on the construction of equiangular frames from graphs abstract we give details of the 1 1 correspondence between equiangular frames of n vectors for r d and graphs with n vertices this has been studied recently for tight equiangular frames because of applications to signal processing and quantum information theory the nontight examples given here which correspond to graphs with more than 2 eigenvalues have the potential for similar applications e g the frame corresponding to the 5 cycle graph is the unique grassmannian frame of 5 vectors in openr 3 further the associated canonical tight frames have a small number of angles in many cases

optimal order preconditioners for the morse ingard equations abstract the morse ingard equations of thermoacoustics morse and ingard 1986 are a system of coupled time harmonic equations for the temperature and pressure of an excited gas they form a critical aspect of modeling trace gas sensors in this paper we analyze a reformulation of the system that has a weaker coupling between the equations than the original form we give a garding type inequality for the system that leads to optimal order asymptotic finite element error estimates we also develop preconditioners for the coupled system these are derived by writing the system as a 2 2 block system with pressure and temperature unknowns segregated into separate blocks and then using either the block diagonal or block lower triangular part of this matrix as a preconditioner consequently the preconditioner requires inverting smaller helmholtz like systems individually for the pressure and temperature rigorous eigenvalue bounds are given for the preconditioned system and these are supported by numerical experiments

hilbert space frames containing a riesz basis and banach spaces which have no subspace isomorphic toc0 abstract we prove that a hilbert space frame fi i icontains a riesz basis if every subfamily fi i j j i is a frame for its closed span secondly we give a new characterization of banach spaces which do not have any subspace isomorphic toc0 this result immediately leads to an improvement of a recent theorem of holub concerning frames consisting of a riesz basis plus finitely many elements

burst erasures and the mean square error for cyclic parseval frames this paper investigates the performance of frames for the linear redundant encoding of vectors when consecutive frame coefficients are lost due to the occurrence of random burst errors we assume that the distribution of bursts is invariant under cyclic shifts and that the burst length statistics are known in analogy with rate distortion theory we wish to find frames of a given size which minimize the mean square reconstruction error for the encoding of vectors in a complex finite dimensional hilbert space we obtain an upper bound for the mean square reconstruction error for a given parseval frame and in the case of cyclic parseval frames we find a family of frames which minimizes this upper bound under certain conditions these minimizers are identical to complex bose chaudhuri hocquenghem codes discussed in the literature the accuracy of our upper bounds for the mean square error is substantiated by complementary lower bounds all estimates are based on convexity arguments and a discrete rearrangement inequality

complex hadamard matrices and equiangular tight frames abstract in this paper we give a new construction of parametric families of complex hadamard matrices of square orders and connect them to equiangular tight frames the results presented here generalize some of the recent ideas of bodmann et al 3 and extend the list of known equiangular tight frames in particular a 36 21 frame coming from a nontrivial cube root signature matrix is obtained for the first time

maximum robustness and surgery of frames in finite dimensions abstract we consider frames in a finite dimensional hilbert space h n where frames are exactly the spanning sets of the vector space we present a method to determine the maximum robustness of a frame we present results on tight subframes and surgery of frames we also answer the question of when length surgery resulting in a tight frame set for h n is possible

optimal reconstruction systems for erasures and for the q potential abstract in this paper we introduce the q potential as an extension of the benedetto fickus frame potential defined on general reconstruction systems and show that protocols are the minimizers of this potential under certain restrictions we extend recent results of b g bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst normalized reconstruction error is computed with respect to a compatible unitarily invariant norm we finally describe necessary and sufficient spectral conditions that we call q fundamental inequalities for the existence of protocols with prescribed properties by relating this problem to klyachko s and fulton s theory on sums of hermitian operators

some properties of g frames in hilbert c modules in this paper we give some new results for g frames in hilbert c modules and then we introduce a bounded a linear operator l by means of this operator l we character the properties of the g frames and g riesz basis in hilbert c modules finally we establish some important equalities and inequalities for frames and g frames in hilbert c modules

