Error Analysis of Frame Reconstruction From Noisy Samples

This paper addresses the problem of reconstructing a continuous function defined on R
^d
from a countable collection of samples corrupted by noise. The additive noise is assumed to be i.i.d. with mean zero and variance sigma
²
. We sample the continuous function / on the uniform lattice (1/m)Z
^d
, and show for large enough m that the variance of the error between the frame reconstruction f
_epsiv,m
from noisy samples of f and the function f satisfy var(f
_epsiv,m
(x) - f(x)) ap(sigma
²
/m
^d
)C
_x
where C
_x
is the best constant for every x isin R
^d
. We also prove a similar result in the case that our data are weighted-average samples of / corrupted by additive noise.