subelliptic peter weyl and plancherel theorems on compact connected semisimple lie groups abstract we will study the connections between the elliptic and subelliptic versions of the peter weyl and plancherel theorems in the case when the sub riemannian structure is generated naturally by the choice of a cartan subalgebra along the way we will introduce and study the subelliptic casimir operator associated to the subelliptic laplacian

regularity of quasi linear equations in the heisenberg group we prove the c regularity of the gradient of weak solutions of a class of quasi linear equations in nilpotent stratified lie groups of step two as applications we prove higher regularity theorems and a liouville type theorem for 1 quasi conformal mappings between domains of the heisenberg group 1997 john wiley sons inc

a christoffel darboux formula and a favard s theorem for orthogonal laurent polynomials on the unit circle let f k z k 0 be the family of orthonormal laurent polynomials on the unit circle which spans d in the ordering induced by p n e n 1 2 from the three term recurrence relation satisfied by f k z k 0 we deduce a christoffel darboux formula particular examples are considered and a favard type theorem is proved a connection with the ordering induced by p n e n 2 is also established

sensitivity analysis for szegő polynomials szeg polynomials are orthogonal with respect to an inner product on the unit circle numerical methods for weighted least squares approximation by trigonometric polynomials conveniently can be derived and expressed with the aid of szeg polynomials this paper discusses the conditioning of several mappings involving szeg polynomials and thereby sheds light on the sensitivity of some approximation problems involving trigonometric polynomials

a limit theorem for szego polynomials with respect to convolution of point masses with the fejer kernel two recently proposed methods for estimating the m frequencies of a trigonometric signal using szego polynomials of fixed degree k m consist of multiplying the moments of the n truncated periodogram by the moments of the poisson kernel and the wrapped gaussian respectively in an effort to address the non convergence of the polynomials as n these methods are seen to be equivalent to convolution of point masses with approximate identities suggesting a general method we characterize the limit polynomial for the case when the approximate identity is the fejer kernel extending recent results of the author for the case of the poisson kernel moreover the limit is seen to be the same as in the former case

an analogue for szego polynomials of the clenshaw algorithm abstract linear combinations of polynomials that are orthogonal with respect to an inner product defined on part of the real axis are commonly evaluated by the clenshaw algorithm we present an analogous algorithm for the evaluation of a linear combination n j 0 j j of polynomials j that are orthogonal with respect to an inner product defined on part of the unit circle the j are known as szeg polynomials and find applications e g in signal processing we also discuss how to express n j 0 j j as a linear combination of monomials

optimal partial regularity for sub elliptic systems with dini continuous coefficients under the superquadratic natural growth the paper is concerned with nonlinear sub elliptic systems with dini continuous coefficients under super quadratic m 2 natural growth conditions in the heisenberg group a general criterion for weak solutions to be regular in the neighborhood of a given point is established by a generalization of method of a harmonic approximation in particular our result is optimal in the sense that in the case of holder continuous coefficients we obtain directly the optimal holder exponent for horizontal gradient of weak solutions on its regular set

generalized szego theory in frequency analysis abstract a sequence of positive borel measures with infinite support on the unit circle is given converging in the weak star topology to a measure with support consisting of a finite number n 0 of mass points the sequences of monic rational functions orthogonal with respect to inner products determined by the measures are studied with the focus being on the asymptotic behavior of the zeros the main result is twofold for a fixed n n 0 n 0 of the zeros interesting zeros tend to the mass points of the limiting measure while the remaining n n 0 zeros uninteresting zeros may not converge at all however by considering subsequences limits may be obtained and the remaining zeros and their limits will be located in a closed disk z r

szego polynomials associated with wiener levinson filters abstract szego polynomials are studied in connection with wiener levinson filters formed from discrete signals xn xn k n 1k 0 our main interest is in the frequency analysis problem of finding the unknown frequencies j when the signal is a trigonometric polynomial x n k j i i j e i j k associated with this signal is the sequence of monic szego polynomials n n z n 0 orthogonal on the unit circle with respect to a distribution function n explicit expressions for the weight function n and associated szego function dn z are given in terms of the z transform xn z of the signal xn several theorems are given to support the following conjecture which was suggested by numerical experiments as n and n increase the 2i 1 zeros of n n z of largest modulus approach the points ei j we conclude by showing that the reciprocal polynomials n n z z n n n 1 z are pade numerators for pade approximants of fixed denominator degree to a meromorphic function related to dn z

