uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

explicit expressions of the nth gradient of 1 r explicit expressions of the nth gradient tensor and partial derivatives of the function 1 r are given in the absolute and in the local bases associated respectively with the cartesian and the spherical coordinates of space they may be used in various multipolar developments of r 1 interaction energies

efficient soft input soft output tree detection via an improved path metric tree detection techniques are often used to reduce the complexity of a posteriori probability app detection in multiantenna wireless communication systems in this paper we introduce an efficient soft input soft output tree detection algorithm that employs a new type of look ahead path metric in the process of branch pruning or sorting while conventional path metrics depend only on symbols on a visited path the new path metric accounts for unvisited parts of the tree in advance through an unconstrained linear estimator and adds a bias term that reflects the contribution of as yet undecided symbols by applying the linear estimate based look ahead path metric to an algorithm that selects the best paths for each level of the tree we develop a new soft input soft output tree detector called an improved soft input soft output algorithm iss ma based on an analysis of the probability of correct path loss we show that the improved path metric offers substantial performance gain over the conventional path metric we also demonstrate through simulations that the proposed iss ma can be a promising candidate for soft input soft output detection in high dimensional systems

quasi linear dynamics in nonlinear schrodinger equation with periodic boundary conditions it is shown that a large subset of initial data with finite energy l2 norm evolves nearly linearly in nonlinear schrodinger equation with periodic boundary conditions these new solutions are not perturbations of the known ones such as solitons semiclassical or weakly linear solutions

regularization pseudofunction and hadamard finite part abstract first we discuss and correlate the various types of regularizations available in the literature for the singular function h x x k where k is an integer and h x is the heaviside function then we present the corresponding regularization for the function r k where r is the radial distance in r n thereafter we express the recently discovered distributional derivatives of this function in terms of pseudofunctional language commonly used in the coulomb gravitational and interparticle potentials where the function 1 r plays a fundamental role

the list sequential liss algorithm and its application for iterative detection decoding turbo schemes we modify sequential decoding which contrary to a posteriori probability app bahl cocke jelinek raviv bcjr decoding enjoys a complexity almost independent of the number of states this novel list sequential liss decoder avoids most of the drawbacks of the classical sequential decoders such as variable workload and erased frames when working within a turbo scheme it uses a metric containing a priori and channel values a metric length bias term for speeding up the tree search a soft extension of paths without increasing the stack size and soft weighting to obtain a soft output we present several turbo applications using the liss including equalization single antenna interference cancellation multiuser and multiple input multiple output detection it is shown that the liss achieves the optimal genie or app performance at a bit error rate after the outer decoder of less than 10 4 but it also works with channels which have a high number of taps constellation points antennas and states where the app bcjr algorithm becomes infeasible

on the complexity of sphere decoding in digital communications sphere decoding has been suggested by a number of authors as an efficient algorithm to solve various detection problems in digital communications in some cases the algorithm is referred to as an algorithm of polynomial complexity without clearly specifying what assumptions are made about the problem structure another claim is that although worst case complexity is exponential the expected complexity of the algorithm is polynomial herein we study the expected complexity where the problem size is defined to be the number of symbols jointly detected and our main result is that the expected complexity is exponential for fixed signal to noise ratio snr contrary to previous claims the sphere radius which is a parameter of the algorithm must be chosen to ensure a nonvanishing probability of solving the detection problem this causes the exponential complexity since the squared radius must grow linearly with problem size the rate of linear increase is however dependent on the noise variance and thus the rate of the exponential function is strongly dependent on the snr therefore sphere decoding can be efficient for some snr and problems of moderate size even though the number of operations required by the algorithm strictly speaking always grows as an exponential function of the problem size

efficient implementation of quasi maximum likelihood detection based on semidefinite relaxation in this paper we develop two quasi maximum likelihood ml channel detectors for multiuser detection semidefinite relaxation sdr detector and phase shift keying psk detector these detectors can deliver near ml bit error rate ber performance with a polynomial worst case complexity the sdr detector for binary phase shift keying bpsk constellation is based on a convex sdr whereas the psk detector for m psk constellations is based on a nonconvex low rank sdr the sdr detector is implemented using a dual scaling interior point method while the psk detector is based on a coordinate descent strategy on a feasible region homotopy we use dynamic dimension reduction and warm start techniques to achieve signal to noise ratio snr sensitive improvements for both detectors numerical simulations of ber performance and running time indicate the effectiveness of the two quasi ml detectors when compared to the conventional sphere decoder and its variants

