sampling and reconstruction of signals in a reproducing kernel subspace of lp rd in this paper we consider sampling and reconstruction of signals in a reproducing kernel subspace of lp rd 1 p associated with an idempotent integral operator whose kernel has certain off diagonal decay and regularity the space of p integrable non uniform splines and the shift invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of lp rd we show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively separated set with sufficiently small gap we also study the exponential convergence consistency and the asymptotic pointwise error estimate of the iterative approximation projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap

remainder terms for some quantum entropy inequalities we consider three von neumann entropy inequalities subadditivity pinsker s inequality for relative entropy and the monotonicity of relative entropy for these we state conditions for equality and we prove some new error bounds away from equality including an improved version of pinsker s inequality

one and a half quantum de finetti theorems when n k systems of an n partite permutation invariant state are traced out the resulting state can be approximated by a convex combination of tensor product states this is the quantum de finetti theorem in this paper we show that an upper bound on the trace distance of this approximation is given by 2 frac kd 2 n where d is the dimension of the individual system thereby improving previously known bounds our result follows from a more general approximation theorem for representations of the unitary group consider a pure state that lies in the irreducible representation u mu nu subset u mu otimes u nu of the unitary group u d for highest weights and let be the state obtained by tracing out u then is close to a convex combination of the coherent states u mu g v mu rangle where g in u d and v mu rangle is the highest weight vector in u

squashed entanglement for multipartite states and entanglement measures based on the mixed convex roof new measures of multipartite entanglement are constructed based on two definitions of multipartite information and different methods of optimizing over extensions of the states one is a generalization of the squashed entanglement where one takes the mutual information of parties conditioned on the state s extension and takes the infimum over such extensions additivity of the multipartite squashed entanglement is proved for both versions of the multipartite information which turn out to be related the second one is based on taking classical extensions this scheme is generalized which enables to construct measures of entanglement based on the mixed convex roof of a quantity which in contrast to the standard convex roof method involves optimization over all decompositions of a density matrix rather than just the decompositions into pure states as one of the possible applications of these results we prove that any multipartite monotone is an upper bound on the amount of multipartite distillable key the findings are finally related to analogous results in classical key agreement

the zak transform and sampling theorems for wavelet subspaces the zak transform is used for generalizing a sampling theorem of g waiter see ieee trans informat theory vol 38 p 881 884 1992 for wavelet subspaces cardinal series based on signal samples f a n n in z with a possibly unequal to 0 waiter s case are considered the condition number of the sampling operator and worst case aliasing errors are expressed in terms of zak transforms of scaling function and wavelet this shows that the stability of the resulting interpolation formula depends critically on a

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

quantum rate distortion coding with auxiliary resources we extend quantum rate distortion theory by considering auxiliary resources that might be available to a sender and receiver performing lossy quantum data compression the first setting we consider is that of quantum rate distortion coding with the help of a classical side channel our result here is that the regularized entanglement of formation characterizes the quantum rate distortion function extending earlier work of devetak and berger we also combine this bound with the entanglement assisted bound from our prior work to obtain the best known bounds on the quantum rate distortion function for an isotropic qubit source the second setting we consider is that of quantum rate distortion coding with quantum side information qsi available to the receiver in order to prove results in this setting we first state and prove a quantum reverse shannon theorem with qsi for tensor power states which extends the known tensor power quantum reverse shannon theorem the achievability part of this theorem relies on the quantum state redistribution protocol while the converse relies on the fact that the protocol can cause only a negligible disturbance to the joint state of the reference and the receiver s qsi this quantum reverse shannon theorem with qsi naturally leads to quantum rate distortion theorems with qsi with or without entanglement assistance

an irregular sampling theorem for functions bandlimited in a generalized sense it is shown that a well known irregular sampling formula for ordinary bandlimited functions is also valid for functions bandlimited in a generalized sense except for some necessary modifications

on feller processes with sample paths in besov spaces under mild regularity assumptions on its domain the infinitesimal generator of a feller process is known to be a pseudo differential operator we give a simple condition on the symbol of the generator in order to characterize the smoothness of the sample paths of real valued feller processes in terms of besov spaces b s pq bbb r our result extends previous papers on the paths of gaussian symmetric alpha stable 6 20 and levy processes 11

