weighted estimates for the helmholtz equation and some applications we characterize the radial functionsvin rnfor which the a priori inequality u1 2 2 k uv1 2 2 c v 1 2 k2 u 2holds with constant independent ofk the condition is forvto have the x rays transform everywhere bounded we apply these estimates to the well posedness of evolution schrodinger equations with time dependent drift terms and to the restriction of the fourier transform to euclidean spheres

toeplitz operators on herglotz wave functions abstract the space of herglotz wave functions in r 2 consists of all the solutions of the helmholtz equation that can be represented as the fourier transform in r 2 of a measure supported in the circle and with density in l 2 s 1 this space has a structure of a hilbert space with reproducing kernel the purpose of this article is to study toeplitz operators with nonnegative radial symbols defined on this space we study the symbols defining bounded and compact toeplitz operators as well as the toeplitz operators belonging to the schatten classes s p

interference alignment with asymmetric complex signaling settling the host madsen nosratinia conjecture it has been conjectured by ho madsen and nosratinia that complex gaussian interference channels with constant channel coefficients have only one degree of freedom regardless of the number of users while several examples are known of constant channels that achieve more than 1 degree of freedom these special cases only span a subset of measure zero in other words for almost all channel coefficient values it is not known if more than 1 degree of freedom is achievable in this paper we settle the host madsen nosratinia conjecture in the negative we show that at least 1 2 degrees of freedom are achievable for all values of complex channel coefficients except for a subset of measure zero for the class of linear beamforming and interference alignment schemes considered in this paper it is also shown that 1 2 is the maximum number of degrees of freedom achievable on the complex gaussian 3 user interference channel with constant channel coefficients for almost all values of channel coefficients to establish the achievability of 1 2 degrees of freedom we use the novel idea of asymmetric complex signaling i e the inputs are chosen to be complex but not circularly symmetric it is shown that unlike gaussian point to point multiple access and broadcast channels where circularly symmetric complex gaussian inputs are optimal for interference channels optimal inputs are in general asymmetric with asymmetric complex signaling we also show that the 2 user complex gaussian x channel with constant channel coefficients achieves the outer bound of 4 3 degrees of freedom i e the assumption of time variations frequency selectivity used in prior work to establish the same result is not needed

rate region boundary of the siso z interference channel with improper signaling this paper provides a complete characterization of the boundary of an achievable rate region called the pareto boundary of the single antenna z interference channel z ic when interference is treated as noise and users transmit complex gaussian signals that are allowed to be improper by considering the augmented complex formulation we derive a necessary and sufficient condition for improper signaling to be optimal this condition is stated as a threshold on the interference channel coefficient which is a function of the interfered user rate and which allows insightful interpretations into the behavior of the achievable rates in terms of the circularity coefficient i e degree of impropriety furthermore the optimal circularity coefficient is provided in closed form the simplicity of the obtained characterization permits interesting insights into when and how improper signaling outperforms proper signaling in the single antenna z ic we also provide an in depth discussion on the optimal strategies and the properties of the pareto boundary

smoothing estimates for the schrodinger equation with unbounded potentials abstract we prove a local in time smoothing estimate for a magnetic schrodinger equation with coefficients growing polynomially at spatial infinity the assumptions on the magnetic field are gauge invariant and involve only the first two derivatives the proof is based on the multiplier method and no pseudodifferential techniques are required

weighted norm inequalities for oscillatory integrals with finite type phases on the line abstract we obtain two weighted l 2 norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type the conditions imposed on the weights involve geometrically defined maximal functions and the inequalities are best possible in the sense that they imply the full l p r l q r mapping properties of the oscillatory integrals our results build on work of carbery soria vargas and the first author

generalized strichartz estimates and scattering for 3d zakharov system we obtain scattering for the 3d zakharov system with non radial small data in the energy space with angular regularity of degree one the main ingredient is a generalized strichartz estimate for the schrodinger equation in the space of l2 angular integrability

