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Register a new userBinary Parseval frames from group orbits
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Binary Parseval frames share many structural properties with real and complex ones. On the other hand, there are subtle differences, for example that the Gramian of a binary Parseval frame is characterized as a symmetric idempotent whose range contains at least one odd vector. Here, we study binary Parseval frames obtained from the orbit of a vector under a group representation, in short, binary Parseval group frames. In this case, the Gramian of the frame is in the algebra generated by the right regular representation. We identify equivalence classes of such Parseval frames with binary functions on the group that satisfy a convolution identity. This allows us to find structural constraints for such frames. We use these constraints to catalogue equivalence classes of binary Parseval frames obtained from group representations. As an application, we study the performance of binary Parseval frames generated with abelian groups for purposes of error correction. We show that $Z_p^q$ is always preferable to $Z_{p^q}$ when searching for best performing codes associated with binary Parseval group frames.
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Parseval frames have particularly useful properties, and in some cases, they can be used to reconstruct signals which were analyzed by a non-Parseval frame. In this paper, we completely describe the degree to which such reconstruction is feasible. Indeed, notice that for fixed frames $cF$ and $cX$ with synthesis operators $F$ and $X$, the operator norm of $FX^*-I$ measures the (normalized) worst-case error in the reconstruction of vectors when analyzed with $cX$ and synthesized with $cF$. Hence, for any given frame $cF$, we compute explicitly the infimum of the operator norms of $FX^*-I$, where $cX$ is any Parseval frame. The $cX$s that minimize this quantity are called Parseval quasi-dual frames of $cF$. Our treatment considers both finite and infinite Parseval quasi-dual frames.
We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzons bound is attained in each unitary geometry of dimension $d = 2^{2l+1}$ over the field $mathbb{F}_{3^2}$. We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.
A very useful identity for Parseval frames for Hilbert spaces was obtained by Balan, Casazza, Edidin, and Kutyniok. In this paper, we obtain a similar identity for Parseval p-approximate Schauder frames for Banach spaces which admits a homogeneous semi-inner product in the sense of Lumer-Giles.
The general construction of frames of p-adic wavelets is described. We consider the orbit of a mean zero generic locally constant function with compact support (mean zero test function) with respect to the action of the p-adic affine group and show that this orbit is a uniform tight frame. We discuss relation of this result to the multiresolution wavelet analysis.
The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains. We consider the symmetric group, $S_n$, one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of emph{global functions} on $S_n$, which are functions whose $2$-norm remains small when restricting $O(1)$ coordinates of the input, and assert that low-degree, global functions have small $q$-norms, for $q>2$. As applications, we show: 1. An analog of the level-$d$ inequality on the hypercube, asserting that the mass of a global function on low-degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group $A_n$. 2. Isoperimetric inequalities on the transposition Cayley graph of $S_n$ for global functions, that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube. 3. Hypercontractive inequalities on the multi-slice, and stabili
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