We consider a filtration of the symmetric function space given by Λ (k) t , the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than k. We introduce symmetric functions called the k-Schur functions, providing an analog for the Schur functions in the subspaces Λ (k) t . We prove several properties for the k-Schur functions including that they form ...

In this paper, we consider the problem of interpolating univariate polynomials over a eld of characteristic zero that are sparse in (a) the Pochhammer basis or, (b) the Chebyshev basis. The polynomials are assumed to be given by black boxes, i.e., one can obtain the value of a polynomial at any point by querying its black box. We describe eecient new algorithms for these problems. Our algorithm...

Here we give a simple proof of a new representation for orthogonal polynomials over triangular domains which overcomes the need to make symmetry destroying choices to obtain an orthogonal basis for polynomials of fixed degree by employing redundancy. A formula valid for simplices with Jacobi weights is given, and we exhibit its symmetries by using the Bernstein–Bézier form. From it we obtain th...

We show that discrete exponentials form a basis of discrete holomorphic functions. On a convex, the discrete polynomials form a basis as well.

The idea of `1-minimization is the basis of the widely adopted compressive sensing method for function approximation. In this paper, we extend its application to high-dimensional stochastic collocation methods. To facilitate practical implementation, we employ orthogonal polynomials, particularly Legendre polynomials, as basis functions, and focus on the cases where the dimensionality is high s...

Denote by Σnm the sum of the m-th powers of the first n positive integers 1m + 2m + . . .+ nm. Similarly let Σrnm be the r-fold sum of the m-th powers of the first n positive integers, defined such that Σn = nm, and then recursively by Σn = Σr1m+Σr2m+ . . .+Σrnm. During the early 17th-century, polynomial expressions for the sums Σrnm and their factorisation and polynomial basis representation p...

We express Wronskian Hermite polynomials in the basis and obtain an explicit formula for coefficients. From this we deduce upper bound modulus of roots case partitions length 2. also derive a general real purely imaginary roots. These bounds are very useful study irreducibility polynomials. Additionally, generalize some our results to larger class

We study the coinvariant ring of the complex reflection group G(r, p, n) as a module for the corresponding rational Cherednik algebra H and its generalized graded affine Hecke subalgebra H. We construct a basis consisting of non-symmetric Jack polynomials, and using this basis decompose the coinvariant ring into irreducible modules for H. The basis consists of certain non-symmetric Jack polynom...