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

faithful squashed entanglement squashed entanglement is a measure for the entanglement of bipartite quantum states in this paper we present a lower bound for squashed entanglement in terms of a distance to the set of separable states this implies that squashed entanglement is faithful that is it is strictly positive if and only if the state is entangled we derive the lower bound on squashed entanglement from a lower bound on the quantum conditional mutual information which is used to define squashed entanglement the quantum conditional mutual information corresponds to the amount by which strong subadditivity of von neumann entropy fails to be saturated our result therefore sheds light on the structure of states that almost satisfy strong subadditivity with equality the proof is based on two recent results from quantum information theory the operational interpretation of the quantum mutual information as the optimal rate for state redistribution and the interpretation of the regularised relative entropy of entanglement as an error exponent in hypothesis testing the distance to the set of separable states is measured in terms of the locc norm an operationally motivated norm giving the optimal probability of distinguishing two bipartite quantum states each shared by two parties using any protocol formed by local quantum operations and classical communication locc between the parties a similar result for the frobenius or euclidean norm follows as an immediate consequence the result has two applications in complexity theory the first application is a quasipolynomial time algorithm solving the weak membership problem for the set of separable states in locc or euclidean norm the second application concerns quantum merlin arthur games here we show that multiple provers are not more powerful than a single prover when the verifier is restricted to locc operations thereby providing a new characterisation of the complexity class qma

weighted sampling and reconstruction in weighted reproducing kernel spaces in this paper we discuss sampling and reconstruction of signals in the weighted reproducing kernel space associated with an idempotent integral operator we show that any signal in that space can be stably reconstructed from its weighted samples taken on a relatively separated set with sufficiently small gap we also develop an iterative reconstruction algorithm for the reconstruction of a signal from its weighted samples taken on a relatively separated set with sufficiently small gap

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

bounds on entanglement distillation and secret key agreement for quantum broadcast channels the squashed entanglement of a quantum channel is an additive function of quantum channels which finds application as an upper bound on the rate at which secret key and entanglement can be generated when using a quantum channel a large number of times in addition to unlimited classical communication this quantity has led to an upper bound of log 1 eta 1 eta on the capacity of a pure loss bosonic channel for such a task where eta is the average fraction of photons that make it from the input to the output of the channel the purpose of this paper is to extend these results beyond the single sender single receiver setting to the more general case of a single sender and multiple receivers a quantum broadcast channel we employ multipartite generalizations of the squashed entanglement to constrain the rates at which secret key and entanglement can be generated between any subset of the users of such a channel along the way developing several new properties of these measures we apply our results to the case of a pure loss broadcast channel with one sender and two receivers

the fidelity of recovery is multiplicative fawzi and renner recently established a lower bound on the conditional quantum mutual information cqmi of tripartite quantum states abc in terms of the fidelity of recovery for i e the maximal fidelity of the state abc with a state reconstructed from its marginal bc by acting only on the c system the for measures quantum correlations by the local recoverability of global states and has many properties similar to the cqmi here we generalize the for and show that the resulting measure is multiplicative by utilizing semi definite programming duality this allows us to simplify an operational proof by brandao et al of the above mentioned lower bound that is based on quantum state redistribution in particular in contrast to the previous approaches our proof does not rely on de finetti reductions

continuity of quantum conditional information we prove continuity of quantum conditional information s 12 2 with respect to the uniform convergence of states and obtain a bound which is independent of the dimension of the second party this can e g be used to prove the continuity of squashed entanglement

wavelets in wandering subspaces mallat s construction via a multiresolution approximation of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions the connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in hilbert space is exploited in the general setting an example of multiresolution approximation generated by cardinal hermite b splines is constructed

