moment conditions and support theorems for radon transforms on affine grassmann manifolds abstract let g p n and g q n be the affine grassmann manifolds of p and q planes in r n respectively and let r p q be the radon transform from smooth functions on g p n to smooth functions on g q n arising from the inclusion incidence relation when p q and dim g p n dim g p n we present a range characterization theorem for r p q via moment conditions we then use this range result to prove a support theorem for r p q this complements a previous range characterization theorem for r p q via differential equations when dim g p n dim g p n we also present a support theorem in this latter case

riesz potentials and integral geometry in the space of rectangular matrices abstract riesz potentials on the space of rectangular n m matrices arise in diverse higher rank problems of harmonic analysis representation theory and integral geometry in the rank one case m 1 they coincide with the classical operators of marcel riesz we develop new tools and obtain a number of new results for riesz potentials of functions of matrix argument the main topics are the fourier transform technique representation of riesz potentials by convolutions with a positive measure supported by submanifolds of matrices of rank m the behavior on smooth and l p functions the results are applied to investigation of radon transforms on the space of real rectangular matrices

nonlinear decoupling via feedback a differential geometric approach the paper deals with the nonlinear decoupling and noninteracting control problems a complete solution to those problems is made possible via a suitable nonlinear generalization of several powerful geometric concepts already introduced in studying linear multivariable control systems the paper also includes algorithms concerned with the actual construction of the appropriate control laws

inequalities between projection functions of convex bodies we obtain injectivity results for certain integral transforms on grassmann manifolds these are used to solve the intermediate cases of a problem of shephard and to give other applications to projection functions of convex bodies of revolution

asymptotic formulas for the dual radon transform and applications caracterisation de la transformation de radon duale d une fonction indefiniment differentiable formules asymptotiques et applications

even minkowski valuations a new integral representation of smooth translation invariant and rotation equivariant even minkowski valuations is established explicit formulas relating previously obtained descriptions of such valuations with the new more accessible one are also derived moreover the action of alesker s hard lefschetz operators on these minkowski valuations is explored in detail

range characterization of radon transforms on quaternionic projective spaces the characterization of the ranges of radon transforms is one of the most important subjects in integral geometry in fact for the radon transforms on euclidean spaces this subject has been studied from the several points of view since john s result 18 in which john showed that the range of the x ray radon transform on the 3 dimensional euclidean space r 3 is characterized as a kernel of some second order ultrahyperbolic differential operator see e g 1 2 4 5 10 13 14 15 in particular gonzalez 4 gave a simple characterization of the range for the radon transform on r moreover recently gonzalez obtained remarkable results on this subject 6

inversion of the radon transform on the laguerre hypergroup by using generalized wavelets abstract we consider the radon transform r 0 on the laguerre hypergroup k 0 r we characterize a space of infinitely differentiable and rapidly decreasing functions together with their derivatives such that r is a bijection from this space onto itself we establish an inversion formula and a plancherel theorem for the operator r finally by using the continuous wavelet transform on the laguerre hypergroup k we deduce another expression for the inverse r 1 of the operator r

invariant differential operators on grassmann manifolds for a lie group g and a closed subgroup h c g let d g h denote the algebra of differential operators d on the manifold g h which are invariant under the action of g in this note we determine this algebra in the case when g is the group m n of rigid motions of the euclidean space r and h is the subgroup of m n leaving a certain p dimensional plane in r invariant here 0 6 p 6 ii 1 and is the space

range characterization of the cosine transform on higher grassmannians abstract we characterize the range of the cosine transform on real grassmannians in terms of the decomposition under the action of the special orthogonal group so n we also give a geometric interpretation of this image in terms of valuations in addition we discuss the non archimedean analogues