the hypoelliptic laplacian on a compact lie group abstract let g be a compact lie group and let g be its lie algebra in this paper we produce a hypoelliptic laplacian on g g which interpolates between the classical laplacian of g and the geodesic flow this deformation is obtained by producing a suitable deformation of the dirac operator of kostant we show that various poisson formulas for the heat kernel can be proved using this interpolation by methods of local index theory the paper was motivated by papers by atiyah and frenkel in connection with localization formulas in equivariant cohomology and with kac s character formulas for affine lie algebras in a companion paper we will use similar methods in the context of selberg s trace formula

limits for szego polynomials in frequency analysis abstract we characterize limits for orthogonal szego polynomials of fixed degree k with respect to certain measures on the unit circle which are weakly convergent to a sum of m k point masses such measures arise for example as a convolution of point masses with an approximate identity it is readily seen that the underlying measures in two recently proposed methods for estimating the m frequencies j of a discrete time trigonometric signal using szego polynomials are of this form we prove existence of szego polynomial limits associated with a general class of weakly convergent measures and prove that for convolution of point masses with the poisson kernel which underlies one of the recently proposed methods the limit has as a factor the szego polynomial with respect to a related measure which we specify since m of the zeros approach the e i j this result uniquely characterizes the limit a similar result is obtained for measures consisting of point masses with additive absolutely continuous part

local risk minimization under transaction costs we propose a new approach to the pricing and hedging of contingent claims under transaction costs in a general incomplete market in discrete time under the assumptions of a bounded mean variance tradeoff substantial risk and a nondegeneracy condition on the conditional variances of asset returns we prove the existence of a locally risk minimizing strategy inclusive of transaction costs for every square integrable contingent claim then we show that local risk minimization is robust under the inclusi on of transaction costs the preceding strategy which is locally risk minimizing inclusive of transaction costs in a model with bid ask spreads on the underlying asset is also locally risk minimizing without transaction costs in a fictitious model which i s frictionless and where the fictitious asset price lies between the bid and ask price processes of the original model in particular our results apply to any nondegenerate model with a finite state space if the transaction cost parameter is sufficiently small

asymptotics for szego polynomials associated with wiener levinson filters abstract pan k asymptotics for szego polynomials associated with wiener levinson filters journal of computational and applied mathematics 46 1993 387 394 both weak and strong convergence theorems are given to support a jones njastad and saff conjecture that unknown frequencies in a periodic discrete time signal can be determined by zeros of szego polynomials with respect to some distribution function

on the regularity of subelliptic p harmonic functions in carnot groups abstract in this paper we prove second order horizontal differentiability and c 1 regularity results for subelliptic p harmonic functions in carnot groups for p close to 2

szego polynomials applied to frequency analysis abstract this paper is concerned with the problem of determining unknown frequencies 1 i using an observed discrete time signal xn xn m arising from a continuous waveform that is the superposition of a finite number of sinusoidal waves with well defined frequencies j periods p j 2 n j n z j 1 i we investigate the wiener levinson method formulated here in terms of szeg polynomials n n z orthogonal on the unit circle with respect to a distribution function n defined by the n observed values of the signal we prove that if n0 denotes the number of critical points eiwj then for every n n0 and n 1 the zeros z j n n of n n z can be arranged so that limn z j n n eiwj for each of the frequencies wj this result confirms one of the main parts of the conjecture given by jones njastad and saff this journal 1990 on the convergence of zeros of the szeg polynomials a related result on the convergence of corresponding two point pade approximants is also given

orthogonal laurent polynomials on the unit circle extended cmv ordering and 2d toda type integrable hierarchies abstract we connect the theory of orthogonal laurent polynomials on the unit circle and the theory of toda like integrable systems using the gauss borel factorization of a cantero moral velazquez moment matrix that we construct in terms of a complex quasi definite measure supported on the unit circle the factorization of the moment matrix leads to orthogonal laurent polynomials on the unit circle and the corresponding second kind functions we obtain jacobi operators 5 term recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas from this point of view in a completely algebraic way we generalize the cantero moral velazquez sequence of laurent monomials recursion relations christoffel darboux kernels and corresponding christoffel darboux formulas in this extended context we introduce continuous deformations of the moment matrix and we show how they induce a time dependent orthogonality problem related to a toda type integrable system which is connected with the well known toeplitz lattice we obtain the lax and zakharov shabat equations using the classical integrability theory tools we explicitly derive the dynamical system associated with the coefficients of the orthogonal laurent polynomials and we compare it with the classical toeplitz lattice dynamical system for the verblunsky coefficients of szeg polynomials for a positive measure discrete flows are introduced and related to darboux transformations finally we obtain the representation of the orthogonal laurent polynomials and their second kind functions using the formalism of miwa shifts in terms of functions and the subsequent bilinear equations