frame theoretic analysis of robust filter bank frames to quantization and erasures this paper presents the theoretic analysis for the robustness of filter bank fb frames in infinite dimensional hilbert space l2 bbz to quantization and erasures as well as studies the design of such robust frames from the perspective of both frame and fb theory first a characterization of the eigenstructure for the frame operator and the induced gram matrix of general fb frames is presented the robustness of fb frames to erasures is investigated in detail especially on the necessary and sufficient condition maximally robust frames are further analyzed by explicitly constructive methods moreover the stability and even the possible fir reconstruction of the subframe by the pseudoinverse are studied thoroughly next we examine the optimal quantized fb frames introducing a novel notion named as frame energy the universal optimality of tight fb frames to quantization is established in contrast to the optimality of only equal norm tight frames shown in previous works the added design freedom obtained by removing the equal norm constraint is explained and illustrated with examples finally the effect of erasures on the quantized fb frames is studied by incorporating the probability of erasures which shows the universal optimality of uniform tight fb frames for one erasure the optimal fb frame is usually not of equal norm and actually its norm distribution follows the reverse waterfilling principle an example of two state erasure channel model is shown to further explain our result followed by an analysis of the successive reconstruction which shows its potential application to frame based multiple description coding

properties of real and complex etfs and their application to the design of low coherence frames certain properties of an equiangular tight frame are contained in the sign or phase of its corresponding gram matrix in this paper we convey new properties for gram matrices of equiangular tight frames and use them to construct new frames with low average and worst case coherences based on these results we also provide properties of the synthesis matrix of equiangular tight frames

harmonic equiangular tight frames comprised of regular simplices abstract an equiangular tight frame etf is a sequence of unit norm vectors in a euclidean space whose coherence achieves equality in the welch bound and thus yields an optimal packing in a projective space a regular simplex is a simple type of etf in which the number of vectors is one more than the dimension of the underlying space more sophisticated examples include harmonic etfs which equate to difference sets in finite abelian groups recently it was shown that some harmonic etfs are comprised of regular simplices in this paper we continue the investigation into these special harmonic etfs we begin by characterizing when the subspaces that are spanned by the etf s regular simplices form an equi isoclinic tight fusion frame eitff which is a type of optimal packing in a grassmannian space we shall see that every difference set that produces an eitff in this way also yields a complex circulant conference matrix next we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set it turns out that these relative difference sets lend themselves to a second related and yet distinct construction of complex circulant conference matrices finally we provide explicit infinite families of etfs to which this theory applies

kirkman equiangular tight frames and codes an equiangular tight frame etf is a set of unit vectors in a euclidean space whose coherence is as small as possible equaling the welch bound also known as welch bound equality sequences such frames arise in various applications such as waveform design and compressed sensing at the moment there are only two known flexible methods for constructing etfs harmonic etfs are formed by carefully extracting rows from a discrete fourier transform steiner etfs arise from a tensor like combination of a combinatorial design and a regular simplex these two classes seem very different the vectors in harmonic etfs have constant amplitude whereas steiner etfs are extremely sparse we show that they are actually intimately connected a large class of steiner etfs can be unitarily transformed into constant amplitude frames dubbed kirkman etfs moreover we show that an important class of harmonic etfs is a subset of an important class of kirkman etfs this connection informs the discussion of both types of frames some steiner etfs can be transformed into constant amplitude waveforms making them more useful in waveform design some harmonic etfs have low spark making them less desirable for compressed sensing we conclude by showing that real valued constant amplitude etfs are equivalent to binary codes that achieve the grey rankin bound and then construct such codes using kirkman etfs

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

duality in reconstruction systems we consider the notion of finite dimensional reconstructions systems rs s which includes the fusion frames as projective rs s we study erasures some geometrical properties of these spaces the spectral picture of the set of all dual systems of a fixed rs the spectral picture of the set of rs operators for the projective systems with fixed weights and the structure of the minimizers of the joint potential in this setting we give several examples

equiangular tight frames that contain regular simplices abstract an equiangular tight frame etf is a type of optimal packing of lines in euclidean space a regular simplex is a special type of etf in which the number of vectors is one more than the dimension of the space they span in this paper we consider etfs that contain a regular simplex that is have the property that a subset of its vectors forms a regular simplex as we explain such etfs are characterized as those that achieve equality in a certain well known bound from the theory of compressed sensing we then consider the so called binder of such an etf namely the set of all regular simplices that it contains we provide a new algorithm for computing this binder in terms of products of entries of the etf s gram matrix in certain circumstances we show this binder can be used to produce a particularly elegant naimark complement of the corresponding etf other times an etf is a disjoint union of regular simplices and we show this leads to a certain type of optimal packing of subspaces known as an equichordal tight fusion frame we conclude by considering the extent to which these ideas can be applied to numerous known constructions of etfs including harmonic etfs