harmonic analysis on grassmannian bundles the harmonic analysis of the grassmannian bundle of k dimensional affine subspaces of rn as a homogeneous space of the euclidean motion group is given explicitly this is used to obtain the diagonalization of various generalizations of the radon transform between such bundles in abstract form the same technique gives the plancherel formula for any unitary representation of a semidirect product g x v v a normal abelian subgroup induced from an irreducible unitary representation of a subgroup of the form h x w introduction let gn k denote the grassmannian manifold of linear k dimensional subspaces of rn and let pn k denote the grassmannian bundle of affine kdimensional subspaces also called k planes so pn k is a bundle over gn k with fibre dimension m where m k n in group theoretic terms gn k g h where g so n and h s o k x 0 m or we could just as well take g 0 n and h o k x 0 m let p denote the standard representation of g on v rn and form the semidirect product g x v the euclidean motion group with group law gi xi o g x gig xi p gl x the eucidean motion group acts naturally on the bundle pn k if we let wr0 denote the subspace spanned by the first k basis elements el ek then the isotropy subgroup of wro is h x w where w is the subspace 7r0 thus pn k g v h x w in this paper we obtain the explicit harmonic analysis on pn k the decomposition of l2 pn k with respect to the invariant measure into a direct integral of irreducible representations of g x v this is accomplished in two stages in the first abstract stage 2 we let g be any compact lie group and p any real representation of g on v and for functions f g x invariant under the right action of a subgroup h x w we obtain the fourier decomposition formula f g x j e fa 6 g x dv a a6g here a is an index set for the orbits in w under the action of h if we choose a point za in each orbit and let gc and ha denote the isotropy subgroup of za in g and h respectively then a x denotes the set of irreducible representations of gca that occur in l2 gca ha the measure dv a is determined by dz da dv a where dz is lebesgue measure on w and da is the invariant measure on the orbit received by the editors march 7 1985 1980 mathematics subject classification primary 43a85 secondary 22d30 44a15

radon inversion on grassmannians via garding gindikin fractional integrals we study the radon transform rf of functions on stiefel and grassmann manifolds we establish a connection between rf and garding gindikin frac tional integrals associated to the cone of positive definite matrices by using this connection we obtain abel type representations and explicit inversion for mulae for rf and the corresponding dual radon transform we work with the space of continuous functions and also with l p spaces

dual radon transforms on affine grassmann manifolds fix 0 p n we will show that the range of this transform is given by smooth functions on g p n annihilated by a system of pfaffian type differential operators we also study aspects of the exceptional case p q n

unified parallel lattice structures for time recursive discrete cosine sine hartley transforms unified efficient computations of the discrete cosine transform dct discrete sine transform dst discrete hartley transform dht and their inverse transforms employing the time recursive approach are considered unified parallel lattice structures that can dually generate the dct and dst simultaneously as well the dht are developed these structures can obtain the transformed data for sequential input time recursively with a throughput rate of one per clock cycle the total number of multipliers required is a linear function of the transform size n with no constraint on n the resulting architectures are regular modular and without global communication so that they are very suitable for vlsi implementation for high speed applications it is shown that the dct dst dht and their inverse transforms share an almost identical lattice structure the tradeoff between time and area for the block data processing is considered

new inversion formulas for radon transforms on affine grassmannians abstract we obtain new inversion formulas for the radon transform and the corresponding dual transform acting on affine grassmann manifolds of planes in r n the consideration is performed in full generality on continuous functions and functions belonging to l p spaces

on dynamic input output linearization of discrete time nonlinear systems this paper studies the problem of linearizing the input output map of an analytic discrete time nonlinear system locally around a given trajectory necessary and sufficient conditions are given for the existence of a regular dynamic state feedback control law under which the input dependent part of the response of a nonlinear system becomes linear in the input and independent of the initial state the proposed conditions are less restrictive than those obtained by lee and marcus for linearizing the input output map via a static state feedback instrumental in the problem solution is the inversion structure algorithm for a discrete time nonlinear system firstly the solvability conditions are expressed in terms of the inversion algorithm secondly the proof of the existence and construction of the dynamic state feedback compensator relies on this algorithm