real orthogonal polynomials in frequency analysis univ estadual paulista ibilce dept ciencias computacao estatist br 15054000 sao paulo brazil

matrix orthogonal laurent polynomials on the unit circle and toda type integrable systems matrix orthogonal laurent polynomials in the unit circle and the theory of toda like integrable systems are connected using the gauss borel factorization of two left and a right cantero morales velazquez block moment matrices which are constructed using a quasi definite matrix measure a block gauss borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal laurent polynomials and matrix szeg polynomials which can be expressed in terms of schur complements of bordered truncations of the block moment matrix the corresponding block extension of the christoffel darboux theory is derived deformations of the quasidefinite matrix measure leading to integrable systems of toda type are studied the integrable theory is given in this matrix scenario wave and adjoint wave functions lax and zakharov shabat equations bilinear equations and discrete flows connected with darboux transformations we generalize the integrable flows of the cafasso s matrix extension of the toeplitz lattice for the verblunsky coefficients of szeg polynomials an analysis of the miwa shifts allows for the finding of interesting connections between christoffel darboux kernels and miwa shifts of the matrix orthogonal laurent polynomials

distribution of roots of random real generalized polynomials the average density of zeros for monic generalized polynomials p n z phi z sum nolimits k 1 n c k f k z with real holomorphic f k and real gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative we obtain compact expressions for both the regular component generated by the complex roots and the singular one real roots of the average density of roots the density of the regular component goes to zero in the vicinity of the real axis like lmz we present the low and high disorder asymptotic behaviors then we particularize to the large n limit of the average density of complex roots of monic algebraic polynomials of the form p n z z n sum nolimits k 1 n c k z n k with real independent identically distributed gaussian coefficients having zero mean and dispersion delta 1 sqrt n lambda the average density tends to a simple universal function of 2nlog z and in the domain coth 2 n sin arg z where nearly all the roots are located for largen

sequences of orthogonal laurent polynomials bi orthogonality and quadrature formulas on the unit circle in this paper the construction of orthogonal bases in the space of laurent polynomials on the unit circle is considered as an application a connection with the so called bi orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on laurent polynomials are studied

strong stieltjes distributions and orthogonal laurent polynomials with applications to quadratures and pade approximation starting from a strong stieltjes distribution general sequences of orthogonal laurent polynomials are introduced and sonic of their most relevant algebraic properties are studied from this perspective the connection between certain quadrature formulas associated with the distribution and two point pade approximants to the stieltjes transform of is revisited finally illustrative numerical examples are discussed

continuation methods for the computation of zeros of szego polynomials abstract let j j 0 be a family of monic polynomials that are orthogonal with respect to an inner product on the unit circle the polynomials j arise in time series analysis and are often referred to as szego polynomials or levinson polynomials knowledge about the location of their zeros is important for frequency analysis of time series and for filter implementation we present fast algorithms for computing the zeros of the polynomials n based on the observation that the zeros are eigenvalues of a rank one modification of a unitary upper hessenberg matrix h n 0 of order n the algorithms first determine the spectrum of h n 0 by one of several available schemes that require only o n 2 arithmetic operations the eigenvalues of the rank one perturbation are then determined from the eigenvalues of h n 0 by a continuation method the computation of the n zeros of n in this manner typically requires only o n 2 arithmetic operations the algorithms have a structure that lends itself well to parallel computation the latter is of significance in real time applications

shadow prices and well posedness in the problem of optimal investment and consumption with transaction costs we revisit the optimal investment and consumption model of davis and norman math oper res 15 1990 pp 676 713 and shreve and soner ann appl probab 4 1994 pp 609 692 following a shadow price approach similar to that of kallsen and muhle karbe ann appl probab 20 2010 pp 1341 1358 making use of the completeness of the model without transaction costs we reformulate and reduce the hamilton jacobi bellman equation for this singular stochastic control problem to a nonstandard free boundary problem for a first order ode with an integral constraint having shown that the free boundary problem has a smooth solution we use it to construct the solution of the original optimal investment consumption problem in a self contained manner and without any recourse to the dynamic programming principle furthermore we provide an explicit characterization of model parameters for which the value function is finite