partial marginalization soft mimo detection with higher order constellations a new method for multiple input multiple output mimo detection with soft output the partial marginalization pm algorithm was recently proposed advantages of the method are that it is straightforward to parallelize and that it offers a fully predictable runtime pm trades performance for computational complexity via a user defined parameter in the limit of high computational complexity the algorithm becomes the map demodulator the pm algorithm also works with soft input but until now it has been unclear how to apply it for other modulation formats than binary phase shift keying bpsk per real dimension in this correspondence we explain how to extend pm with soft input to general signaling constellations while maintaining the low complexity advantage of the original algorithm

universal binary semidefinite relaxation for ml signal detection semidefinite relaxation sdr provides a computationally efficient polynomial time approximation of the maximum likelihood detector however most of the existing works mainly focus on particular signal constellations in this paper we propose a universal binary semidefinite relaxation scheme that can handle arbitrary signal constellations in polynomial time the proposed scheme first binarizes the original signal space to a linearly constrained binary space and then solves the detection problem through sdr colorblack a specialized dual barrier method is provided to solve the sdr more efficiently in addition we propose to apply on the fly decision feedback to further reduce the computational complexity and improve the detection performance the proposed binary sdr together with on the fly decision feedback scheme can provide comparable or better solutions compared to existing sdr methods specialized to specific constellations such as 16 qam and 8 psk in terms of computational complexity and symbol error rate furthermore the proposed scheme is universal and can solve any other constellations such as 12 qam 32 qam or m psk

the eigenvalue problem for the focusing nonlinear schrodinger equations new solvable cases abstract in this paper we study the semi classical limit of the zakharov shabat eigenvalue problem for the focusing of nls with some specific initial data in all these cases the eigenvalue problem is reduced to connection problems for the hypergeometric equation and for other classical equations the special initial data suppl prog theor phys 55 1974 284 is contained in our family of initial data parameterized by a real parameter as a particular case 0 we find that beyond a certain value of the parameter the pure point spectrum becomes empty and all the scattering information is contained in the reflection coefficient

distributional solutions of the hypergeometric differential equation abstract we present the distributional solutions to the hypergeometric differential equation these solutions are obtained in the form of infinite series of the dirac delta functions and its derivatives we employ these solutions to observe their interesting features furthermore the form of these solutions is the same as the ones required for the weight distributions for a certain class of orthogonal polynomials

sumis near optimal soft in soft out mimo detection with low and fixed complexity the fundamental problem of interest here is soft input soft output multiple input multiple output mimo detection we propose a method referred to as subspace marginalization with interference suppression sumis that yields unprecedented performance at low and fixed deterministic complexity our method provides a well defined tradeoff between computational complexity and performance apart from an initial sorting step consisting of selecting channel matrix columns the algorithm involves no searching nor algorithmic branching hence the algorithm has a completely predictable run time and allows for a highly parallel implementation we numerically assess the performance of sumis in different practical settings full partial channel state information sequential iterative decoding and low high rate outer codes we also comment on how the sumis method performs in systems with a large number of transmit antennas

a unified framework for tree search decoding rediscovering the sequential decoder we consider receiver design for coded transmission over linear gaussian channels we restrict ourselves to the class of lattice codes and formulate the joint detection and decoding problem as a closest lattice point search clps here a tree search framework for solving the clps is adopted in our framework the clps algorithm is decomposed into the preprocessing and tree search stages the role of the preprocessing stage is to expose the tree structure in a form matched to the search stage we argue that the forward and feedback matrix filters of the minimum mean square error decision feedback equalizer mmse dfe are instrumental for solving the joint detection and decoding problem in a single search stage it is further shown that mmse dfe filtering allows for solving underdetermined linear systems and using lattice reduction methods to diminish complexity at the expense of a marginal performance loss for the search stage we present a generic method based on the branch and bound bb algorithm and show that it encompasses all existing sphere decoders as special cases the proposed generic algorithm further allows for an interesting classification of tree search decoders sheds more light on the structural properties of all known sphere decoders and inspires the design of more efficient decoders in particular an efficient decoding algorithm that resembles the well known fano sequential decoder is identified the excellent performance complexity tradeoff achieved by the proposed mmse dfe fano decoder is established via simulation results and analytical arguments in several multiple input multiple output mimo and intersymbol interference isi scenarios

