TY - JOUR
T1 - Riemann Localisation on the Sphere
AU - Wang, Yu Guang
AU - Sloan, Ian H.
AU - Womersley, Robert S.
PY - 2018/2
Y1 - 2018/2
N2 - This paper first shows that the Riemann localisation property holds for the Fourier-Laplace series partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimension. By Riemann localisation on the sphere Sd ⊂ Rd+1, d ≥ 2, we mean that for a suitable subset X of Lp (Sd), 1 ≤ p ≤ ∞, the Lp-norm of the Fourier local convolution of ƒ ∈ X converges to zero as the degree goes to infinity. The Fourier local convolution of ƒ at x ∈ Sd is the Fourier convolution with a modified version of ƒ obtained by replacing values of ƒ by zero on a neighbourhood of x. The failure of Riemann localisation for d > 2 can be overcome by considering a filtered version: we prove that for a sphere of any dimension and sufficiently smooth filter the corresponding local convolution always has the Riemann localisation property. Key tools are asymptotic estimates of the Fourier and filtered kernels.
AB - This paper first shows that the Riemann localisation property holds for the Fourier-Laplace series partial sum for sufficiently smooth functions on the two-dimensional sphere, but does not hold for spheres of higher dimension. By Riemann localisation on the sphere Sd ⊂ Rd+1, d ≥ 2, we mean that for a suitable subset X of Lp (Sd), 1 ≤ p ≤ ∞, the Lp-norm of the Fourier local convolution of ƒ ∈ X converges to zero as the degree goes to infinity. The Fourier local convolution of ƒ at x ∈ Sd is the Fourier convolution with a modified version of ƒ obtained by replacing values of ƒ by zero on a neighbourhood of x. The failure of Riemann localisation for d > 2 can be overcome by considering a filtered version: we prove that for a sphere of any dimension and sufficiently smooth filter the corresponding local convolution always has the Riemann localisation property. Key tools are asymptotic estimates of the Fourier and filtered kernels.
KW - Filtered polynomial approximation
KW - Riemann-Lebesgue lemma
KW - Localization
KW - Dirichlet kernel
KW - Jacobi weights
KW - POINTWISE FOURIER INVERSION
KW - EIGENFUNCTION-EXPANSIONS
KW - COMPACT MANIFOLDS
KW - PINSKY PHENOMENON
KW - WAVELETS
KW - NEEDLETS
KW - FRAMES
UR - http://www.scopus.com/inward/record.url?scp=84982840967&partnerID=8YFLogxK
UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84982840967&origin=recordpage
U2 - 10.1007/s00041-016-9496-4
DO - 10.1007/s00041-016-9496-4
M3 - 21_Publication in refereed journal
VL - 24
SP - 141
EP - 183
JO - Journal of Fourier Analysis and Applications
JF - Journal of Fourier Analysis and Applications
SN - 1069-5869
IS - 1
ER -