sampling and quasi optimal approximation for signals in a reproducing kernel space of homogeneous type abstract this paper considers the reconstruction of signals in a reproducing kernel space of homogeneous type from finite samples first a pre reconstruction operator based on finite samples and probability measure is proposed and its bounded property is studied secondly the stability and an iterative algorithm with exponential convergence are established for sampling and recovering signals in a subspace of homogeneous reproducing kernel space then we show that the proposed algorithm also provides a quasi optimal approximation to signals in a reproducing kernel space of homogeneous type finally some numerical simulations are given to reconstruct signals on an interval

growth and holder conditions for the sample paths of feller processes let a d a be the infinitesimal generator of a feller semigroup such that cc n d a and a cc n is a pseudo differential operator with symbol p x satisfying p c 1 2 and imp x c0rep x we show that the associated feller process xt t 0 on n is a semimartingale even a homogeneous diffusion with jumps in the sense of 21 and characterize the limiting behaviour of its trajectories as t 0 and to this end we introduce various indices e g x 0 lim x y 2 p y 0 or x 0 liminf x y 2 e 1 p y e 0 and obtain a s x that lim t 0t 1 s t xs x 0 or according to x or x similar statements hold for the limit inferior and superior and also for t our results extend the constant coefficient i e levy case considered by w pruitt 27

quantum discord a measure of the quantumness of correlations two classically identical expressions for the mutual information generally differ when the systems involved are quantum this difference defines the quantum discord it can be used as a measure of the quantumness of correlations separability of the density matrix describing a pair of systems does not guarantee vanishing of the discord thus showing that absence of entanglement does not imply classicality we relate this to the quantum superposition principle and consider the vanishing of discord as a criterion for the preferred effectively classical states of a system i e the pointer states

a sharp continuity estimate for the von neumann entropy we derive an inequality relating the entropy difference between two quantum states to their trace norm distance sharpening a well known inequality due to fannes in our inequality equality can be attained for every prescribed value of the trace norm distance

path regularity for feller semigroups via gaussian kernel estimates and generalizations to arbitrary semigroups on c0 given 1 1 we present a dyadic growth condition open image in new window on the finite dimensional distributions of operator semigroups on c0 e which for 0 and feller semigroups assures that the corresponding feller process has paths in local holder spaces and in weighted besov spaces of order we show that for operator semigroups satisfying gaussian kernel estimates of order m 1 condition open image in new window holds for all open image in new window and even for all open image in new window in the case of feller semigroups such gaussian kernel estimates are typical for feller semigroups on fractals of walk dimension m and for semigroups generated by elliptic operators on d of order m d

irregular sampling for spline wavelet subspaces spline wavelets spl psi sub m t are important in time frequency localization due to i spl psi sub m can be arbitrarily close to the optimal case as m is sufficiently large ii spl psi sub m has compact support and simple analytic expression which lead to effective computation although the spline wavelet subspaces are so simple walter s well known sampling theorem does not hold if the order of spline m is even moreover when irregular sampling is considered in these spaces it is hard to determine the sampling density which is a serious problem in applications in this correspondence a general sampling theorem is obtained for m spl ges 3 in the sense of iterative construction and the sampling density spl delta sub m is estimated

on the stability of multivariate trigonometric systems abstract kadec s 1 4 theorem says that if n n z is a sequence of real numbers for which n n l 1 4 then ei n n z forms a riesz basis for l2 s favier r zalik c chui and x shi extended this result to the multivariate case but their results lead to very small stability bounds in this paper we give an optimal stability bound for the multivariate trigonometric systems moreover for the case of fourier frames in l2 d we also give the stability bounds

