Szeg? type polynomials and para-orthogonal polynomials

Szeg? type polynomials with respect to a linear functional M for which the moments M[tn]=μ−n are all complex, μ−n=μn and Dn≠0 for n?0, are considered. Here, Dn are the associated Toeplitz determinants. Para-orthogonal polynomials are also studied without relying on any integral representation. Relation between the Toeplitz determinants of two different types of moment functionals are given. Starting from the existence of polynomials similar to para-orthogonal polynomials, sufficient conditions for the existence of Szeg? type polynomials are also given. Examples are provided to justify the results.     

Approximation of analytic functions by polynomials “close” to polynomials of best approximation

The rate of approximation of analytic functions at interior points of compact sets with connected complement by polynomials close to polynomials of best approximation is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 208–214, February, 1992.     

When do linear combinations of orthogonal polynomials yield new sequences of orthogonal polynomials?

Given {P_n}_n≥0 a sequence of monic orthogonal polynomials, we analyze their linear combinations with constant coefficients and fixed length, i.e., 

     

Bootstrapping and Askey–Wilson polynomials

The mixed moments for the Askey–Wilson polynomials are found using a bootstrapping method and connection coefficients. A similar bootstrapping idea on generating functions gives a new Askey–Wilson generating function. Modified generating functions of orthogonal polynomials are shown to generate polynomials satisfying recurrences of known degree greater than three. An important special case of this hierarchy is a polynomial which satisfies a four term recurrence, and its combinatorics is studied.     

Near-rings of polynomials overΩ-groups

IfV is a variety of -groups and ifGV then one can study the algebraG ^V [x] of polynomials overG andx inV. With respect to addition and substitution,G ^V [x] is a near-ring. Its zero-symmetric part can excellently be used to describe generated ideals. Also, we study maximal left ideals and get a general result on the structure ofG ^V [V].This study was kindly supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung (Projekt Nr. 3479).     

Sensitivity analysis for Szegő polynomials

Szegő polynomials are orthogonal with respect to an inner product on the unit circle. Numerical methods for weighted least-squares approximation by trigonometric polynomials conveniently can be derived and expressed with the aid of Szegő polynomials. This paper discusses the conditioning of several mappings involving Szegő polynomials and, thereby, sheds light on the sensitivity of some approximation problems involving trigonometric polynomials. This Research supported in part by NSF grant DMS-0107858.     

Bernstein–Sato polynomials of hyperplane arrangements

We show that the Bernstein–Sato polynomial (that is, the b-function) of a hyperplane arrangement with a reduced equation is calculable by combining a generalization of Malgrange’s formula with the theory of Aomoto complexes due to Esnault, Schechtman, Terao, Varchenko, and Viehweg in certain cases. We prove in general that the roots are greater than \(-2\) and the multiplicity of the root \(-1\) is equal to the (effective) dimension of the ambient space. We also give an estimate of the multiplicities of the roots in terms of the multiplicities of the arrangement at the dense edges, and provide a method to calculate the Bernstein–Sato polynomial at least in the case of 3 variables with degree at most 7 and generic multiplicities at most 3. Using our argument, we can terminate the proof of a conjecture of Denef and Loeser on the relation between the topological zeta function and the Bernstein–Sato polynomial of a reduced hyperplane arrangement in the 3 variable case.     

“Hidden symmetry” of Askey-Wilson polynomials

Conclusions We have shown that the Askey-Wilson polynomials of general form are generated by the algebra AW(3), which has a fairly simple structure and is the q-analog of a Lie algebra with three generators. The main properties of these polynomials (weight function, recursion relation, etc.) can be obtained directly from analysis of the representations of the algebra.In this paper, we have considered finite-dimensional representations of the algebra AW(3) and the Aksey-Wilson polynomials of discrete argument corresponding to these representations. A separate analysis is required for the infinite-dimensional representations, which generate polynomials of a continuous argument (these polynomials were investigated in detail in the review [2]). Also of interest is investigation of representations of the algebra AW(3) for complex values of the basic parameter and of the structure parameters.In our view, the algebra AW(3) by itself warrants careful study on account of several remarkable properties (in the first place, the duality with respect to the operators K₀, K₁) not present in the currently very popular quantum algebras of the type SU_q(2).We assume that the algebra AW(3) is an algebra of dynamical or hidden symmetry in all problems in which exponential or hyperbolic spectra and the corresponding q-polynomials arise. We hope that in time the algebra AW(3) will come to play the same role in q-problems as Lie algebras play in exactly solvable problems of quantum mechanics.Donetsk State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 89, No. 2, pp. 190–204, November, 1991     

