Type of Document Dissertation Author Acosta Reyes, Ernesto Author's Email Address ernesto.acosta@vanderbilt.edu URN etd-03292009-145945 Title Non-linear optimal signal models and stability of sampling-reconstruction Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Professor Akram Aldroubi Committee Chair Professor Douglas Hardin Committee Member Professor Gieri Simonett Committee Member Professor Larry Schumaker Committee Member Professor Nilanjan Sarkar Committee Member Keywords

- optimal non-linear signal models
- Stability of sampling-reconstruction
Date of Defense 2009-03-26 Availability unrestricted AbstractThis dissertation has two main goals: To study the stability of sampling-reconstructionmodels, and to study the existence of optimal non-linear signal models. In the ¯rst part,

we describe and quantify admissible perturbation of the sampling set X(Jitter), or the

measuring devices (the way the sampling is performed, that is, we average a signal by

¯nite Borel measures), or the generator of the shift-invariant space (class to which belongs

the signal to be sampled and reconstructed from its samples on X). We also study the

reconstruction of a signal belonging to a shift-invariant space from its samples using an

iterative algorithm, and we show that the sequence de¯ned by the algorithm converges to

the original signal geometrically fast. Furthermore, we show that the reconstruction error

due to perturbations we describe is controlled continuously by the perturbation errors.

In the second part, we guarantee the existence of a signal model M =

S

i2I Ci from

observed data F = ff1; : : : ; fmg ½ H that minimize the quantity e(F; fC1; : : : ;Clg) =

Pm

j=1 min1·i·l d2(fj ;Ci), where H is a separable Hilbert space (Problem 1). Su±cient

conditions are given over the class C = fCigi2I in terms of the weak operator topology

in order to guarantee the existence of a minimizer to Problem 1 for any set of data F.

Moreover, we consider the problem when the class C is de¯ned in terms of a collection

of unitary operators applied to a convex subset of H, and we obtain an algorithm for

constructing collections of closed subspaces for which we a priori know that Problem 1 can

be solved. As a consequence, we obtain the well-known qualitative version of the Eckart-

Young Theorem.

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