binary parseval frames from group orbits abstract 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 if p is an odd prime then the group z p q is always preferable to z p q when searching for best performing codes associated with binary parseval group frames

causal compensation for erasures in frame representations in a variety of signal processing and communications contexts erasures occur inadvertently or can be intentionally introduced as part of a data reduction strategy this paper discusses causal compensation for erasures in frame representations of signals the approach described assumes linear synthesis of the signal using a prespecified frame but no specific generation mechanism for the coefficients under this assumption it is demonstrated that erasures can be compensated for using low complexity causal systems if the transmitter is aware of the occurrence of the erasure an optimal compensation is to project the erasure error to the remaining coefficients it is demonstrated that the same compensation can be executed using a transmitter receiver combination in which the transmitter is not aware of the erasure occurrence the transmitter precompensates using projections as if assuming erasures will occur the receiver undoes the compensation for the coefficients that have not been erased thus maintaining the compensation only of the erased coefficients the stability of the resulting systems is explored and stability conditions are derived it is shown that stability for any erasure pattern can be enforced by optimizing a constrained quadratic program at the system design stage the paper concludes with examples and simulations that verify the theoretical results and illustrate key issues in the algorithms

linearly connected sequences and spectrally optimal dual frames for erasures abstract in the case that a frame is prescribed for applications and erasures occur in the process of data transmissions we examine optimal dual frames for the recovery from single erasures in contrast to earlier papers we consider the spectral radius of the error operator instead of its operator norm as a measure of optimality this notion of optimality is natural when the neumann series is used to recover the original data in an iterative manner we obtain a complete characterization of spectrally one erasure optimal dual frames in terms of the redundancy distribution of the prescribed frame our characterization relies on the connection between erasure optimal frames and the linear connectivity property of the frame we prove that the linear connectivity property is equivalent to the intersection dependent property and is also closely related to the well known concept of a k independent set additionally we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iterative reconstruction work

signal reconstruction from the magnitude of subspace components we consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems in the deterministic setting a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions that require at least a quadratic number of subspaces moreover we address reconstruction under the erasure of a subset of the norms using the concepts of p fusion frames and list decoding we propose an algorithm that outputs a finite list of candidate signals one of which is the correct one in the random setting we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program

p woven frames abstract the known family of woven collections of frames have interesting properties and applications yet they cannot be easily constructed the present paper introduces new collections of frames sharing many properties with the original ones and having simpler construction the frames i i i and i i i for a hilbert space h are called partition woven or briefly p woven if there exists a nontrivial subset of i such that the family g i i i represented as the triple i i and called a p weaven frame is a frame defined by g i i if i and g i i otherwise the paper also studies the class of cp woven collections of frames which is a subset of the p woven collections of frames and satisfy a stronger condition the set of all woven frames is a strict subset of the set of all cp woven frames which is further a strict subset of the set of all p woven frames several examples and counterexamples are studied to show whether or not a pair of frames is p woven or cp woven

optimal filter banks for signal reconstruction from noisy subband components conventional design techniques for analysis and synthesis filters in subband processing applications guarantee perfect reconstruction of the original signal from its subband components the resulting filters however lose their optimality when additive noise due for example to signal quantization disturbs the subband sequences we propose filter design techniques that minimize the reconstruction mean squared error mse taking into account the second order statistics of signals and noise in the case of either stochastic or deterministic signals a novel recursive pseudo adaptive algorithm is proposed for efficient design of these filters analysis and derivations are extended to 2 d signals and filters using powerful kronecker product notation a prototype application of the proposed ideas in subband coding is presented simulations illustrate the superior performance of the proposed filter banks versus conventional perfect reconstruction filters in the presence of additive subband noise

equiangular tight frames from paley tournaments abstract we prove the existence of equiangular tight frames having n 2 d 1 elements drawn from either c d or c d 1 whenever n is either 2 k 1 for k n or a power of a prime such that n 3 mod 4 we also find a simple explicit expression for the prime power case by establishing a connection to a 2 d element equiangular tight frame based on quadratic residues

the pythagorean theorem i the finite case the pythagorean theorem and variants of it are studied the variations evolve to a formulation in terms of noncommutative conditional expectations on von neumann algebras that displays the theorem as the basic result of noncommutative metric euclidean geometry the emphasis in the present article is finite dimensionality both discrete and continuous

