sobolev spaces on lie groups embedding theorems and algebra properties abstract let g be a noncompact connected lie group denote with a right haar measure and choose a family of linearly independent left invariant vector fields x on g satisfying hormander s condition let be a positive character of g and consider the measure whose density with respect to is in this paper we introduce sobolev spaces l p adapted to x and 1 p 0 and study embedding theorems and algebra properties of these spaces as an application we prove local well posedness and regularity results of solutions of some nonlinear heat and schrodinger equations on the group

on metric dimensions of discrete time systems in this paper metric dimensions of a class of discrete time systems are derived the n width and e dimension of the class in l1 kernel norm are obtained and a corresponding minimal e net is constructed by time domain truncation of system kernels asymptotic estimates of the n width and e dimension of the class in h norm are established

a note on metric dimension and feedback in discrete time the notions of a in dimension and in entropy were introduced into feedback theory of g zames 1976 1979 who proposed them as measures of the complexity of identification and the effectiveness of feedback in reducing identification costs the 1979 paper was concerned mainly with continuous time systems but also included a summary of discrete time results which contained certain errata the authors elaborate the discrete time case supply some omitted proofs and correct the errata

estimates for functions of the laplace operator on homogeneous trees in this paper we study the heat equation on a homogeneous graph relative to the natural nearest neighbour laplacian we find pointwise estimates for the heat and resolvent kernels and the lp lq mapping properties of the corresponding operators in recent years random walks on graphs have been studied as analogues of diffusions on manifolds see e g t coulhon cn m kanai k1 k2 and n th varopoulos v1 v2 more recently i chavel and e a feldman cf and m m h pang p considered the semigroup e t 0 in this paper we consider the heat semigroup associated to the canonical laplace operator on a homogeneous tree which is perhaps the basic example of a graph where the cardinality of the ball of radius r grows exponentially in r we study the properties of the heat and resolvent operators and discover very close analogies with diffusions on hyperbolic spaces in line with the philosophy of the above mentioned authors a homogeneous tree x of degree q 1 is defined to be a connected graph with no loops in which every vertex is adjacent to q 1 other vertices we denote by d the natural distance on x d x y being the number of edges between the vertices x and y and by l x the lebesgue space with respect to counting measure on x on x the canonical laplace operator l is defined thus lf x f x 1 q 1 y x d x y 1 f y x x it is easily seen to be bounded on l x for every p in 1 and self adjoint on l x denote by 2 l the l x spectrum of l and by p the associated spectral resolution of the identity such that lf 2 l dp f f l x and denote inf 2 l by b2 for in 0 1 and with re 0 the heat operator and the resolvent of l are the operators defined by the formulae htf 2 l e t dp f t r received by the editors october 4 1996 1991 mathematics subject classification primary 43a85 secondary 20e08 43a90 22e35

bellman functions and dimension free lp estimates for the riesz transforms in bessel settings abstract in this article we prove dimension free l p boundedness of riesz transforms associated with a bessel differential operator we obtain explicit estimates of the l p norms for the bessel riesz transforms in terms of p establishing a linear behavior with respect to p we use the bellman function technique to prove a bilinear dimension free inequality involving poisson semigroups defined through this bessel operator

the complexity of model classes and smoothing noisy data abstract we consider the problem of smoothing a sequence of noisy observations using a fixed class of models via a deterministic analysis we obtain necessary and sufficient conditions on the noise sequence and model class that ensure that a class of natural estimators gives near optimal smoothing in the case of i i d random noise we show that the accuracy of these estimators depends on a measure of complexity of the model class involving covering numbers our formulation and results are quite general and are related to a number of problems in learning prediction and estimation as a special case we consider an application to output smoothing for certain classes of linear and nonlinear systems the performance of output smoothing is given in terms of natural complexity parameters of the model class such as bounds on the order of linear systems the l1 norm of the impulse response of stable linear systems or the memory of a lipschitz nonlinear system satisfying a fading memory condition

on the rate of rational approximation to functions with a finite number of poles abstract the rate of the best rational approximation and interpolation of a meromorphic function on a compact faber set is described in terms of its growth convergence along the rows as well as the columns of the walsh array is considered emphasis is on precise o estimates instead of limit relations

on the littlewood paley stein g function we consider semigroups tt which are contractive on lp m for all p e q q and q e 1 2 we give an example on symmetric spaces of the noncompact type which shows that the littlewood paley stein g function associated to the infinitesimal generator of tt may be unbounded on lq m and on l m we prove that variants of the g function are bounded on these lebesgue spaces in this paper we present some new results concerning the lp boundedness of the littlewood paley stein g function suppose that tt is a co semigroup of operators on lp m where m is a a finite measure space and let a e n the littlewood paley stein function ga associated to the given semigroup is defined by the rule ga f 1 ta ttfi2 dt for all f for which the right hand side makes sense in classical cases where tt is the poisson semigroup acting on lp r or lp t the g function is an important tool in fourier analysis for instance it is frequently used to prove multiplier theorems and pointwise convergence results the reader is referred to 1 1 for a survey of its role in a general context the g function was considered by e m stein he proved that if tt is a symmetric diffusion semigroup then ga satisfies the estimate 1 aplifiip liga f iip bp fllp vf e lp m whenever p e 1 oo a proof of this result can be found in 10 a simpler and more general approach via transference techniques is presented in 2 a third proof is in 4 and 5 it is of an entirely different nature in that it is based on the fact that generators of symmetric diffusion semigroups and even more general semigroups 3 have an hx functional calculus one of the interesting applications of g functions is to functional calculus for infinitesimal generators of co semigroups stein proved a marcinkiewicz type result for those as a corollary of the theory he developed for the g function the relationship between g functions and functional calculus for the infinitesimal generator of tt was investigated in 4 and 5 our treatment here owes received by the editors march 7 1994 originally communiated to the proceedings of the ams by j marshall ash 1991 mathematics subject classification primary 42b25 47a60 secondary 43a85