integral geometry on grassmann manifolds and calculus of invariant differential operators abstract in this paper we mainly deal with two problems in integral geometry the range characterizations and construction of inversion formulas for radon transforms on higher rank grassmann manifolds the results will be described explicitly in terms of invariant differential operators we will also study the harmonic analysis on grassmann manifolds using the method of integral geometry in particular we will give eigenvalue formulas and radial part formulas for invariant differential operators

radon transform on symmetric matrix domains let k r c h be the field of real complex or quaternionic numbers and mp q k the vector space of all p q matrices let x be the matrix unit ball in mn r r k consisting of contractive matrices as a symmetric space x g k o n r r o n r o r u n r r u n r u r and respectively sp n r r sp n r sp r the matrix unit ball y0 in mr r r with r n 1 is a totally geodesic submanifold of x and let y be the set of all g translations of the submanifold y0 the set y is then a manifold and an affine symmetric space we consider the radon transform rf y for functions f c 0 x defined by integration of f over the subset y and the dual transform rt f x x x for functions f y on y for 2r n 2r r with a certain evenness condition in the case k r we find a g invariant differential operator m and prove it is the right inverse of rtr rtr mf cf for f c 0 x c 0 the operator f rtr f is an integration of f against a singular function determined by the root systems of x and y0 we study the analytic continuation of the powers of the function and we find a bernstein sato type formula generalizing earlier work of the author in the set up of the berezin transform when x is a rank one domain of hyperbolic balls in kn 1 and y0 is the hyperbolic ball in kr 1 1 r n we obtain an inversion formula for the radon transform namely mrtr f cf this generalizes earlier results of helgason for non compact rank one symmetric spaces for the case r n 1

decoupling theory of general multivariable analytic non linear systems in this paper the decoupling problem of the general muitivariable analytic nonlinear system of the form xdot x u y h x u is discussed where the state space is a real analytic manifold sufficient and necessary conditions for decoupling by state feedback combined with compensation and state feedback only are represented a sufficient and necessary condition for stable decoupling is also derived for decouplable systems the synthesis procedure is represented

processes of flats induced by higher dimensional processes stationary processes of k flats in mathbb e d can be thought of as point processes on the grassmannian mathcal l kd of k dimensional subspaces of mathbb e d if such a process is sampled by a d k j dimensional space f it induces a process of j flats in f in this work we will investigate the possibility of determining the original k process from knowledge of the intensity measures of the induced j processes we will see that this is impossible precisely when 1 k d 1 and j 0 2 r 2 1 where r is the rank of the manifold mathcal l kd we will show how the problem is equivalent to the study of the kernel of various integral transforms these will then be investigated using harmonic analysis on grassmannian manifolds

kinematic formulas on the quaternionic plane we introduce different bases for the vector space of sp 2 sp 1 invariant translation invariant continuous valuations on the quaternionic plane and determine a complete set of kinematic formulas

characterization of allowable perturbations for robust decoupling of affine non linear systems for affine non linear systems we study the effects of uncertainties in systems vector fields on decoupled linear input output behaviour a characterization of an allowable class of perturbations which preserve the decoupling properties of the nominal system is given an immediate consequence of this characterization indicates that decoupling position control of rigid robotic manipulators is not robust under any perturbations in robot dynamics the result is applied to simple hamiltonian control systems to characterize the allowable perturbation of total energy function

null spaces and ranges for the classical and spherical radon transforms abstract let r be the classical radon transform that integrates a function over hyperplanes in r n and let sm be the transform that integrates a function over spheres containing the origin in r n we prove continuity results for both transforms and explicitly give the null space of r for a class of square integrable functions on the exterior of a ball in r n as well as the null space of sm for square integrable functions on a ball we show sm l 2 r n l 2 r n is one one and we characterize the range of sm on classes of smooth functions and square integrable functions by certain moment conditions if g x is a schwartz function on r n that is zero to infinite order at x 0 we prove moment conditions sufficient for g to be in the range of sm c r n we apply our results on sm to existence and uniqueness theorems for solutions to a characteristic initial value problem for the darboux partial differential equation