block skew circulant matrices in complex valued signal processing two main lines of approach can be identified in the recent literature on improper signals and widely linear operations the augmented complex formulation based on the signal and its complex conjugate is considered as more insightful since it leads to convenient mathematical formulations for many considered problems moreover it allows an easy distinction between proper and improper signals as well as between linear and widely linear operations on the other hand the composite real representation using the real and imaginary parts of the signal is closer to the actual implementation and it allows to readily reuse results that have originally been derived for real valued signals or proper complex signals in this work we aim at getting the best of both worlds by introducing mathematical tools that make the composite real representation more powerful and elegant the proposed approach relies on a decomposition of real matrices into a block skew circulant and a block hankel skew circulant component by means of various application examples from the field of signal processing for communications we demonstrate the usefulness of the proposed framework

lp boundedness of the wave operator for the one dimensional schrodinger operator given a one dimensional perturbed schrodinger operator h d2 dx2 v x we consider the associated wave operators w defined as the strong l2 limits lim s to pm infty e ish e ish 0 we prove that w are bounded operators on lp for all 1 p provided 1 x 2 v x in l 1 or else 1 x v x in l 1 and 0 is not a resonance for p we obtain an estimate in terms of the hilbert transform some applications to dispersive estimates for equations with variable rough coefficients are given

band limited functions lp convergence we consider the set bp q functions of lp tl whose fourier spectrum lies in i2 i2 we prove that the prolate spheroidal wave func tions constitute a basis of this space if and only if 4 3 p 4 the result is obtained as a consequence of the analogous problem for the spherical bessel functions the proof rely on a weighted inequality for the hubert transform

derivation of fixed interval smoothing algorithm using covariance information in distributed parameter systems this paper proposes a recursive least mean squared error fixed interval smoothing algorithm in distributed parameter systems it is assumed that the state space model of the signal to be estimated is unknown and the algorithm only requires the second order moments of the signal and the white noise perturbing its observations practical application of the proposed algorithm is illustrated with a restoration image problem

proper complex random processes with applications to information theory the covariance of complex random variables and processes when defined consistently with the corresponding notion for real random variables is shown to be determined by the usual complex covariance together with a quantity called the pseudo covariance a characterization of uncorrelatedness and wide sense stationarity in terms of covariance and pseudo covariance is given complex random variables and processes with a vanishing pseudo covariance are called proper it is shown that properness is preserved under affine transformations and that the complex multivariate gaussian density assumes a natural form only for proper random variables the maximum entropy theorem is generalized to the complex multivariate case the differential entropy of a complex random vector with a fixed correlation matrix is shown to be maximum if and only if the random vector is proper gaussian and zero mean the notion of circular stationarity is introduced for the class of proper complex random processes a discrete fourier transform correspondence is derived relating circular stationarity in the time domain to uncorrelatedness in the frequency domain an application of the theory is presented

on the wave equation with a large rough potential we prove an optimal dispersive l decay estimate for a three dimensional wave equation perturbed with a large nonsmooth potential belonging to a particular kato class the proof is based on a spectral representation of the solution and suitable resolvent estimates for the perturbed operator

data aided and blind stochastic gradient algorithms for widely linear mmse mai suppression for ds cdma in this paper three novel stochastic gradient algorithms for adjustment of the widely linear wl minimum mean squared error mmse filter for multiple access interference mai suppression for direct sequence code division multiple access ds cdma are introduced and analyzed in particular we derive a data aided wl least mean square lms algorithm a blind wl minimum output energy moe algorithm and a wl blind lms blms algorithm we give analytical expressions for the steady state signal to interference plus noise ratios sinrs of the proposed wl algorithms and we also investigate their speed of convergence wherever possible comparisons with the corresponding linear adaptive algorithms are made both analytical considerations and simulations show in good agreement the superiority of the novel wl adaptive algorithms nevertheless all proposed wl algorithms require a slightly lower computational complexity than their linear counterparts