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

on the faber transform and efficient numerical rational approximation algorithms are presented for numerical type m n rational approximation m geqq n 1 which can be effectively applied either in general simply connected regions of the complex plane or as a special case on intervals of the real line in the former case particularly the methods are much more efficient than those previously proposed but they are also competitive even for intervals the approximations are based on the observation that the faber transform of a rational function is itself rational and are of two types a class of pade like approximants and approximations of rational caratheodory fejer type based on the singular value decomposition of a hankel matrix of faber coefficients which generalise those recently introduced for the unit disc by trefethen numer math 37 1981 pp 297 320 these latter particularly give rise to approximations which are sufficiently near to best for practical purposes

on the consequences of a mihlin hormander functional calculus maximal and square function estimates we prove that the existence of a mihlin hormander functional calculus for an operator l implies the boundedness on l p of both the maximal multiplier operators and the continuous square functions build on spectral multipliers of l the considered multiplier functions are finitely smooth and satisfy an integral condition at infinity in particular multipliers of compact support are admitted

control oriented system identification a worst case deterministic approach in h sub infinity the authors formulate and solve two related control oriented system identification problems for stable linear shift invariant distributed parameter plants in each of these problems the assumed a priori information is minimal consisting only of a lower bound on the relative stability of the plant an upper bound on a certain gain associated with the plant and an upper bound on the noise level the first of these problems involves identification of a point sample of the plant frequency response from a noisy finite output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest this problem leads naturally to the second problem which involves identification of the plant transfer function in h sub infinity from a finite number of noisy point samples of the plant frequency response concrete plans for identification algorithms are provided for each of these two problems

sobolev algebras on lie groups and riemannian manifolds we prove that on any connected unimodular lie group g the space l p g l g where l p g is the sobolev space of order 0 associated with a sublaplacian is an algebra under pointwise product this generalizes results due to strichartz in the euclidean case to bohnke in the case of stratified groups and others a global version of this fact holds for groups with polynomial growth we give similar results for riemannian manifolds with ricci curvature bounded from below respectively nonnegative

identification and adaptive control towards a complexity based general theory abstract two recent developments will be surveyed here which are pointing the way towards an input output theory of h l 1 adaptive feedback the solution of problems involving 1 feedback performance exact optimization under large plant uncertainty on the one hand the two disc problem of h and 2 optimally fast identification in h on the other taken together these are yielding adaptive algorithms for slowly varying data in h l 1 at a conceptual level these results motivate a general input output theory linking identification adaptation and control learning in such a theory the definition of adaptation is based on system performance under uncertainty and is independent of internal structure presence or absence of variable parameters or even feedback

how much uncertainty can be dealt with by feedback feedback is used primarily for reducing the effects of the plant uncertainty on the performance of control systems and as such understanding the following questions is of fundamental importance how much uncertainty can be dealt with by feedback what are the limitations of feedback how does the feedback performance depend quantitatively on the system uncertainty how can the capability of feedback be enhanced if a priori information about the system structure is available as a starting point toward answering these questions a typical class of first order discrete time dynamical control systems with both unknown nonlinear structure and unknown disturbances is selected for our investigation and some concrete answers are obtained in the paper in particular we find that in the space of unknown nonlinear functions the generalized lipschitz norm is a suitable measure for characterizing the size of the structure uncertainty and that the maximum uncertainty that can be dealt with by the feedback mechanism is described by a ball with radius 3 2 spl radic 2 in this normed function space

harmonic analysis on semigroups wherever t 0 and 1 p so e m stein st 3 developed a littlewoodpaley theory for such semigroups with some additional hypotheses r r coifman r rochberg and g weiss crw pointed out that some of stein s estimates could be obtained more simply by the use of transference techniques which were developed by coifman and weiss cw our object is to push these transference methods to present an alternative and simpler approach to some of the principal results of stein s book we shall also use these results to prove estimates for the wave equation

besov and triebel lizorkin spaces on lie groups in this paper we develop a theory of besov and triebel lizorkin spaces on general noncompact connected lie groups endowed with a sub riemannian structure such spaces are defined by means of hypoelliptic sub laplacians with drift and endowed with a measure whose density with respect to a right haar measure is a continuous positive character of the group we prove several equivalent characterizations of their norms we establish comparison results also involving sobolev spaces of recent introduction and investigate their complex interpolation and algebra properties