We investigate shift invariant subspaces of $L^2(G)$, where $G$ is a locally compact abelian group. We show that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame. For a second countable locally compact abelian group $G$ we prove a useful Hilbert space isomorphism, introduce range funct...

Frames are the foundation of the linear operators used in the decomposition and reconstruction of signals, such as the discrete Fourier transform, Gabor, wavelets, and curvelet transforms. The emergence of sparse representation models has shifted of the emphasis in frame theory toward sparse l1-minimization problems. In this paper, we apply frame theory to the sparse representation of signals i...

We evaluate the Berkeley parser on text from an online discussion forum. We evaluate the parser output with and without gold tokens and spellings (using Sparseval and Parseval), and we compile a list of problematic phenomena for this domain. The Parseval f-score for a small development set is 77.56. This increases to 80.27 when we apply a set of simple transformations to the input sentences and...

Recent efforts have shown that the reconstruction performance could be improved with optimized sensing matrix according to a given dictionary for a compressed sensing (CS) system. The existed optimizing conditions are mainly used to address the worst-case performance of CS recovery. Considering the quality of a sensing matrix with respect to the mean squared error (MSE) performance of the Oracl...

Motivated by recent developments in quantum simulation of synthetic dimensions, e.g., optical lattices ultracold atoms, we discuss here d-dimensional periodic, gapped systems for d ≤ 4, with a focus on the topology occupied energy states. We perform this analysis asking whether spectral subspace below gap can be spanned smooth and periodic Bloch functions, corresponding to localized Wannier fun...

Methods for constructing finite Parseval frames by using Walsh matrices and the discrete Vilenkin–Chrestenson transform are described.