Abstract
Discrete transforms of Weyl orbit functions on finite fragments of shifted dual root lattices are established. The congruence classes of the dual weight lattices intersected with the fundamental domains of the affine Weyl groups constitute the point sets of the transforms. The shifted weight lattices intersected with the fundamental domains of the extended dual affine Weyl groups form the sets of labels of Weyl orbit functions. The coinciding cardinality of the point and label sets and corresponding discrete orthogonality relations of Weyl orbit functions are demonstrated. The explicit counting formulas for the numbers of elements contained in the point and label sets are calculated. The forward and backward discrete Fourier-Weyl transforms, together with the associated interpolation and Plancherel formulas, are presented. The unitary transform matrices of the discrete transforms are exemplified for the case A 2 .
Highlights
The purpose of this article is to develop discrete Fourier-Weyl transforms [1,2,3,4,5,6] on finite fragments of shifted dual root lattices that correspond to affine Weyl groups
The kernels of the discrete transforms are formed by four types of complex-valued Weyl orbit functions [7,8] that are labeled by shifted weight lattices [9]
The basic boundary behavior of each Weyl orbit function on the borders of the fundamental domains is determined by the underlying sign homomorphism [2], § 4
Summary
The purpose of this article is to develop discrete Fourier-Weyl transforms [1,2,3,4,5,6] on finite fragments of shifted dual root lattices that correspond to affine Weyl groups. Each developed discrete transform manifests a unique boundary behavior that depends on the type of orbit function, together with underlying lattice shifts of its point and label sets. The extended affine Weyl group and its fundamental domain determine the labels of orbit functions that correspond to the dual-root lattice. The entire group-theoretical formalism of shifted discrete transforms potentially generalizes the 16 classical types of discrete cosine and sine transforms together with 32 multivariate (anti)symmetric trigonometric transformations that are related to systems A1 and Cn , respectively, to the entire collection of crystallographic root systems. Comments and follow-up questions are included in the last section
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