radial weights and mixed norm inequalities for the disc multiplier in this paper we show that the l2 weighted inequalities with radial weights of the multiplier for the ball are controlled by the maximal function defined by averages on arbitrary rectangles this solves the case of radial weights of a problem proposed by e stein in 1978 it also gives an elegant and simple proof of the boundedness of the multiplier for the ball on the spaces of mixed norm lradp lang2 for 2n n 1 p 2n n 1 obtained by a cordoba and by g mockenhaupt additional work is presented in order to extend these mixed norm estimates to the end points 2n n 1 and 2n n 1 in the sharp form of restricted weak type inequalities

strichartz estimates for the magnetic schrodinger equation abstract we prove global scale invariant strichartz estimates for the linear magnetic schrodinger equation with small time dependent magnetic field this is done by constructing an appropriate parametrix as an application we show a global regularity type result for schrodinger maps in dimensions n 6

improper signaling for symbol error rate minimization in k user interference channel the rate maximization for the k user interference channels ics has been investigated extensively in the literature however the practical problem of minimizing the error probability with given signal modulations and or data rates of the users is less studied in this paper by utilizing additional degrees of freedom from the improper signaling versus the conventional proper signaling we seek to optimize the precoding matrices for the k user single input single output siso ics to minimize pair wise error probability pep and symbol error rate ser with two proposed algorithms respectively compared with conventional proper signaling and other state of the art improper signaling designs our proposed improper signaling schemes achieve notable error rate improvement in siso ics under both the additive white gaussian noise awgn and cellular system setups with or without channel coding our study provides another viewpoint for optimizing transmissions in ics and further justifies the practical benefit of improper signaling in interference limited communication systems

receivers with widely linear processing for frequency selective channels we propose several equalization schemes based on widely linear processing wlp the received signal and its complex conjugate are separately filtered and the results are linearly combined it is shown that wlp yields a gain in performance if the noiseless received signal can be interpreted as the convolution of a real valued data sequence and an equivalent complex valued intersymbol interference channel impulse response such a model applies to e g amplitude shift keying offset quadrature amplitude modulation and binary minimum shift keying type modulation we consider receivers without and with decision feedback finite impulse response filters are derived for these structures which are optimum with respect to the zero forcing and minimum mean squared error mmse criteria respectively in the mmse case adaptive algorithms for filter adjustment are given infinite filter orders are investigated in order to obtain analytical performance results furthermore suboptimum trellis based detection with widely linear preprocessing is briefly discussed it is demonstrated analytically and by numerical examples that widely linear schemes may outperform conventional schemes significantly depending on the considered application

fixed interval smoothing problem from uncertain observations with correlated signal and noise this paper presents recursive filtering and fixed interval smoothing algorithms from observations corrupted by additive and multiplicative noises additive noise is a white process correlated with the signal and multiplicative noise is modelled by a sequence of independent bernoulli random variables it is assumed that both autocovariance function of signal and crosscovariance function about signal and observation noise are expressed in a semi degenerate kernel form the algorithms are obtained by an innovation approach without using the state space model but only covariance information of signal and observation noise and probability that signal exists in the observed values

fixed interval smoothing for markovian switching systems a suboptimal approach to the fixed interval smoothing problem for markovian switching systems is examined a smoothing algorithm is developed that uses two multiple model filters where one of the filters propagates in the forward time direction and the other one propagates in the backward time direction a backward time filtering algorithm based on the interacting multiple model concept is also developed results from a simulation example are given to illustrate the performance of the smoothing algorithm with respect to that of filtering the example involves radar tracking of a mach 1 aircraft

linear smoothing for discrete time systems in the presence of correlated disturbances and uncertain observations this paper considers the smoother with the least mean squared error lmse within the class of linear smoothers for discrete linear systems whose observations may contain noise alone and the probability of this occurring is known systems with uncertain observations also we assume that state and measurement noises are correlated the present results together with hermoso linares s results on prediction and filtering give a complete treatment of the lmse linear estimation problem for such systems

