Multivariate approximation by translates of the Korobov function on Smolyak grids
For a set W ⊂ Lp(Td), 1 < p < ∞, of multivariate periodic functions on the torus Td and a given function ϕ ∈ Lp(Td), we study the approximation in the Lp(Td)-norm of functions f ∈ W by arbitrary linear combinations of n translates of ϕ. For W = Ur p (Td) and ϕ = κr,d, we prove upper bounds...
Saved in:
| Main Author: | |
|---|---|
| Format: | Book Book chapter Dataset |
| Published: |
Journal of Complexity
2016
|
| Subjects: | |
| Online Access: | http://repository.vnu.edu.vn/handle/VNU_123/11124 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Institution: | Vietnam National University, Hanoi |
| Summary: | For a set W ⊂ Lp(Td), 1 < p < ∞, of multivariate periodic functions on the torus Td and a
given function ϕ ∈ Lp(Td), we study the approximation in the Lp(Td)-norm of functions f ∈ W
by arbitrary linear combinations of n translates of ϕ. For W = Ur
p (Td) and ϕ = κr,d, we prove
upper bounds of the worst case error of this approximation where Ur
p (Td) is the unit ball in the
Korobov space Kr
p(Td) and κr,d is the associated Korobov function. To obtain the upper bounds,
we construct approximation methods based on sparse Smolyak grids. The case p = 2, r > 1/2,
is especially important since Kr
2 (Td) is a reproducing kernel Hilbert space, whose reproducing
kernel is a translation kernel determined by κr,d. We also provide lower bounds of the optimal
approximation on the best choice of ϕ. |
|---|