upper bounds for s distance sets and equiangular lines abstract the set of points in a metric space is called an s distance set if pairwise distances between these points admit only s distinct values two distance spherical sets with the set of scalar products 0 1 are called equiangular the problem of determining the maximum size of s distance sets in various spaces has a long history in mathematics we suggest a new method of bounding the size of an s distance set in compact two point homogeneous spaces via zonal spherical functions this method allows us to prove that the maximum size of a spherical two distance set in r n n 7 is n n 1 2 with possible exceptions for some n 2 k 1 2 3 k n we also prove the universal upper bound 2 3 n a 2 for equiangular sets with 1 a and employing this bound prove a new upper bound on the size of equiangular sets in all dimensions finally we classify all equiangular sets reaching this new bound

a note on equiangular tight frames we settle a conjecture of joseph renes about the existence and construction of certain equiangular tight frames

on some identities and inequalities for frames in hilbert spaces we give new parseval type identities and inequalities for frames in hilbert spaces our results generalize the remarkable results obtained recently by r balan p g casazza d edidin and g kutyniok

filter bank fusion frames in this paper we characterize and construct novel oversampled filter banks implementing fusion frames a fusion frame is a sequence of orthogonal projection operators whose sum can be inverted in a numerically stable way when properly designed fusion frames can provide redundant encodings of signals which are optimally robust against certain types of noise and erasures however up to this point few implementable constructions of such frames were known we show how to construct them using oversampled filter banks in this work we first provide polyphase domain characterizations of filter bank fusion frames we then use these characterizations to construct filter bank fusion frame versions of discrete wavelet and gabor transforms emphasizing those specific finite impulse response filters whose frequency responses are well behaved

frames for vector spaces and affine spaces a finite frame for a finite dimensional hilbert space is simply a spanning sequence we show that the linear functionals given by the dual frame vectors do not depend on the inner product and thus it is possible to extend the frame expansion and other elements of frame theory to any finite spanning sequence for a vector space the corresponding coordinate functionals generalise the dual basis the case when the vectors are linearly independent and are characterised by the fact that the associated gramian matrix is an orthogonal projection existing generalisations of the frame expansion to banach spaces involve an analogue of the frame bounds and frame operator the potential applications of our results are considerable whenever there is a natural spanning set for a vector space computations can be done directly with it in an efficient and stable way we illustrate this with a diverse range of examples including multivariate spline spaces generalised barycentric coordinates and vector spaces over the rationals such as the cyclotomic fields

on the existence of equiangular tight frames abstract an equiangular tight frame etf is a d n matrix that has unit norm columns and orthogonal rows of norm n d its key property is that the absolute inner products between pairs of columns are i identical and ii as small as possible etfs have applications in communications coding theory and sparse approximation numerical experiments indicate that etfs arise for very few pairs d n and it is an important challenge to develop restrictions on the pairs for which they can exist this article uses field theory to provide detailed conditions on real and complex etfs in particular it describes restrictions on harmonic etfs a specific type of complex etf that appears in applications finally the article offers empirical evidence that these conditions are sharp or nearly sharp especially in the real case

nilpotent bridging for unions of two bases in a recent article the authors discovered a new efficient method for perfect reconstruction from frame erasures called nilpotent bridging in order to perform the reconstruction it is necessary to invert a matrix called the bridge matrix which consists of inner products of the frame vectors indexed by the erasure set with vectors from the corresponding dual frame indexed by a set known as the bridge set which is disjoint from the erasure set if no search procedure is required to find a bridge set of the same size as the erasure set for which the bridge matrix is invertible the dual frame pair is said to satisfy the full skew spark property equivalently a frame satisfies the full skew spark property if every subset of the complement of the erasure set with the same cardinality as the erasure set yields an invertible bridge matrix in this paper we consider dual frame pairs which consist of a union of two bases and the union of their corresponding dual bases in finite dimensions it is shown that if we restrict the class of bridge sets to block bridge sets then a bridge set search is also unnecessary for most unions of two bases and most unions of two orthonormal bases by most we mean the set of unions of two bases resp orthonormal bases for which no bridge set search is necessary contains an open dense set in the set of all unions of two bases resp orthonormal bases so while not every bridge set of the same cardinality as the erasure set yields an invertible bridge matrix the bridge sets which have the special structure of block bridge sets are very likely to yield an invertible bridge matrix these unions of two bases are said to satisfy the block skew spark property in infinite dimensions we also show that the block skew spark property holds for a dense but not necessarily open set of unions of two riesz resp orthonormal bases lastly applications to shannon whittaker sampling theory are discussed and several open questions are posed pertaining to sampling theory

