Central Splitting of A2 Discrete Fourier–Weyl Transforms
Abstract
:1. Introduction
2. Splitting Weight and Point Sets
2.1. Root and Weight Lattices
2.2. Splitting Weight Sets
2.3. Splitting Point Sets
3. Weight Lattice Fourier–Weyl Transforms
3.1. C- and S-Functions
3.2. Discrete Orthogonality
3.3. Splitting Transforms
4. Central Splitting
4.1. Central Splitting of Discrete Transforms
4.2. Decompositions of Unitary Transform Matrices
4.3. Decompositions of Transform Matrices and
5. Concluding Remarks
- The decompositions of the weight lattice Fourier–Weyl transforms into the splitting transforms are considered as a point of departure in exploring the fast bivariate discrete Fourier transforms involving the (anti)symmetric orbit functions of the Weyl group . The central splitting method offers an advantageous approach to computational efficiency, using the reduction of the initial weight lattice Fourier–Weyl transform into three smaller-weight lattice splitting transforms. Even though the recursive method to determine the further splitting of currently developed two-variable cosine and sine transforms has not been determined yet, the demonstrated reduced weight transform is considered as a stepping stone towards the fast discrete transforms. Moreover, analogously to the interpolation tests conducted for cosine and sine discrete transforms [2,17,18], the developed discrete splitting transforms also manifest excellent interpolation properties.
- For the crystallographic reflection group , repeatedly using the central splitting of one-variable discrete cosine and sine transforms produces the standard versions of the fast split-radix transforms [4]. In this case, the possibility of rearranging the splitting point sets into the original format governed by the affine Weyl group ensures the central splitting method’s recursive behavior. The decompositions of the transform matrices, similar to the formulated unitary matrix decompositions in Theorem 2, have been rigorously proven for the one-dimensional sine and cosine transforms in [30]. However, in the case of the group, since a further splitting of the points in the kite-shaped domain has not been formulated yet, the recursive central splitting method remains an unsolved problem.
- Given the importance of multi-dimensional digital data processing [16,37,38,39,40], a central-splitting mechanism could be potentially developed for other compact simple Lie groups with non-trivial elements of the center, such as with its center provided by a cyclic group of elements, , and with the center given by a cyclic group of order 2, whose center contains 4 elements, that equivalently to has 3 elements of the center. Such an approach would be considered as a first step to a general multidimensional fast transform. Since a similar behavior of the central splitting in the case of the finite reflection group is expected, the extension of the developed Fourier–Weyl transforms to higher-dimensional cases should be treated independently.
- Another family of orbit functions, known as E-functions, is obtained by symmetrizing multivariate exponential terms over even subgroups of a considered Weyl group. Such functions are developed in [13], and their corresponding Fourier–Weyl transforms together with continuous interpolations are examined in detail [41]. There is one type of the E-functions for the root systems with the roots of one length [3]. For the root systems with two lengths of simple roots, the six types of E-functions, together with their even complex-valued dual weight lattice Fourier–Weyl transforms, are formulated in [14]. Hence, instead of the discrete transforms based on the C- and S-functions, the central splitting of the transforms developed by means of the E-functions represents an open problem. Furthermore, the Fourier–Weyl transforms with their kernel represented by the combinations of different types of Weyl orbit functions have not been previously explored.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Hrivnák, J.; Myronova, M.; Patera, J. Central Splitting of A2 Discrete Fourier–Weyl Transforms. Symmetry 2020, 12, 1828. https://doi.org/10.3390/sym12111828
Hrivnák J, Myronova M, Patera J. Central Splitting of A2 Discrete Fourier–Weyl Transforms. Symmetry. 2020; 12(11):1828. https://doi.org/10.3390/sym12111828
Chicago/Turabian StyleHrivnák, Jiří, Mariia Myronova, and Jiří Patera. 2020. "Central Splitting of A2 Discrete Fourier–Weyl Transforms" Symmetry 12, no. 11: 1828. https://doi.org/10.3390/sym12111828
APA StyleHrivnák, J., Myronova, M., & Patera, J. (2020). Central Splitting of A2 Discrete Fourier–Weyl Transforms. Symmetry, 12(11), 1828. https://doi.org/10.3390/sym12111828