asymptotic taylor expansion for fourier hyperfunctions a necessary and sufficient condition is proved that the asymptotic taylor expansion for a fourier hyperfunction converges in the space of fourier hyperfunctions

layered tabu search algorithm for large mimo detection and a lower bound on ml performance in this letter we are concerned with low complexity detection in large multiple input multiple output mimo systems with tens of transmit receive antennas our new contributions in this letter are two fold first we propose a low complexity algorithm for large mimo detection based on a layered low complexity local neighborhood search second we obtain a lower bound on the maximum likelihood ml bit error performance using the local neighborhood search the advantages of the proposed ml lower bound are i it is easily obtained for mimo systems with large number of antennas because of the inherent low complexity of the search algorithm ii it is tight at moderate to high snrs and iii it can be tightened at low snrs by increasing the number of symbols in the neighborhood definition the proposed detection algorithm based on the layered local neighborhood search achieves bit error performances which are quite close to this lower bound for large number of antennas and higher order qam

soft input soft output single tree search sphere decoding soft input soft output siso detection algorithms form the basis for iterative decoding the computational complexity of siso detection often poses significant challenges for practical receiver implementations in particular in the context of multiple input multiple output mimo wireless communication systems in this paper we present a low complexity siso sphere decoding algorithm based on the single tree search paradigm proposed originally for soft output mimo detection in studer soft output sphere decoding algorithms and vlsi implementation ieee j sel areas commun vol 26 no 2 pp 290 300 feb 2008 the new algorithm incorporates clipping of the extrinsic log likelihood ratios llrs into the tree search which results in significant complexity savings and allows to cover a large performance complexity tradeoff region by adjusting a single parameter furthermore we propose a new method for correcting approximate llrs resulting from sub optimal detectors which often significantly improves detection performance at low additional computational complexity

on semiclassical zero dispersion limit solutions of the focusing nonlinear schrodinger equation we calculate the leading order term of the solution of the focusing nonlinear cubic schrodinger equation nls in the semiclassical limit for a certain one parameter family of initial conditions this family contains both solitons and pure radiation in the pure radiation case our result is valid for all times t 0 we utilize the riemann hilbert problem formulation of the inverse scattering problem to obtain the leading order term of the solution error estimates are provided 2004 wiley periodicals inc

semiclassical limit of the focusing nls whitham equations and the riemann hilbert problem approach abstract the main goal of this paper is to put together a the whitham theory applicable to slowly modulated n phase nonlinear wave solutions to the focusing nonlinear schrodinger fnls equation and b the riemann hilbert problem approach to particular solutions of the fnls in the semiclassical small dispersion limit that develop slowly modulated n phase nonlinear wave in the process of evolution both approaches have their own merits and limitations understanding of the interrelations between them could prove beneficial for a broad range of problems involving the semiclassical fnls

generalized function solutions of differential and functional differential equations recently there has been considerable interest in problems concerning the existence of solutions to differential and functional differential equations fde in various spaces of generalized functions many important areas in theoretical and mathematical physics theory of partial differential equations quantum electrodynamics operational calculus and functional analysis use the methods of the distributions theory yet for ordinary differential equations ode and especially fde research in this direction is insufficiently developed and remains restricted to isolated results for some secondorder equations or special higher order systems we note in particular papers l 6 which contain references to previous work it is weil known that normal linear homogeneous systems of ode with infinitely smooth coefficients have no generalized function solutions other than the classical solutions in contrast to this case for equations with singularities in the coefficients new solutions in generalized functions may appear and some classical solutions may disappear the number m is called the order of the distribution

