The Padua points are a family of points on the square [−1, 1] given by explicit formulas that admits unique Lagrange interpolation by bivariate polynomials. Interpolation polynomials and cubature formulas based on the Padua points are studied from an ideal theoretic point of view, which leads to the discovery of a compact formula for the interpolation polynomials. The L convergence of the inter...

DISSERTATION zur Erlangung des akademischen Grades doctor rerum naturalium Dedicated to the memory of KARIN GATERMANN Table 1: List of mathematical symbols C field of complex numbers R field of real numbers Z ring of integer numbers C[x] ring of polynomials with coefficients in the field of complex numbers I ideal I def, tor deformed toric ideal V (I) variety of an ideal V (I def, tor) variety ...

Knop and Sahi simultaneously introduced a family of non-homogeneous, non-symmetric polynomials, Gα(x; q, t). The top homogeneous components of these polynomials are the non-symmetric Macdonald polynomials, Eα(x; q, t). An appropriate Hecke algebra symmetrization of Eα yields the Macdonald polynomials, Pλ(x; q, t). A search for explicit formulas for the polynomials Gα(x; q, t) led to the main re...

We define a family of polynomials of the form ∑ f(σ)x1,σ(1) · · ·xn,σ(n) in terms of the Kazhdan-Lusztig basis {C′ w(1) |w ∈ Sn} for the symmetric group algebra C[Sn]. Using this family, we obtain nonnegativity properties of polynomials of the form ∑ cI,I′∆I,I′(x)∆I,I′(x). In particular, we show that the application of certain of these polynomials to Jacobi-Trudi matrices yields symmetric funct...

In this work, we develop fast algorithms for computations involving finite expansions in Gegenbauer polynomials. We describe a method to convert a linear combination of Gegenbauer polynomials up to degree n into a representation in a different family of Gegenbauer polynomials with generally O(n log(1/ε)) arithmetic operations where ε is a prescribed accuracy. Special cases where source or targe...

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Litt...

It is well known that the family of Hahn polynomials {hα,β n (x;N)}n≥0 is orthogonal with respect to a certain weight function up to N . In this paper we present a factorization for Hahn polynomials for a degree higher than N and we prove that these polynomials can be characterized by a ∆-Sobolev orthogonality. We also present an analogous result for dual-Hahn, Krawtchouk, and Racah polynomials...

where φn(x,y) are the two-variable polynomials which will be shown to be a suitable generalization of the Hermite-Kampé de Fériet (HKdF) family [1] or a particular case of the Boas-Buck polynomials [2]. As it is well known, the HKdF polynomials are generated by (1.1) when f(x) reduces to an exponential function, while in the case of Boas-Buck polynomials, the argument of f should be replaced by...

Abstract. In this work, hierarchical Jacobi-based expansions are explored for the static analysis of multilayered beams, plates, and shells as structural theories well shape functions. Jacobi polynomials, denoted P_p^((γ,θ) ), belong to family classical orthogonal polynomials depend on two scalars parameters

We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several explicit formulas for these polynomials, and investigate their relations with Schubert polynomials, harmonic polynomials, Demazure characters, and generalized Litt...