capacity of second order cyclostationary complex gaussian noise channels in this paper we derive the capacity of a continuous time single input single output siso frequency selective band limited linear time invariant lti channel whose output is corrupted by a second order cyclostationary socs complex gaussian noise by using a pair of invertible linear conjugate linear time varying operators called a properizing frequency shift p fresh vectorizer and a p fresh scalarizer it is shown that whether the complex noise is proper or improper the siso channel can always be converted to an equivalent multiple input multiple output mimo lti channel whose output is now corrupted by a proper complex vector wide sense stationary noise a variational problem is then formulated in the frequency domain to find the optimal input distribution that maximizes the throughput of the equivalent mimo channel it turns out that the optimal input to the siso channel obtained through a procedure similar to the water filling is an socs complex gaussian random process with the same cycle period as the noise it is shown that this procedure named cyclic water filling significantly outperforms ordinary water filling by effectively utilizing the spectral correlation of the cyclostationary noise

h_ infty fixed interval smoothing estimation for time delay systems this paper is concerned with the h fixed interval smoothing estimation for time delay systems which include continuous time case and discrete time case in the case of discrete time systems the problem can be solved by using the conventional state augmentation approach however this approach is not suitable for the continuous time case in this paper we will propose a unified approach to study the h fixed interval smoothing problem for both continuous time and discrete time systems with l output delays by introducing a suitable stochastic linear time delay models in an indefinite space it is shown that the h fixed interval smoother can be obtained by calculating h2 fixed interval smoother for time delay systems in an indefinite space therefore based on the orthogonal projection theory in an indefinite space the h fixed interval smoothers for time delay systems are designed by performing l 1 riccati equations with the same dimension as the original systems moreover a necessary and sufficient conditions for the existence of the h fixed interval smoother will also be given

interference alignment with asymmetric complex signaling on mimo x channels it is known that with conventional spatial interference alignment i e linear interference alignment without symbol extension the achievable degrees of freedom dof of mimo x channels is no more than lfloor d outer rfloor where d outer denotes the outer bound of dof of mimo x channels in this paper we propose a spatial interference alignment and zero forcing framework in combination with asymmetric complex signaling which can achieve the dof of 1 2 lfloor 2d outer rfloor for arbitrary number of antennas on each terminal note that 1 2 lfloor 2d outer rfloor is equal to or larger than lfloor d outer rfloor the result shows that the technique of asymmetric complex signaling which was originally proposed for single antenna systems can be useful for mimo channels in particular when symbol extensions are not allowed

a bilinear extension inequality in two dimensions abstract we provide sharp decay estimates for circular averages of a certain bilinear extension operator on l 2 s 1 l 2 s 1

linear estimation for discrete time systems in the presence of time correlated disturbances and uncertain observations this note considers optimum linear recursive estimation for linear discrete time systems with uncertain observations in cases where state and measurement noises are correlated the different kinds of estimation problems treated include one stage prediction and filtering

characterization of the pareto boundary for the two user symmetric gaussian interferencechannel we characterize the pareto boundary of the achiev able rate region of the two user symmetric gaussian interference channel when transmitters control their transmission power and receivers have successive interference cancelation capability by identifying the shape of achievable rate region according to the strength of interference channels we find combinations of decod ing and power allocation strategies achieving the pareto boundary we also characterize the pareto boundary of the achievable rate region when time sharing is additionally considered index terms interference channel achievable rate region pareto boundary successive interference cancelation

a stein conjecture for the circle we discuss the manner in which one might expect directional maximal functions to control the fourier extension operator via l 2 weighted inequalities we prove a general inequality of this type for the extension operator restricted to circles in the plane

applications of the funk hecke theorem to smoothing and trace estimates abstract for a wide class of kato smoothing estimates with radial weights the funk hecke theorem is used to generate a new expression for the optimal constant in terms of the fourier transform of the weight from which several applications are given for example we are able to easily establish a unified theorem assuming natural power like asymptotic estimates for the fourier transform of the weight from which many well studied smoothing estimates immediately follow as well as sharpness of the decay and smoothness exponents furthermore observing that the weight has an everywhere positive fourier transform in many well studied cases our approach allows sharper information regarding the optimal constant and extremisers substantially extending earlier work of simon these observations are very closely related to the mizohata takeuchi conjecture regarding the equivalence of weighted l 2 bounds for the fourier extension operator on the sphere and the uniform boundedness of the x ray transform of the weight for radial weights this has been independently established by barcelo ruiz vega and carbery soria we provide a short alternative proof in three and higher dimensions of this equivalence when the fourier transform of the weight is positive with the optimal relationship between constants finally our approach works for the closely connected trace theorems on the sphere where analogous results are given including the optimal constant and characterisation of extremisers for the inhomogeneous h s r d l 2 s d 1 trace theorem

