Projection Constants of Polynomial Spaces via Representation Theory
Abstract
We develop a representation-theoretic framework for computing projection constants of finite-dimensional subspaces of C(K), where K is a compact homogeneous G-space. Within this framework, we characterize the finite-dimensional G-invariant subspaces for which the G-equivariant projection is unique. These are precisely the finite orthogonal sums of full isotypic components in the Peter--Weyl decomposition of L2(K). For such subspaces, averaging shows that this unique G-equivariant projection is minimal; it is the restriction to C(K) of the L2(K)-orthogonal projection. Consequently, their projection constants are given by the L1-norm of an explicit reproducing-kernel slice. We apply this method to spaces of d-homogeneous polynomials in high dimension. The examples range from Fourier analysis on the torus, through spherical harmonic analysis on real and complex Euclidean spheres, to genuinely noncommutative harmonic analysis on the unitary group, where representation theory becomes indispensable. For all these families, we obtain precise high-dimensional asymptotics at the square-root-of-dimension scale.
Related
- Projection Constants and Banach-Mazur Distance in Codimension Two
- Stability of maximal relative projection constants
- From Sidelnikov-Welch bounds to projection constants
- Maximal Projection Constants and Extremal Vector Configurations: Some Conjectures and Examples
- Polynomial growth of Bohnenblust--Hille constants on the Hamming cube
BibTeX
@misc{defant2026projection,
title = {{Projection Constants of Polynomial Spaces via Representation Theory}},
author = {Andreas Defant and Daniel Galicer and Martín Mansilla and Mieczysław Mastyło and Santiago Muro and Pablo Zadunaisky},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22050},
eprint = {2609.22050},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22050},
}
RIS
TY - UNPB
TI - Projection Constants of Polynomial Spaces via Representation Theory
AU - Andreas Defant
AU - Daniel Galicer
AU - Martín Mansilla
AU - Mieczysław Mastyło
AU - Santiago Muro
AU - Pablo Zadunaisky
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22050
UR - https://arxiv.org/abs/2609.22050
AB - We develop a representation-theoretic framework for computing projection constants of finite-dimensional subspaces of C(K), where K is a compact homogeneous G-space. Within this framework, we characterize the finite-dimensional G-invariant subspaces for which the G-equivariant projection is unique. These are precisely the finite orthogonal sums of full isotypic components in the Peter--Weyl decomposition of L_2(K). For such subspaces, averaging shows that this unique G-equivariant projection is minimal; it is the restriction to C(K) of the L_2(K)-orthogonal projection. Consequently, their projection constants are given by the L_1-norm of an explicit reproducing-kernel slice. We apply this method to spaces of d-homogeneous polynomials in high dimension. The examples range from Fourier analysis on the torus, through spherical harmonic analysis on real and complex Euclidean spheres, to genuinely noncommutative harmonic analysis on the unitary group, where representation theory becomes indispensable. For all these families, we obtain precise high-dimensional asymptotics at the square-root-of-dimension scale.
ER -