a distributional theory for asymptotic expansions the purpose of this chapter is to present the distributional theory of asymptotic expansions for functions of one variable this chapter and the next where the multidimensional expansions are studied are the central part of the book

semidefinite relaxation approaches to soft mimo demodulation for higher order qam signaling the soft demodulator is a key component of receivers that employ iterative demodulation and decoding for multiple input multiple output systems with binary or qpsk signaling the semidefinite relaxation sdr approach to list based soft demodulation provides a desirable balance between the performance and computational cost in this paper that approach is extended to systems with higher order qam signaling for the class of list sdr demodulators the key to the extension is a quadratic approximation of the a priori information and for the more computationally efficient class of single sdr demodulators the key to the extension is an algebraic expression derived herein for the probability that the sdr randomization procedure generates each candidate symbol to implement these demodulators customized primal dual and dual scaling algorithms are developed to solve the underlying semidefinite programs the performance of the proposed demodulators is compared with that of some existing demodulators by using error rate simulations the empirical distribution of the computational cost and exit chart analyses in our comparisons the proposed single sdr demodulator provides performance that comes close to that of the list sequential demodulator at a computational cost that is close to that of the minimum mean square error parallel soft interference canceller

the asymptotic expansion of certain multi dimensional generalized functions abstract we apply regularization of divergent integrals in the derivation of the asymptotic expansion of certain multi dimensional generalized functions we further present several illustrations to demonstrate that the asymptotic development of generalized functions provides a lucid formulation of many concepts in asymptotic analysis such as the expansion of oscillatory integrals and the expansion of certain series

finite gap integration of the derivative nonlinear schrodinger equation a riemann hilbert method abstract in this paper we retrieve finite gap solutions of the gerdjikov ivanov type derivative nonlinear schrodinger equation by using the algebro geometric method and the riemann hilbert method we show that the baker akhiezer function of the derivative nonlinear schrodinger equation can be described in terms of two solvable matrix riemann hilbert problems on with 2 and 3 symmetry conditions based on the technique developed in the study of long time asymptotics our main tools include matrix baker akhiezer function asymptotic analysis algebraic curve and riemann theta function matrix riemann hilbert problem and associated deformation procedures

semiclassical limit of focusing nls for a family of square barrier initial data the small dispersion limit of the focusing nonlinear schrodinger equation nls exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic scales the non self adjoint scattering problem and ill posed limiting whitham equations associated to focusing nls make rigorous asymptotic results difficult previous studies have focused on special classes of analytic initial data for which the limiting elliptic whitham equations are wellposed in this paper we consider another exactly solvable family of initial data the family of square barriers 0 x q l l for real amplitudes q using riemann hilbert techniques we obtain rigorous pointwise asymptotics for the semiclassical limit of focusing nls globally in space and up to an o 1 maximal time in particular we show that the discontinuities in our initial data regularize by the immediate generation of genus one oscillations emitted into the support of the initial data to the best of our knowledge this is the first case in which the genus structure of the semiclassical asymptotics for focusing nls have been calculated for nonanalytic initial data 2013 wiley periodicals inc

quasi maximum likelihood multiuser detection using semi definite relaxation with application to synchronous cdma the maximum likelihood ml multiuser detector is well known to exhibit better bit error rate ber performance than many other multiuser detectors unfortunately ml detection mld is a nondeterministic polynomial time hard np hard problem for which there is no known algorithm that can find the optimal solution with polynomial time complexity in the number of users in this paper a polynomial time approximation method called semi definite sd relaxation is applied to the mld problem with antipodal data transmission sd relaxation is an accurate approximation method for certain np hard problems the sd relaxation ml sdr ml detector is efficient in that its complexity is of the order of k3 5 where k is the number of users we illustrate the potential of the sdr ml detector by showing that some existing detectors such as the decorrelator and the linear minimum mean square error detector can be interpreted as degenerate forms of the sdr ml detector simulation results indicate that the ber performance of the sdr ml detector is better than that of these existing detectors and is close to that of the true ml detector even when the cross correlations between users are strong or the near far effect is significant

