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Sums of Laplace eigenvalues: rotations and tight frames in higher dimensions
 J. Math. Phys
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Isometric tight frames
 Electr. J. Lin. Algeb
, 2002
"... Abstract. A d × n matrix, n ≥ d, whose columns have equal length and whose rows are orthonormal is constructed. This is equivalent to finding an isometric tight frame of n vectors in R d (or C d), or writing the d × d identity matrix I = d ∑n n i=1 Pi, wherethePiare rank 1 orthogonal projections. Th ..."
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Abstract. A d × n matrix, n ≥ d, whose columns have equal length and whose rows are orthonormal is constructed. This is equivalent to finding an isometric tight frame of n vectors in R d (or C d), or writing the d × d identity matrix I = d ∑n n i=1 Pi, wherethePiare rank 1 orthogonal projections. The simple inductive procedure given shows that there are many such isometric tight frames. Key words. Isometric tight frame, normalised tight frame, uniform tight frame. AMS subject classifications. 42C15, 52B15, 42C40
THE ROLE OF FRAME FORCE IN QUANTUM DETECTION
"... Abstract. A general method is given to solve tight frame optimization problems, borrowing notions from classical mechanics. In this paper, we focus on a quantum detection problem, where the goal is to construct a tight frame that minimize an error term, which in quantum physics has the interpretati ..."
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Abstract. A general method is given to solve tight frame optimization problems, borrowing notions from classical mechanics. In this paper, we focus on a quantum detection problem, where the goal is to construct a tight frame that minimize an error term, which in quantum physics has the interpretation of the probability of a detection error. The method converts the frame problem into a set of ordinary differential equations using concepts from classical mechanics and orthogonal group techniques. The minimum energy solutions of the differential equations are proven to correspond to the tight frames that minimize the error term. Because of this perspective, several numerical methods become available to compute the tight frames. Beyond the applications of quantum detection in quantum mechanics, solutions to this frame optimization problem can be viewed as a generalization of classical matched filtering solutions. As such, the methods we develop are a generalization of fundamental detection techniques in radar. 1.
On computing all harmonic frames of n vectors in C^d
, 2006
"... There are a finite number of inequivalent isometric frames (equal–norm tight frames) of n vectors for C^d which are generated from a single vector by applying an abelian group G of symmetries. Each of these socalled harmonic frames can be obtained by taking d rows of the character table of G; often ..."
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There are a finite number of inequivalent isometric frames (equal–norm tight frames) of n vectors for C^d which are generated from a single vector by applying an abelian group G of symmetries. Each of these socalled harmonic frames can be obtained by taking d rows of the character table of G; often in many different ways, which may even include using different abelian groups. Using an algorithm implemented in the algebra package Magma, we determine which of these are equivalent. The resulting list of all harmonic frames for various choices of n and d is freely available, and it includes many properties of the frames such as: a simple description, which abelian groups generate it, identification of the full group of symmetries, the minimum, average and maximum distance between vectors in the frame, and whether it is real or complex, lifted or unlifted. Additional attributes aimed at specific applications include: a measure of the cross correlation (Grassmannian frames), the number of erasures (robustness to erasures), and the diversity product of the full group of its symmetries (multiple–antenna code design). Some outstanding frames are identified and discussed, and a number of questions are answered by considering the examples on the list.
GROUP RECONSTRUCTION SYSTEMS
 ELA
, 2011
"... We consider classes of reconstruction systems (RS’s) for finite dimensional real or complex Hilbert spaces H, called group reconstruction systems (GRS’s), that are associated with representations of finite groups G. These GRS’s generalize frames with high degree of symmetry, such as harmonic or geom ..."
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We consider classes of reconstruction systems (RS’s) for finite dimensional real or complex Hilbert spaces H, called group reconstruction systems (GRS’s), that are associated with representations of finite groups G. These GRS’s generalize frames with high degree of symmetry, such as harmonic or geometrically uniform ones. Their canonical dual and canonical Parseval are shown to be GRS’s. We establish simple conditions for oneerasure robustness. Projective GRS’s, that can be viewed as fusion frames, are also considered. We characterize the Gram matrix of a GRS in terms of block group matrices. Unitary equivalences and unitary symmetries of RS’s are studied. The relation between the irreducibility of the representation and the tightness of the GRS is established. Taking into account these results, we consider the construction of Parseval, projective and oneerasure robust GRS’s.
Orthogonal polynomials on the disc
 J. Approx. Theory
"... This article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproductio ..."
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This article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit:
Stability of Phase Retrievable Frames
 proceedings of SPIE 2013
"... In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set F of m vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then th ..."
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In this paper we study the property of phase retrievability by redundant sysems of vectors under perturbations of the frame set. Specifically we show that if a set F of m vectors in the complex Hilbert space of dimension n allows for vector reconstruction from magnitudes of its coefficients, then there is a perturbation bound ρ so that any frame set within ρ from F has the same property. In particular this proves the recent construction in15 is stable under perturbations. By the same token we reduce the critical cardinality conjectured in11 to proving a stability result for non phaseretrievable frames.
ELA ISOMETRIC TIGHT FRAMES∗
"... Abstract. A d × n matrix, n ≥ d, whose columns have equal length and whose rows are orthonormal is constructed. This is equivalent to finding an isometric tight frame of n vectors in R d (or C ..."
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Abstract. A d × n matrix, n ≥ d, whose columns have equal length and whose rows are orthonormal is constructed. This is equivalent to finding an isometric tight frame of n vectors in R d (or C