orthogonal matrix polynomials and applications orthogonal matrix polynomials on the real line or on the unit circle have properties which are natural generalizations of properties of scalar orthogonal polynomials appropriately modified for the matrix calculus we show that orthogonal matrix polynomials both on the real line and on the unit circle appear at various places and we describe some of them the spectral theory of doubly infinite jacobi matrices can be described using orthogonal 2 x 2 matrix polynomials on the real line scalar orthogonal polynomials with a sobolev inner product containing a finite number of derivatives can be studied using matrix orthogonal polynomials on the real line orthogonal matrix polynomials on the unit circle are related to unitary block hessenberg matrices and are very useful in multivariate time series analysis and multichannel signal processing finally we show how orthogonal matrix polynomials can be used for gaussian quadrature of matrix valued functions parallel algorithms for this purpose have been implemented using pvm and some examples are worked out

on the uniqueness of unbounded viscosity solutions arising in an optimal terminal wealth problem with transaction costs we study the uniqueness of viscosity solutions of a hamilton jacobi bellman equation which arises in a portfolio optimization problem in which an investor maximizes expected utility of terminal wealth in the presence of proportional transaction costs our main contribution is that the comparison theorem can be applied to prove the uniqueness of the value function in the portfolio optimization problem for logarithmic and power utility

c1 partial regularity for sub elliptic systems with dini continuous coefficients in carnot groups abstract in this paper we consider nonlinear sub elliptic systems with dini continuous coefficients for the case 1 m 2 in carnot groups based on a generalization of the technique of a harmonic approximation introduced by duzaar grotowski kronz and a sobolev poincare type inequality established in carnot groups a c 1 partial regularity result for weak solutions of sub elliptic systems with natural growth terms is established in particular our result is optimal in the sense that in the case of holder continuous coefficients it yields directly the optimal holder exponent for horizontal gradients of weak solutions on its regular set

on the construction of szego polynomials abstract the chebyshev and stieltjes procedures are algorithms for computing recursion coefficients for polynomials that are orthogonal with respect to an inner product defined on part of the real axis the chebyshev procedure is an implementation of a map from moments to recursion coefficients of orthogonal polynomials the modified chebyshev procedure is an implementation of a map from modified moments to recursion coefficients the latter map is generally much better conditioned than the former one the conditioning of these maps has been studied by gautschi this paper is concerned with analogues of the chebyshev and stieltjes procedures when the inner product is defined on the unit circle polynomials orthogonal with respect to such an inner product are known as szeg polynomials and the analogue of the chebyshev procedure is known as schur s algorithm this algorithm implements a map from moments to recursion coefficients for szeg polynomials our analysis shows that this map generally is much better conditioned than the map implemented by the chebyshev procedure and suggests that schur s algorithm is as insensitive to errors in the data as the modified chebyshev procedure thus roughly moments associated with inner products defined on the unit circle correspond to modified moments associated with inner products defined on the real axis

the subelliptic heat kernel on su 2 representations asymptotics and gradient bounds the lie group su 2 endowed with its canonical subriemannian structure appears as a three dimensional model of a positively curved subelliptic space the goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

partial regularity for discontinuous sub elliptic systems with vmo coefficients involving controllable growth terms in heisenberg groups abstract in this paper we consider discontinuous sub elliptic systems with vmo coefficients for the controllable growth case of p 2 in the heisenberg group based on a generalization of the technique of a harmonic approximation with the superquadratic growth case a partial holder continuity result for weak solutions of discontinuous sub elliptic systems with vmo coefficients is established in particular the primary model covered by our analysis is the non degenerate p sub laplace system involving super quadratic controllable growth terms

signal recovery by discrete approximation and a prony like method abstract we introduce an algorithm which combines ideas of prony s approach to recover signals from given samples with approximation methods we solve two overdetermined systems of linear equations with linear programming methods and calculate the zeros of a suitable prony like polynomial we get the bandwidth m the frequencies as well as the amplitudes and some other characteristics of the signal especially it is reconstructed if sufficient many at least 2 m samples are given if we have too few samples or if they are too much noised we get an approximation of the original noiseless signal even if we have sufficient many but possible erroneous samples the obtained signal interpolates at least m of them usually the low noised or noiseless ones and of course all samples if the signal is recovered the described method behaves well to moderate sampling errors and is resistant to outliers in the samples which can be detected filtered off or corrected during the calculations to improve the quality of the computed signal

