TY - UNPB
T1 - Stable decomposition of homogeneous Mixed-norm Triebel-Lizorkin spaces
AU - Nielsen, M
PY - 2022/6/8
Y1 - 2022/6/8
N2 - We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel-Lizorkin spaces in an anisotropic setting on $\bR^d$. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel-Lizorkin spaces. In the second part of the paper we study nonlinear $m$-term nonlinear approximation with the constructed basis in the mixed-norm setting, where the behaviour, in general, for $d\geq 2$, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for $m$-term approximation can still be derived.
AB - We construct smooth localized orthonormal bases compatible with homogeneous mixed-norm Triebel-Lizorkin spaces in an anisotropic setting on $\bR^d$. The construction is based on tensor products of so-called univariate brushlet functions that are constructed using local trigonometric bases in the frequency domain. It is shown that the associated decomposition system form unconditional bases for the homogeneous mixed-norm Triebel-Lizorkin spaces. In the second part of the paper we study nonlinear $m$-term nonlinear approximation with the constructed basis in the mixed-norm setting, where the behaviour, in general, for $d\geq 2$, is shown to be fundamentally different from the unmixed case. However, Jackson and Bernstein inequalities for $m$-term approximation can still be derived.
KW - math.FA
KW - 42B35, 42C15, 41A17, 41A65
U2 - 10.48550/arXiv.2206.03724
DO - 10.48550/arXiv.2206.03724
M3 - Preprint
BT - Stable decomposition of homogeneous Mixed-norm Triebel-Lizorkin spaces
PB - arXiv
ER -