steiner equiangular tight frames abstract we provide a new method for constructing equiangular tight frames etfs the construction is valid in both the real and complex settings and shows that many of the few previously known examples of etfs are but the first representatives of infinite families of such frames it provides great freedom in terms of the frame s size and redundancy this method also explicitly constructs the frame vectors in their native domain as opposed to implicitly defining them via their gram matrix moreover in this domain the frame vectors are very sparse the construction is extremely simple a tensor like combination of a steiner system and a regular simplex this simplicity permits us to resolve an open question regarding etfs and the restricted isometry property rip we show that the rip behavior of some etfs is unfortunately no better than their coherence indicates

geometry of the welch bounds abstract a geometric perspective involving grammian and frame operators is used to derive the entire family of welch bounds this perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k tensors tight frames homogeneous polynomials and t designs in particular a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k tensors it is also shown that tightness of the bounds requires tight frames the lack of tight frames of symmetric k tensors in many cases however leads to consideration of sets that come as close as possible to attaining the bounds the geometric derivation is then extended in the setting of generalized or continuous frames the welch bounds for finite sets and countably infinite sets become special cases of this general setting

nullspaces and frames in this paper we give new characterizations of riesz and conditional riesz frames in terms of the properties of the nullspace of their synthesis operators on the other hand we also study the oblique dual frames whose coefficients in the reconstruction formula minimize different weighted norms

equiangular tight frames and fourth root seidel matrices in this paper we construct complex equiangular tight frames etfs in particular we study the grammian associated with an etf whose off diagonal entries consist entirely of fourth roots of unity these etfs are classified and we also provide some computational techniques which give rise to previously undiscovered etfs

lattices from equiangular tight frames abstract we consider the set of all linear combinations with integer coefficients of the vectors of a unit equiangular tight k n frame and are interested in the question whether this set is a lattice that is a discrete additive subgroup of the k dimensional euclidean space we show that this is not the case if the cosine of the angle of the frame is irrational we also prove that the set is a lattice for n k 1 and that there are infinitely many k such that a lattice emerges for n 2 k we dispose of all cases in dimensions k at most 9 in particular we show that a 7 28 frame generates a strongly eutactic lattice and give an alternative proof of roland bacher s recent observation that this lattice is perfect

on quasi orthogonal signatures for cdma systems sum capacity optimal signatures in synchronous code division multiple access cdma systems are functions of the codebook length as well as the number of active users a new signature set must be assigned every time the number of active users changes this correspondence considers signature sets that are less sensitive to changes in the number of active users equiangular signature sequences are proven to solve a certain max min signal to interference plus noise problem which results from their interference invariance unions of orthonormal bases have subsets that come close to satisfying the welch bound bounds on the maximum number of bases with minimum maximum correlation are derived and a new construction algorithm is provided connections are made between these signature design problems grassmannian line packing frame theory and algebraic geometry

optimal dual frames for erasures for any given frame for the purpose of encoding in a finite dimensional hilbert space we investigate its dual frames that are optimal for erasures for the purpose of decoding we show that in general the canonical dual is not necessarily optimal moreover optimal dual frames are not necessarily unique we present some sufficient conditions under which the canonical dual is the unique optimal dual frame for the erasure problem as an application we get that the canonical dual is the only optimal dual when either the frame is induced by a group representation or the frame is uniform tight

frames of uniform subframe bounds with applications to erasures abstract we introduce a class of frames called usb frames frames of uniform subframe bounds which mean that any subsequence of the frame is a frame for the subspace it spans and the frame bounds depend only on the cardinality of the subsequence we give several necessary and sufficient conditions for a frame to be a usb frame as an application we show that a tight usb frame is optimal for erasures in the sense that the reconstruction error is independent of the position of erasures we give several explicit construction of usb frames

optimal dual frames for communication coding with probabilistic erasures assume that a frame is preselected for encoding in a communication system we investigate the optimal dual frames for signal reconstruction decoding which minimize the maximal error when the probabilistic erasures occur in the transmission process from the perspective of mathematical theory of frames we set up a probability model under which we define the probability optimal po dual frames for a given frame when the frame based coding involves probabilistic erasures we obtain a sufficient and necessary condition under which the canonical dual frame is the unique po dual frame additionally we derive some general sufficient conditions for which the canonical dual frame is either not optimal or it is optimal but not the unique optimal one we present two simulation examples to compare the reconstruction effects when both the po dual frames and the general optimal go dual frames are used for reconstruction