that critical values at infinity depend continuously on t, then we prove that the geometric monodromy representations of the are all equivalent.
Received: January 14, 2002 相似文献
4.
Diego Dominici 《The Ramanujan Journal》2008,15(3):303-338
We analyze the Krawtchouk polynomials K
n
(x,N,p,q) asymptotically. We use singular perturbation methods to analyze them for N→∞, with appropriate scalings of the two variables x and n. In particular, the WKB method and asymptotic matching are used. We obtain asymptotic approximations valid in the whole domain
[0,N]×[0,N], involving some special functions. We give numerical examples showing the accuracy of our formulas.
相似文献
5.
Diego Dominici 《Central European Journal of Mathematics》2007,5(2):280-304
We analyze the Charlier polynomials C
n
(χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving
some special functions. We give numerical examples showing the accuracy of our formulas.
相似文献
6.
K.H. Kwon D.W. Lee F. Marcellán S.B. Park 《Annali di Matematica Pura ed Applicata》2001,180(2):127-146
Given an orthogonal polynomial system {Q
n
(x)}
n=0
∞, define another polynomial system by where α
n
are complex numbers and t is a positive integer. We find conditions for {P
n
(x)}
n=0
∞ to be an orthogonal polynomial system. When t=1 and α1≠0, it turns out that {Q
n
(x)}
n=0
∞ must be kernel polynomials for {P
n
(x)}
n=0
∞ for which we study, in detail, the location of zeros and semi-classical character.
Received: November 25, 1999; in final form: April 6, 2000?Published online: June 22, 2001 相似文献
7.
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
8.
M. Felten 《Acta Mathematica Hungarica》2008,118(3):227-263
Lubinsky and Totik’s decomposition [11] of the Cesàro operators σ
n
(α,β)
of Jacobi expansions is modified to prove uniform boundedness in weighted sup norms, i.e., ‖w
(a,b)
σ
n
(α,β)
‖∞ ≦ C‖w
(a,b)
f‖∞, whenever α,β ≧ −1/2 and a, b are within the square around (α/2 + 1/4, α/2 + 1/4) having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and
various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second
paper [7] provides some necessary estimations.
相似文献
9.
Vadim B. Kuznetsov Vladimir V. Mangazeev Evgeny K. Sklyanin 《Indagationes Mathematicae》2003,14(3-4):451
Applying Baxter's method of the Q-operator to the set of Sekiguchi's commuting partial differential operators we show that Jack polynomials Pλ(1/g) (χ1, …, χn) …, χn) are eigenfunctions of a one-parameter family of integral operators Qz. The operators Qz are expressed in terms of the Dirichlet-Liouville n-dimensional beta integral. From a composition of n operators Qzk we construct an integral operator Sn factorising Jack polynomials into products of hypergeometric polynomials of one variable. The operator Sn admits a factorisation described in terms of restricted Jack polynomials Pλ(1/g) (x1, …, xk, 1, … 1). Using the operator Qz for z = 0 we give a simple derivation of a previously known integral representation for Jack polynomials. 相似文献
10.
V. E. Tarakanov 《Mathematical Notes》1997,62(6):731-738
For the Legendre-Sobolev orthonormal polynomials
depending on the parametersM≥0,N≥0, weighted and uniform estimates on the orthogonality interval are obtained. It is shown that, unlike the Legendre orthonormal
polynomials, the polynomials
forM>0,N>0 decay at the rate ofn
−3/2 at the points 1 and −1. The values of
are calculated.
Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 871–880, December, 1997.
Translated by N. K. Kulman 相似文献
11.
I. I. Kachurik 《Ukrainian Mathematical Journal》1998,50(8):1201-1211
We obtain algebraic relations (identities) for q-numbers that do not contain q
α-factors. We derive a formula that expresses any q-number [x] in terms of the q-number [2]. We establish the relationship between the q-numbers [n] and the Fibonacci numbers, Chebyshev polynomials, and other special functions. The sums of combinations of q-numbers, in particular, the sums of their powers, are calculated. Linear and bilinear generating functions are found for
“natural” q-numbers.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1055–1063, August, 1998. 相似文献
12.
Warren P. Johnson 《The Ramanujan Journal》2007,13(1-3):167-201
More than 200 years ago, Pfaff found two generalizations of Leibniz’s rule for the nth derivative of a product of two functions. Thirty years later Cauchy found two similar identities, one equivalent to one
of Pfaff’s and the other new. We give simple proofs of these little-known identities and some further history. We also give
applications to Abel-Rothe type binomial identities, Lagrange’s series, and Laguerre and Jacobi polynomials. Most importantly,
we give extensions that are related to the Pfaff/Cauchy theorems as Hurwitz’s generalized binomial theorems are to the Abel-Rothe
identities. We apply these extensions to Laguerre and Jacobi polynomials as well.
Dedicated to Dick Askey on the occasion of his 70th birthday.
2000 Mathematics Subject Classification Primary—05A19; Secondary—33C45 相似文献
13.