solving large scale sparse semidefinite programs for combinatorial optimization we present a dual scaling interior point algorithm and show how it exploits the structure and sparsity of some large scale problems we solve the positive semidefinite relaxation of combinatorial and quadratic optimization problems subject to boolean constraints we report the first computational results of interior point algorithms for approximating maximum cut semidefinite programs with dimension up to 3 000

iterative tree search detection for mimo wireless systems this paper presents a reduced complexity soft input soft output detection scheme called iterative tree search detection for multiple input multiple output wireless communication systems employing turbo processing at the receiver in this scheme a reduced search space is selected with the aid of the m algorithm and qam signal constellations with block partitionable labels are used in order to make the detection complexity per bit almost independent of the modulation order as well as asymptotically linear in the number of transmit antennas results from computer simulations are presented which demonstrate the capability of the scheme to approach optimal performance at considerably reduced complexity

an interior point method for semidefinite programming we propose a new interior point based method to minimize a linear function of a matrix variable subject to linear equality and inequality constraints over the set of positive semidefinite matrices

reduced complexity soft output mimo sphere detectors part i algorithmic optimizations optimum soft output so multiple input multiple output mimo tree search detection algorithms pose significant implementation challenges due to their nondeterministic processing throughput and high computational complexity in this two part work we present extensive algorithmic and architectural optimizations of the sphere decoding algorithm targeted at achieving practical tradeoffs between desired link performance and affordable computational complexity the algorithmic optimizations in this part span the tree search traversal scheme leaf processing step internal node pruning and skipping step child enumeration based on a state machine adaptive radius scaling for llr clipping qr decomposition based on minimum cumulative residuals and multitree configurations the optimizations demonstrate that a 64 qam so mimo detector for lte is capable of attaining almost ml performance with an snr loss of only 0 85 db at 1 bler by visiting at most 200 tree nodes

optimal weighted average power similar tests for the covariance structure in the linear regression model this paper suggests a procedure for the construction of optimal weighted average power similar tests for the error covariance matrix of a gaussian linear regression model when the alternative model belongs to the exponential family the paper uses a saddlepoint approximation to construct simple test statistics for a large class of problems and overcomes the computational burden of evaluating the complicated integrals arising in the derivation of optimal weighted average power tests extensions to panel data models are considered applications are given to tests for error autocorrelation in the linear regression model and in a panel data framework

taylor expansion for generalized functions a necessary and sufficient condition is given that the asymptotic taylor expansion for a distribution for an ultradistribution is the taylor series convergent in the space of distributions of ultradistributions

soft quasi maximum likelihood detection for multiple antenna wireless channels the paper addresses soft maximum likelihood ml detection for multiple antenna wireless communication channels we propose a soft quasi ml detector that maximizes the log likelihood function by deploying a semi definite relaxation sdr given perfect channel state information at the receiver the quasi ml sdr detector closely approximates the performance of the optimal ml detector in both coded and uncoded multiple input multiple output mimo channels with quadrature phase shift keying qpsk modulation and frequency flat rayleigh fading the complexity of the quasi ml sdr detector is much less than that of the optimal ml detector thus offering more favorable performance complexity characteristics in contrast to the existing sphere decoder the new quasi ml detector enjoys guaranteed polynomial worst case complexity the two detectors exhibit quite comparable performance in a variety of ergodic qpsk mimo channels but the complexity of the quasi ml detector scales better with increasing number of transmit and receive antennas especially in the region of low signal to noise ratio snr

performance assessment of mimo bicm demodulators based on mutual information we provide a comprehensive performance comparison of soft output and hard output demodulators in the context of non iterative multiple input multiple output bit interleaved coded modulation mimo bicm coded bit error rate ber widely used in literature for demodulator comparison has the drawback of depending strongly on the error correcting code being used this motivates us to propose the mutual information of the equivalent modulation channel comprising modulator wireless channel and demodulator as a code independent performance measure we present extensive numerical results for spatially independent identically distributed i i d ergodic and quasi static fading channels under perfect and imperfect channel state information these results reveal that the performance ranking of mimo demodulators is rate dependent and provide new insights regarding mimo bicm system design i e the choice of antenna configuration symbol constellation and demodulator for a given target rate