portfolio selection with transaction costs in this paper optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log normal diffusion this problem was solved by merton and others when transactions between bank and stock are costless here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted it is shown that the optimal buying and selling policies are the local times of the two dimensional process of bank and stock holdings at the boundaries of a wedge shaped region which is determined by the solution of a nonlinear free boundary problem an algorithm for solving the free boundary problem is given

the subelliptic heat kernel on the cr sphere we study the heat kernel of the sub laplacian l on the cr sphere mathbb s 2n 1 an explicit and geometrically meaningful formula for the heat kernel is obtained as a by product we recover in a simple way the green function of the conformal sub laplacian l n 2 that was obtained by geller j differ geom 15 417 435 1980 and also get an explicit formula for the sub riemannian distance the key point is to work in a set of coordinates that reflects the symmetries coming from the fibration mathbb s 2n 1 rightarrow mathbb cp n

subelliptic estimates on compact semisimple lie groups abstract in this paper we consider a natural subelliptic structure in semisimple compact and connected lie groups and estimate the constant in the so called subelliptic friedrichs knapp stein inequality which has implications in the regularity theory of p energy minimizers

vertex operator solutions to the discrete kp hierarchy vertex operators which are disguised darboux maps transform solutions of the kp equation into new ones in this paper we show that the bi infinite sequence obtained by darboux transforming an arbitrary kp solution recursively forward and backwards yields a solution to the discrete kp hierarchy the latter is a kp hierarchy where the continuous space x variable gets replaced by a discrete n variable the fact that these sequences satisfy the discrete kp hierarchy is tantamount to certain bilinear relations connecting the consecutive kp solutions in the sequence at the grassmannian level these relations are equivalent to a very simple fact which is the nesting of the associated infinite dimensional planes flag the discrete kp hierarchy can thus be viewed as a container for an entire ensemble of vertex or darboux generated kp solutions it turns out that many new and old systems lead to such discrete semi infinite solutions like sequences of soliton solutions with more and more solitons sequences of calogero moser systems having more and more particles just to mention a few examples this is developed in 3 in this paper as another example we show that the q kp hierarchy maps via a kind of fourier transform into the discrete kp hierarchy enabling us to write down a very large class of solutions to the q kp hierarchy this was also reported in a brief note 4

quantitative and constructive aspects of the generalized koenig s and de montessus s theorems for pade approximants abstract the generalized koenig s theorem and de montessus s theorem are two classical results concerning the convergence of the rows of the pade table for meromorphic functions employing a technique that was recently developed for the analysis of vector extrapolation methods refined versions of these theorems are proved in the present work specifically complete expansions for the numerators and denominators of pade approximants are derived these expansions are then used to obtain 1 precise asymptotic rates of convergence of the poles of the pade approximants to the corresponding poles simple or multiple of the meromorphic function in question and 2 the precise asymptotic behavior of the error in the relevant pade approximants one important feature of the asymptotic results derived in this work is that these are expressed in terms of a very small number of parameters approximations of optimal accuracy to multiple poles and the principal parts of the corresponding laurent expansions are also constructed in addition the convergence problem for the case in which the only singularities on the circle of meromorphy are poles is solved completely through the solution of a nonlinear integer programming problem

asymptotic properties of zeros of orthogonal rational functions abstract a sequence x m m 0 1 2 of real numbers gives rise to an absolutely continuous measure n t on the unit circle given by d t n d 12 n 1 m 0 x m t m e im 2 where n is a natural number t 0 1 each measure n t determines a sequence n n t of monic orthogonal rational functions with prescribed poles outside the closed unit disk in particular these rational functions may be polynomials a measure 0 with support consisting of the points 1 n0 on the unit circle is given let n n0 under a suitable condition on weak convergence of the measures and with a proper ordering of the zeros zk n n t of n n t it can be shown that lim t 1 lim t z k n n t k k 1 n 0 and that there exists a positive number n z k n n t n for k n0 1 n for all n and all t 0 1 the theory is applied to the problem of determining unknown frequencies in a trigonometric signal x m ij iajeimwj m 0 1 2