LetSp(n, R) be the sympletic group, and letK
n
* be its maximal compact subgroup. ThenG=Sp(n,R)/K
n
* can be realized as the Siegel domain of type one. The square-integrable representation ofG gives the admissible wavelets AW and wavelet transform. The characterization of admissibility condition in terms of the Fourier
transform is given. The Bergman kernel follows from the viewpoint of coherent state. With the Laguerre polynomials, Hermite
polynomials and Jacobi polynomials, two kinds of orthogonal bases for AW are given, and they then give orthogonal decompositions
ofL
2-space on the Siegel domain of type one ℒ(ℋ
n
, |y| *dxdy).
Project supported in part by the National Natural Science Foundation of China (Grant No. 19631080). 相似文献
14.
I. I. Sharapudinov 《Mathematical Notes》1997,62(4):501-512
Suppose that 0<δ≤1,N=1/δ, and α, ga≥0, is an integer. For the classical Meixner polynomials
orthonormal on the gird {0, δ, 2δ, ...} with weight ρ(x)=(1-e
−δ)αг(Nx+α+ 1)/г(Nx+1), the following asymptotic formula is obtained:
. The remainderv
n,N
α
(z) forn≤λN satisfies the estimate
where Λ
k
α
(x) are the Laguerre orthonormal polynomials. As a consequence, a weighted estimate, for the Meixner polynomial
on the semiaxis [0, ∞) is obtained.
Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 603–616, October, 1997.
Translated by N. K. Kulman 相似文献
15.
For n -dimensional subspaces E
n
, F
n
of L
1
(-1,1) with E
n
spanned by Chebyshev polynomials of the second kind and F
n
the set of Müntz polynomials with , , it is shown that the relative projection constants satisfy
(E
n
, L
1
(-1,1))
C log n and
(F
n
, L
1
(-1,1)) = O(1) , . The spaces L
1
w(α,β)
, where w
α,β
is the weight function of the Jacobi polynomials and , are also studied. The Jacobi partial sum projections, which are used in connection with E
n
, are not minimal.
September 26, 1996. 相似文献
16.
A. Dubickas 《Lithuanian Mathematical Journal》1999,39(3):245-250
Let us consider the set of polynomials with integer coefficients of a given degree and of bounded height. We prove that among
all polynomials in this set with no integer roots the polynomialx
n-H(x
n−1+x
n−2+...+1) has a root closest to an integer.
Partially supported by the Lithuanian State Science and Studies Foundation.
Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp.
310–316, July–September, 1999. 相似文献
17.
Let Hn be the nth Hermite polynomial, i.e., the nth orthogonal on
polynomial with respect to the weight w(x)=exp(−x2). We prove the following: If f is an arbitrary polynomial of degree at most n, such that |f||Hn| at the zeros of Hn+1, then for k=1,…,n we have f(k)Hn(k), where · is the
norm. This result can be viewed as an inequality of the Duffin and Schaeffer type. As corollaries, we obtain a Markov-type inequality in the
norm, and estimates for the expansion coefficients in the basis of Hermite polynomials. 相似文献
18.
George E. Andrews 《The Ramanujan Journal》2007,13(1-3):311-318
In a letter dated March 3, 1971, L. Carlitz defined a sequence of polynomials, Φ
n
(a,b; x, y; z), generalizing the Al-Salam & Carlitz polynomials, but closely related thereto. He concluded the letter by stating: “It would
be of interest to find properties of Φ
n
(a, b; x, y; z) when all the parameters are free.” In this paper, we reproduce the Carlitz letter and show how a study of Carlitz’s polynomials
leads to a clearer understanding of the general 3Φ2 (a, b, c; d; e; q, z).
Dedicated to my friend, Richard Askey.
2000 Mathematics Subject Classification Primary—33D20.
G. E. Andrews: Partially supported by National Science Foundation Grant DMS 0200047. 相似文献
19.
Erwin Miña-Díaz 《Constructive Approximation》2009,29(3):421-448
Let φ(z) be an analytic function on a punctured neighborhood of ∞, where it has a simple pole. The nth Faber polynomial F
n
(z) (n=0,1,2,…) associated with φ is the polynomial part of the Laurent expansion at ∞ of [φ(z)]
n
. Assuming that ψ (the inverse of φ) conformally maps |w|>1 onto a domain Ω bounded by a piecewise analytic curve without cusps pointing out of Ω, and under an additional assumption concerning the “Lehman expansion” of ψ about those points of |w|=1 mapped onto corners of ∂
Ω, we obtain asymptotic formulas for F
n
that yield fine results on the limiting distribution of the zeros of Faber polynomials.
相似文献
20.
Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations 总被引:1,自引:1,他引:0
Some families of orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n have recently been introduced (see [Internat. Math. Res. Notices 10 (2004) 461–484]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues’ formulas of the type (ΦnW)(n)W-1, where Φ is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues’ formula, well suited to the matrix case, appears in [Internat. Math. Res. Notices 10 (2004) 482].In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues’ formula and show that scalar type Rodrigues’ formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar-type Pearson equation as well as that of a noncommutative version of it. 相似文献