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

numerical analysis of the non uniform sampling problem we give an overview of recent developments in the problem of reconstructing a band limited signal from nonuniform sampling from a numerical analysis view point it is shown that the appropriate design of the finite dimensional model plays a key role in the numerical solution of the nonuniform sampling problem in the one approach often proposed in the literature the finite dimensional model leads to an ill posed problem even in very simple situations the other approach that we consider leads to a well posed problem that preserves important structural properties of the original infinite dimensional problem and gives rise to efficient numerical algorithms furthermore a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy nonuniformly spaced samples we also discuss the design of efficient regularization methods for ill conditioned reconstruction problems numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods

multipartite quantum correlations and local recoverability characterizing genuine multipartite quantum correlations in quantum physical systems has historically been a challenging problem in quantum information theory more recently however the total correlation or multipartite information measure has been helpful in accomplishing this goal especially with the multipartite symmetric quantum msq discord piani et al 2008 phys rev lett 100 090502 doi 10 1103 physrevlett 100 090502 1 and the conditional entanglement of multipartite information cemi yang et al 2008 phys rev lett 101 140501 doi 10 1103 physrevlett 101 140501 2 here we apply a recent and significant improvement of strong subadditivity of quantum entropy fawzi renner 2014 in order to develop these quantities further in particular we prove that the msq discord is nearly equal to zero if and only if the multipartite state for which it is evaluated is approximately locally recoverable after performing measurements on each of its systems furthermore we prove that the cemi is a faithful entanglement measure i e it vanishes if and only if the multipartite state for which it is evaluated is a fully separable state along the way we provide an operational interpretation of the msq discord in terms of the partial state distribution protocol which in turn as a special case gives an interpretation for the original discord quantity finally we prove an inequality that could potentially improve upon the fawzi renner inequality in the multipartite context but it remains an open question to determine whether this is so 1 http dx doi org 10 1103 physrevlett 100 090502 2 http dx doi org 10 1103 physrevlett 101 140501

optimal quantum source coding with quantum side information at the encoder and decoder consider many instances of an arbitrary quadripartite pure state of four quantum systems abcd alice holds the ac part of each state bob holds b while r represents all other parties correlated with abc alice is required to redistribute the c systems to bob while asymptotically preserving the overall purity we prove that this is possible using q qubits of communication and e ebits of shared entanglement between alice and bob provided that q ges 1 2i c d b and q e ges h c b proving the optimality of the luo devetak outer bound the optimal qubit rate provides the first known operational interpretation of quantum conditional mutual information we also show how our protocol leads to a fully operational proof of strong subaddivity and uncover a general organizing principle in analogy to thermodynamics that underlies the optimal rates

the recovery of irregularly sampled band limited functions via tempered splines abstract we show that band limited functions can be recovered from their values on certain irregularly distributed discrete sampling sets as the limits of the piecewise polynomial spline interpolants when the order of the splines goes to infinity this is an extension of the classical case when the sampling set is a lattice which was considered by collatz quade schoenberg and others

squashed entanglement an additive entanglement measure in this paper we present a new entanglement monotone for bipartite quantum states its definition is inspired by the so called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions we derive certain properties of the new measure which we call squashed entanglement it is a lower bound on entanglement of formation and an upper bound on distillable entanglement furthermore it is convex additive on tensor products and superadditive in general continuity in the state is the only property of our entanglement measure which we cannot provide a proof for we present some evidence however that our quantity has this property the strongest indication being a conjectured fannes type inequality for the conditional von neumann entropy this inequality is proved in the classical case

no local broadcasting theorem for multipartite quantum correlations we prove that the correlations present in a multipartite quantum state have an operational quantum character even if the state is unentangled as long as it does not simply encode a multipartite classical probability distribution said quantumness is revealed by the new task of local broadcasting i e of locally sharing preestablished correlations which is feasible if and only if correlations are stricly classical our operational approach leads to natural definitions of measures for quantumness of correlations it also reproduces the standard no broadcasting theorem as a special case

