Cubature Formulas of Multivariate Polynomials Arising from Symmetric Orbit Functions

Abstract

Abstract:The paper develops applications of symmetric orbit functions, known from irreducible representations of simple Lie groups, in numerical analysis. It is shown that these functions have remarkable properties which yield to cubature formulas, approximating a weighted integral of any function by a weighted finite sum of function values, in connection with any simple Lie group. The cubature formulas are specialized for simple Lie groups of rank two. An optimal approximation of any function by multivariate polynomials arising from symmetric orbit functions is discussed.

Dual-Root Lattice Discretization of Weyl Orbit Functions

“…Existence of generalization of the dualweight lattice orthogonality of the selected subset of Macdonald polynomials [7] to the dual-root lattice poses an open problem. The related polynomial interpolation and approximation methods, cubature formulas [15] and their comparison to the weight and dual weight versions deserve further study. • Besides developing novel discrete transforms on generalized and composed grids, other fundamentally different options are provided by existence of functions invariant with respect to some subgroups of the given Weyl group.…”

“…Investigating Weyl orbit functions as special functions, the results range from generalizations of continuous multivariate Fourier transforms in [20,21] to generalized Chebyshev polynomial methods [24,29]. Discrete Fourier methods are comprehensively studied for Weyl orbit functions [14,16,17] as well as for their multivariate Chebyshev polynomial generalizations [7,15,27,29]. The refinement of the dual weight lattice intersected with the fundamental domain of the affine Weyl group form a finite point set on which the majority of the discrete Fourier and Chebyshev methods is developed [14,16,24,27].…”

Discrete cosine and sine transforms generalized to honeycomb lattice

“…• The families of C− and S−functions induce two kinds of discretely orthogonal generalized Chebyshev polynomials. Cubature formulas for numerical integration are among the recently studied associated polynomial methods [14,26,27]. As a linear combination of C− and S− functions, each case of the honeycomb functions generates a set of polynomials discretely orthogonal on points forming a deformed honeycomb pattern inside the Steiner's hypocycloid.…”

“…Discrete transforms of the Weyl orbit functions over finite sets of points and their applications are, of recent, intensively studied [13][14][15][16][17]24]. The majority of these methods arise in connection with point sets taken as the fragments of the refined dual-weight lattice.…”

Fast Cosine Transform for FCC Lattices

“…The multivariate Chebyshev polynomials [15] are now classical, but have found applications only in the last years. Apart from the applications in algebraic signal processing they are applied in the discretization of partial differential equations in [16], [17], for the derivation of cubature formulas in [18], [19], [20] and developing discrete transforms in [21].…”