complete characterization of the pareto boundary for the miso interference channel in this correspondence we study the achievable rate region of the multiple input single output miso interference channel under the assumption that all receivers treat the interference as additive gaussian noise our main result is an explicit parametrization of the pareto boundary for an arbitrary number of users and antennas the parametrization describes the boundary in terms of a low dimensional manifold for the two user case we show that a single real valued parameter per user is sufficient to achieve all points on the pareto boundary and that any point on the pareto boundary corresponds to beamforming vectors that are linear combinations of the zero forcing zf and maximum ratio transmission mrt beamformers we further specialize the results to the miso broadcast channel bc a numerical example illustrates the result

magnetic virial identities weak dispersion and strichartz inequalities we show a family of virial type identities for the schrodinger and wave equations with electromagnetic potentials as a consequence some weak dispersive inequalities in space dimension n 3 involving morawetz and smoothing estimates are proved finally we apply them to prove strichartz inequalities for the wave equation with a non trapping electromagnetic potential with almost coulomb decay

on radial weights for the spherical summation operator abstract in this paper we give a sufficient condition for radial weights such that the spherical summation operator is bounded on l 2 r n x dx n 2 when applied to power weights x x x these conditions are also necessary moreover in the general case we shall show that our condition is almost necessary furthermore we give an example which shows that the conjecture in anderson proc amer math soc 83 no 2 1981 269 275 fails to hold

transmit optimization with improper gaussian signaling for interference channels this paper studies the achievable rates of gaussian interference channels with additive white gaussian noise awgn when improper or circularly asymmetric complex gaussian signaling is applied for the gaussian multiple input multiple output interference channel mimo ic with the interference treated as gaussian noise we show that the user s achievable rate can be expressed as a summation of the rate achievable by the conventional proper or circularly symmetric complex gaussian signaling in terms of the users transmit covariance matrices and an additional term which is a function of both the users transmit covariance and pseudo covariance matrices the additional degrees of freedom in the pseudo covariance matrix which is conventionally set to be zero for the case of proper gaussian signaling provide an opportunity to further improve the achievable rates of gaussian mimo ics by employing improper gaussian signaling to this end this paper proposes widely linear precoding which efficiently maps proper information bearing signals to improper transmitted signals at each transmitter for any given pair of transmit covariance and pseudo covariance matrices in particular for the case of two user gaussian single input single output interference channel siso ic we propose a joint covariance and pseudo covariance optimization algorithm with improper gaussian signaling to achieve the pareto optimal rates by utilizing the separable structure of the achievable rate expression an alternative algorithm with separate covariance and pseudo covariance optimization is also proposed which guarantees the rate improvement over conventional proper gaussian signaling

the capacity of the gaussian interference channel under strong interference corresp the capacity region of a gaussian interference channel with two separate messages is obtained for the case of moderately strong interference it is shown that the region coincides with the one where both messages are required in both receiving terminals

morrey campanato estimates for helmholtz equations abstract we derive uniform weighted l 2 and morrey campanato type estimates for helmholtz equations in a medium with a variable index which is not necessarily constant at infinity our technique is based on a multiplier method with appropriate weights which generalize those of morawetz for the wave equation we also extend our method to the wave equation

