Type of Document 
Dissertation 
Author 
Leonetti, Casey Clark

Author's Email Address 
casey.leonetti@alumni.vanderbilt.edu 
URN 
etd03292007180005 
Title 
Reconstruction from ErrorAffected Sampled Data in ShiftInvariant Spaces 
Degree 
PhD 
Department 
Mathematics 
Advisory Committee 
Advisor Name 
Title 
Akram Aldroubi 
Committee Chair 
Benoit Dawant 
Committee Member 
Douglas P. Hardin 
Committee Member 
Guoliang Yu 
Committee Member 
Larry Schumaker 
Committee Member 

Keywords 
 jitter
 noise
 sampling
 shiftinvariant space
 frame
 fourier

Date of Defense 
20070328 
Availability 
unrestricted 
Abstract
In the following chapters we provide error estimates for signals reconstructed from corrupt data. Two different types of error are considered. First, we address 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 0 and variance σ^{2}. We sample the continuous function f 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_{ε} from noisy samples of f and the function f satisfy var(f_{ε} (x)f(x))≈ (σ^{2}/m^{d})C_{x}. Second, we address the problem of nonuniform sampling and reconstruction in the presence of jitter. In sampling applications, the set X={x_{j}: j ∈ J} on which a signal f is sampled is not precisely known. Two main questions are considered. First, if sampling a function f on the countable set X leads to unique and stable reconstruction of f, then when does sampling on the set X'={x_{j}+δ_{j}: j ∈ J} also lead to unique and stable reconstruction? Second, if we attempt to recover a sampled function f using the reconstruction operator corresponding to the sampling set X (because the precise sample points are unknown), is the recovered function a good approximation of the original f?

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