Kazhdan–Lusztig polynomials of boolean elements

We give closed combinatorial product formulas for Kazhdan–Lusztig polynomials and their parabolic analogue of type q in the case of boolean elements, introduced in (Marietti in J. Algebra 295:1–26, 2006), in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincaré polynomials of the Schubert varieties of boolean elements.     

On Sikkema-Kantorovič polynomials of order K

Let a_n≥0 and F(u)∈C [0,1], Sikkema constructed polynomials: , ifα _n ≡0, then B_n (0, F, x) are Bernstein polynomials. Let , we constructe new polynomials in this paper: Q _n ^(k) (α _n ,f(t))=d ^k /dx ^k B _n+k (α _n ,F _k (u),x), which are called Sikkema-Kantorovic polynomials of order k. Ifα _n ≡0, k=1, then Q_n ⁽¹⁾ (0, f(t), x) are Kantorovič polynomials P_n(f). Ifα _n =0, k=2, then Q_n ⁽²⁾, (0, f(t), x) are Kantorovič polynomials of second order (see Nagel). The main result is: Theorem 2. Let 1≤p≤∞, in order that for every f∈L^P [0, 1], , it is sufficient and necessary that , § 1. Let f(t) de a continuous function on [a, b], i. e., f∈C [a, b], we define^{[1–2],[8–10]}: . As usual, for the space L^p [a,b](1≤p<∞), we have and L[a, b]=l¹[a, b]. Letα _n ⩾0and F(u)∈C[0,1],Sikkema-Bernstein polynomials ^{[3] [4]}. The author expresses his thanks to Professor M. W. Müller of Dortmund University at West Germany for his supports.     

On the restricted Chebyshev–Boubaker polynomials

Using the language of Riordan arrays, we study a one-parameter family of orthogonal polynomials that we call the restricted Chebyshev–Boubaker polynomials. We characterize these polynomials in terms of the three term recurrences that they satisfy, and we study certain central sequences defined by their coefficient arrays. We give an integral representation for their moments, and we show that the Hankel transforms of these moments have a simple form. We show that the (sequence) Hankel transform of the row sums of the corresponding moment matrix is defined by a family of polynomials closely related to the Chebyshev polynomials of the second kind, and that these row sums are in fact the moments of another family of orthogonal polynomials.     

Approximation with Bernstein–Szegö polynomials

We present approximation kernels for orthogonal expansions with respect to Bernstein–Szegö polynomials. Theconstruction is derived from known results for Chebyshev polynomials of the first kind and does not pose any restrictions on the Bernstein–Szegö polynomials.     

Orthogonal polynomials—centroid of their zeroes

Let c _n,k (k=1,...,n) the n zeroes of the monic orthogonal polynomials family P _n (x). The centroid of these zeroes: $s_n=\frac1n \sum\limits^n_{k=1}c_{n,k}$ controls globally the distribution of the zeroes, and it is relatively easy to obtain information on s _n , like bounds, inequalities, parameters dependence, ..., from the links between s _n , the coefficients of the expansion of P _n (x), and the coefficients β _n , γ _n in the basic recurrence relation satisfied by P _n (x). After a review of basic properties of the centroid on polynomials, this work gives some results on the centroid of a large class of orthogonal polynomials.     

Polynomial zerofinders based on Szegő polynomials

The computation of zeros of polynomials is a classical computational problem. This paper presents two new zerofinders that are based on the observation that, after a suitable change of variable, any polynomial can be considered a member of a family of Szegő polynomials. Numerical experiments indicate that these methods generally give higher accuracy than computing the eigenvalues of the companion matrix associated with the polynomial.