probabilistic analysis of semidefinite relaxation for binary quadratic minimization we consider semidefinite programming relaxation sdr of a binary quadratic minimization problem this np hard problem arises naturally in the maximum likelihood detection of discrete signals for digital communications we analyze the average performance of the sdr algorithm for a class of randomly generated binary quadratic minimization problems although the sdr worst case approximation ratio is unbounded for this np hard problem our analysis shows that sdr can provide in polynomial time a provably near optimal solution achieving a constant factor approximation of the optimal objective value in probability moreover this constant factor remains bounded with increasing problem size our proof is based on an asymptotic analysis of karush kuhn tucker optimality conditions using random matrix theory

the korteweg de vries equation with small dispersion higher order lax levermore theory we utilize the inverse scattering transformation to study the rapid oscillations which arise in the solution of the initial value problem begin gathered u t 6u u x in 2 u xxx 0 hfill u left x 0 right v left x right hfill end gathered when 0 we refine the lax levermore theory which gives the weak limit of the solution as 0 by introducing a quantum condition this allows us to calculate the waveform of the local oscillations up to phase shifts when is small

on the long time limit of semiclassical zero dispersion limit solutions of the focusing nonlinear schrodinger equation pure radiation case in a previous paper 13 we calculated the leading order term q0 x t e of the solution ofq x t e the focusing nonlinear cubic schrodinger nls equation in the semiclassical limit e 0 for a certain one parameter family of initial conditions this family contains both solitons and pure radiation in the pure radiation case our result is valid for all times t 0 the aim of the present paper is to calculate the long term behavior of the semiclassical solution q x t e in the pure radiation case as before our main tool is the riemann hilbert problem rhp formulation of the inverse scattering problem and the corresponding system of moment and integral conditions known also as a system of modulation equations c 2006 wiley periodicals inc

exact solutions of semiclassical non characteristic cauchy problems for the sine gordon equation abstract the use of the sine gordon equation as a model of magnetic flux propagation in josephson junctions motivates studying the initial value problem for this equation in the semiclassical limit in which the dispersion parameter e tends to zero assuming natural initial data having the profile of a moving 2 kink at time zero we analytically calculate the scattering data of this completely integrable cauchy problem for all e 0 sufficiently small and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small e this sequence of exact solutions is analogous to that of the well known n soliton or higher order soliton solutions of the focusing nonlinear schrodinger equation we then use plots obtained from a careful numerical implementation of the inverse scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit in the limit e 0 one observes the appearance of nonlinear caustics i e curves in space time that are independent of e but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases in the appendices we give a self contained account of the cauchy problem from the perspectives of both inverse scattering and classical analysis picard iteration specifically appendix a contains a complete formulation of the inverse scattering method for generic l 1 sobolev initial data and appendix b establishes the well posedness for l p sobolev initial data which in particular completely justifies the inverse scattering analysis in appendix a

low complexity decoding via reduced dimension maximum likelihood search in this paper we consider a low complexity detection technique referred to as a reduced dimension maximum likelihood search rd mls rd mls is based on a partitioned search which approximates the maximum likelihood ml estimate of symbols by searching a partitioned symbol vector space rather than that spanned by the whole symbol vector the inevitable performance loss due to a reduction in the search space is compensated by 1 the use of a list tree search which is an extension of a single best searching algorithm called sphere decoding and 2 the recomputation of a set of weak symbols i e those ignored in the reduced dimension search for each strong symbol candidate found during the list tree search through simulations on m quadrature amplitude modulation qam transmission in frequency nonselective multi input multioutput mimo channels we demonstrate that the rd mls algorithm shows near constant complexity over a wide range of bit error rate ber 10 1 10 4 while limiting performance loss to within 1 db from ml detection

on asymptotic expansions of twisted products the series development of the quantum mechanical twisted product is studied the series is shown to make sense as a moment asymptotic expansion of the integral formula for the twisted product either pointwise or in the distributional sense depending on the nature of the factors a condition is given that ensures convergence and is stronger than previously known results possible applications are examined