radon transforms on grassmann manifolds if n p q 1 the affine grassmann manifolds g p n of p planes and g q n of q planes in rn are dual homogeneous spaces of the euclidean motion group m n with g p n and g q n being incident if and only if meets at a right angle the resulting integral transform from functions on g p n to functions on g q n generalizes both the classical radon transform and its dual we show that this transform intertwines the actions of certain m n invariant differential operators on g p n and g q n and we prove an inversion formula for this transform when n is odd generalizing the radon inversion formula and obtaining in particular an inversion formula for the dual transform

radon transforms on higher grassmannians on definit une transformee de radon r a partir des fonctions cr k n les grassmanniens des k plans projectifs dans cp n aux fonctions sur gr l n si f c gr k n et l gr l n alors rf l est l integrale de f h sur tous les k plans h qui sont dans l si r t est la transformee duale on montre sous de bonnes hypotheses sur k et l que r t r est inversible par un polynome dans les operateurs de casimir de u n 1 le groupe des isometries de cp n

flatness and defect of non linear systems introductory theory and examples we introduce flat systems which are equivalent to linear ones via a special type of feedback called endogenous their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of kalman s controllability the distance to flatness is measured by a non negative integer the defect we utilize differential algebra where flatness and defect are best defined without distinguishing between input state output and other variables many realistic classes of examples are flat we treat two popular ones the crane and the car with n trailers the motion planning of which is obtained via elementary properties of plane curves the three non flat examples the simple double and variable length pendulums are borrowed from non linear physics a high frequency control strategy is proposed such that the averaged systems become flat

projection functions area measures and the alesker fourier transform abstract dual to koldobsky s notion of j intersection bodies the class of j projection bodies is introduced generalizing minkowski s classical notion of projection bodies of convex bodies a fourier analytic characterization of j projection bodies in terms of their area measures of order j is obtained in turn this yields an equivalent characterization of j projection bodies involving alesker s fourier type transform on translation invariant smooth spherical valuations as applications of these results several basic properties of j projection bodies are established and new non trivial examples are constructed

even valuations on convex bodies the notion of eveil valuation is introduced as a natural generalization of volume on cornpact convex subsets of euclidean space a recent characterization theorem for volume leads in turn to a connection between even valuations on compact convex sets and continuous functions on grassmannians this connection can be described in part using generating distributions for symmetric compact convex sets we also explore some consequences of these characterization results in convex and integral geometry recent interest in volume as a valuation on compact conlvex sets stems from hilbert s third problem which is actually an ancient problem recast in modern terms hilbert asked if two polytopes p and q can be each cut into a finite number of pieces p pn and q qm with each pi congruent to qi by a rigid euclidean motion provided that p and q have the same volume 18 this question was answered in the negative by max dehn 2 4 29 who found a functionlal on polytopes that is invariant under dissections over rigid motions while varying in value among polytopes of equal volume in other words the dehn invariant is a simple rigid motion invariant valuation on polytopes that is not equal to volume under any normalization dehn s solution left open the question of exactly what conditions oil p and q imply equidissectability over the group of rigid motions although this problem was solved by hadwiger in the case where only translations and no rotations nor reflections are permitted see 2 16 17 26 29 in the course of studying this and related problems hadwiger discovered a characterization of euclidean volume as a continuous rigid motion invariant simple valuation on compact convex sets that is a continuous rigid motion invariant valuation that vanishes on convex sets of less than full dimension this result led in turn to a complete characterization of all continuous rigid motion inivariant valuations on compact convex sets in 1r as consisting of a real n 4 dimensional vector space spanned by the intrinsic volumes or quermassintegrals 16 also 20 21 31 since many standard functionals and integral operators can be interpreted as invariant valuations such as intrinsic volumes mean projections crofton and kinematic formulas what came to be known as hadwiger s chtracterization theorem proved to be a valuable tool for generating quick and effortless proofs of many formulas and equations in integral geometry unfortunately hadwiger s original proof was long and difficult 161 while seeking a shorter proof of hadwiger s volume characterization the author discovered received by the editors june 24 1996 and in revised form september 29 1997 1991 mathematics subject classification primary 52a22 52a38 52a39 52b45 research supported in part by nsf grants dms 9022140 to msri and dms 9626688 to the author 1999 american mathematical society 71 this content downloaded from 207 46 13 52 on mon 24 oct 2016 04 15 57 utc all use subject to http about jstor org terms