complex valued random vectors and channels entropy divergence and capacity recent research has demonstrated significant achievable performance gains by exploiting circularity noncircularity or properness improperness of complex valued signals in this paper we investigate the influence of these properties on important information theoretic quantities such as entropy divergence and capacity we prove two maximum entropy theorems that strengthen previously known results the proof of the first maximum entropy theorem is based on the so called circular analog of a given complex valued random vector the introduction of the circular analog is additionally supported by a characterization theorem that employs a minimum kullback leibler divergence criterion in the proof of the second maximum entropy theorem results about the second order structure of complex valued random vectors are exploited furthermore we address the capacity of multiple input multiple output mimo channels regardless of the specific distribution of the channel parameters noise vector and channel matrix if modeled as random we show that the capacity achieving input vector is circular for a broad range of mimo channels including coherent and noncoherent scenarios finally we investigate the situation of an improper and gaussian distributed noise vector we compute both capacity and capacity achieving input vector and show that improperness increases capacity provided that the complementary covariance matrix is exploited otherwise a capacity loss occurs for which we derive an explicit expression

on a quantum chaos theorem of r schrader and m taylor abstract we extend results of r schrader and m taylor on the semi classical asymptotics of eigenfunctions of a quantum hamiltonian for a particle in a gauge field our method is based on the guillemin sternberg uribe theory of fuzzy ladders and reductions of hamiltonian systems with symmetry

on the sum rate of the gaussian mimo z channel and the gaussian mimo x channel in this paper we study the gaussian mimo z channel and the gaussian mimo x channel the mimo x channel xc consists of two multiple antenna transmit receive pairs where each transmitter communicates with both receivers the mimo z channel zc is obtained from the mimo x channel by eliminating one of the links and its corresponding message first we derive a sum rate upper bound for the mimo z channel and compare it with an existing bound in literature next we consider the mimo x channel and propose a new sum rate upper bound by utilizing the sum rate upper bound for the mimo zc subsequently we derive another upper bound for the mimo xc by assuming receiver cooperation and deriving the worst noise covariance matrix for the resulting two user mac we compare the above two upper bounds for the mimo xc with the maddahali motahari khandani mmk scheme then we consider some consequences of the above results for the mimo interference channel finally we present some numerical results the numerical results suggest that the proposed sum rate capacity upper bounds are tighter than existing bounds

on the dimension of divergence sets of dispersive equations we refine results of carleson sjogren and sjolin regarding the pointwise convergence to the initial data of solutions to the schrodinger equation we bound the hausdorff dimension of the sets on which convergence fails for example with initial data in h 1 mathbb r 3 the sets of divergence have dimension at most one

asymptotic frequency shift properizer for block processing of improper complex second order cyclostationary random processes in this paper the block processing of a discrete time dt improper complex second order cyclostationary socs random process is considered in particular it is of interest to find a preprocessing operation that enables the adoption of conventional signal processing techniques and algorithms developed for the filtering of proper complex signals and that leads to computationally efficient near optimal postprocessing an invertible linear conjugate linear lcl operator named the dt frequency shift fresh properizer is first proposed it is shown that the dt fresh properizer converts a dt improper complex socs random process input to an equivalent dt proper complex socs random process output by utilizing the information only about the cycle period of the input an invertible lcl block processing operator named the asymptotic fresh properizer is then proposed that mimics the operation of the dt fresh properizer but processes a finite number of consecutive samples of a dt improper complex socs random process it is shown that the output of the asymptotic fresh properizer is not proper but asymptotically proper and that its frequency domain covariance matrix converges to a highly structured block matrix with diagonal blocks as the block size tends to infinity two representative estimation and detection problems are presented to demonstrate that asymptotically optimal low complexity postprocessors can be easily designed by exploiting these asymptotic second order properties of the output of the asymptotic fresh properizer

on a calderon zygmund commutator type estimate abstract in this paper we present a calderon zygmund commutator type estimate this estimate enables us to prove an embedding result concerning weighted function spaces

new design of fixed interval smoother using covariance information in linear stochastic continuous time systems this paper proposes recursive least squares fixed interval smoothing and filtering algorithms in linear continuous time stochastic systems the estimators require the information of the autocovariance function of the signal the variance of white observation noise and the observed value the autocovariance function of the signal is expressed in the form of the semi degenerate kernel