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

continuity bounds on the quantum relative entropy ii the quantum relative entropy is frequently used as a distance measure between two quantum states and inequalities relating it to other distance measures are important mathematical tools in many areas of quantum information theory we have derived many such inequalities in previous work the present paper is a follow up on this and provides a sharp upper bound on the relative entropy in terms of the trace norm distance and of the smallest eigenvalues of both states concerned the result obtained here is more general than the corresponding one from our previous work as a corollary we obtain a sharp upper bound on the regularised relative entropy introduced by lendi farhadmotamed and van wonderen

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

reconstruction of band limited signals from local averages the sampling theorem says that every band limited signal is uniquely determined by its sampled values provided the sampling points satisfy certain conditions however sampled values obtained in practice may not be the exact values of a signal at sampling points but only averages of the signal near these points grochenig 1992 proved that band limited signals can be reconstructed exactly from local averages if the sampling density is large enough we study the reconstruction of band limited signals from local averages with symmetric averaging functions we study the aliasing error arising when a non band limited signal is reconstructed from local averages and give explicit error bounds since the classical point sampling can be viewed as a limiting case of average sampling we indeed give new aliasing error bounds for both regular and irregular sampling

on the connection between exponential bases and certain related sequences in l2 π π a sequence of vectors fn in a separable hilbert space h is a frame if there are positive constants a b such that a f 2 n f fn 2 b f 2 for all f h and fn is a riesz sequence if it is a riesz basis in the closure of the space spanned by the vectors fn the latter of the following two questions has been raised by khrushchev nikolskii and pavlov can every frame of complex exponentials ei nx in l2 be made into a riesz basis by removing from ei nx a suitable collection of the functions ei nx can every riesz sequence ei nx in l2 be made into a riesz basis by adjoining to ei nx a suitable collection of exponentials ei nx ei nx we show that this is indeed so for all frames and riesz sequences ei nx studied so far and we prove that we can always solve both problems in a certain weak sense however our main conclusion is that the answer to both questions is no

generalized sampling theorems in multiresolution subspaces it is well known that under very mild conditions on the scaling function multiresolution subspaces are reproducing kernel hilbert spaces rkhss this allows for the development of a sampling theory in this paper we extend the existing sampling theory for wavelet subspaces in several directions we consider periodically nonuniform sampling sampling of a function and its derivatives oversampling multiband sampling and reconstruction from local averages all these problems are treated in a unified way using the perfect reconstruction pr filter bank theory we give conditions for stable reconstructions in each of these cases sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported except in trivial cases this restriction no longer applies for the generalizations we study here due to the existence of fir pr banks in fact with nonuniform sampling oversampling and reconstruction from local averages we can guarantee compactly supported synthesizing functions moreover local averaging schemes have additional nice properties robustness to the input noise and compression capabilities we also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis after this we extend the sampling theory to random processes we require autocorrelation functions to belong to some subspace related to wavelet subspaces it turns out that we cannot recover random processes themselves unless they are bandlimited but only their power spectral density functions we consider both uniform and nonuniform sampling

a lower bound of quantum conditional mutual information in this paper a lower bound of quantum conditional mutual information is obtained by employing the peierls bogoliubov inequality and the golden thompson inequality comparison with the bounds obtained by other researchers indicates that our result is independent of any measurements it may give some new insights into squashed entanglement and perturbations of markov chain states

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

the growth of random walks and levy processes let x i be a sequence of independent identically distributed non degenerate random variables taking values in mathbb r d and s n sum n i 1 x i m n max 1 leqq i leqq n s i define for x 0 g x p x 1 x k x x 2 e x 1 2 1 x 1 leq x m x x 1 e x 1 1 x 1 leq x and h x g x k x m x then if beta sup alpha lim sup x alpha h x 0 delta sup alpha lim inf x alpha h x 0 it is proved that n 1 alpha m n rightarrow 0 for alpha delta while the lim inf is 0 and the lim sup is infty for beta alpha delta some alternative characterizations of the indices beta delta are obtained as well as the analogous results for levy processes