on the shift property of dct s and dst s the relationship between the family of dct s and dst s of original sequence and shifted sequences is developed while these properties are not as simple as the case for the dft they are still useful for processing long streams of data sequences where time varying filtering is required

efficient computation of local geometric moments local moments have attracted attention as local features in applications such as edge detection and texture segmentation the main reason for this is that they are inherently integral based features so that their use reduces the effect of uncorrelated noise the computation of local moments when viewed as a neighborhood operation can be interpreted as a convolution of the image with a set of masks nevertheless moments computed inside overlapping windows are not independent and convolution does not take this fact into account by introducing a matrix formulation and the concept of accumulation moments this paper presents an algorithm which is computationally much more efficient than convolving and yet as simple

radon transform on real complex and quaternionic grassmannians letgn k k be the grassmannian manifold of k dimensional k subspaces in k where k r c h is the field of real complex or quaternionic numbers for 1 k k n 1 we define the radon transform rf gn k k for functions f on gn k k as an integration over all when k k n we give an inversion formula in terms of the garding gindikin fractional integration and the cayley type differential operator on the symmetric cone of positive k k matrices over k this generalizes the recent results of grinberg rubin for real grassmannians

norm estimates for k plane transforms and geometric inequalities abstract the article is devoted to remarkable interrelation between the norm estimates for k plane transforms in weighted and unweighted l p spaces and geometric integral inequalities for cross sections of measurable sets in r n a similar interrelation is studied for more general j plane to k plane transforms on affine grassmannians and their compact modifications the article contains a series of new integral geometric inequalities with sharp constants explicit equalities conjectures and open problems

on the range of the radon d plane transform and its dual we present direct group theoretic proofs of the range theorem for the radon d plane transform on r the original proof by richter involves extensive use of local coordinate calculations on g d n the grassmann manifold of affine a planes in r we show that moment conditions are not sufficient to describe this range when d n in contrast to the compactly supported case finally we show that the dual rf plane transform maps g g d n surjectively onto f r

analytic derivation of the finite wordlength effect of the twiddle factors in recursive implementation of the sliding dft analytic derivation of the erroneous effect is presented for the sliding dft which is implemented in a recursive way with the finite bit approximation of the twiddle factors the derivation is based on the statistical exploration of the error dynamic equation that describes the error propagation of the recursion the analysis result is obtained in a closed form equation of the noise to signal power ratio employing the zero mean white gaussian signal as the target input and is verified with data obtained from computer simulation

radon transforms on affine grassmannians we develop an analytic approach to the radon transform f f f where f is a flinction on the affine grassmann manifold g n k of k dimensional planes in r n and is a k dimensional plane in the similar manifold g n k k k for f e l p g n k we prove that this transform is finite almost everywhere on g n k if and only if 1 n 1 and interpreted as a direct quasi orthogonal radon transform for another pair of affine grassmannians as a consequence we obtain that the radon transform and the dual radon transform are injective simultaneously if and only if k k n 1 the investigation is carried out for locally integrable and continuous functions satisfying natural weak conditions at infinity

a cordic based unified systolic architecture for sliding window applications of discrete transforms a cordic based unified systolic architecture for sliding window applications of the discrete fourier transform dft the discrete hartley transform dht the discrete cosine transform dct and the discrete sine transform dst is proposed compared to earlier works the proposed scheme offers significant reduction in hardware particularly for dht for an n point dht it requires only n 2 1 processing elements each consisting of one cordic processor and two adders

the gaussian limit for high dimensional spherical means abstract we show that the limit of integrals along slices of a high dimensional sphere is a gaussian integral on a corresponding finite codimension affine subspace in infinite dimensions