Linear independence of wavelet systems and translations of functions in Lp(R2)

Edgar and Rosenblatt have proven that any nonzero function f in Lp( Rn ) with 1 ≤ p ≤ 2n/(n-1) has linearly independent translates. In fact, restricting to n=2, they showed that there is a nonzero function f in Lp( R2 ) for all p>4, with linearly dependent translates. In this dissertation we will show that in order for a set of translations of a non-trivial function f to be linearly dependent and to be in Lp( R2 ) for all p>4, the cardinality of the translation set must be greater than or equal to four. We also show that for any integer K ≥ 4, there exists a function f in Lp( R2 ) for all p>4 such that there is a linearly dependent set of translations of f with cardinality K. This dissertation will also discuss linear independence in the context of wavelet systems, following the work of Christensen and Linder, who showed that, under certain hypotheses, wavelet systems can be decomposed into finitely many linearly independent sets. We will discuss our work on this problem which focuses on extending the result to higher dimensions.