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 sigma². 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(sigma²/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.