functions with bounded spectrum let 0 o has the same behavior as the sequence sup q i 0 in other words if we know all far points of supp ff we can wholly describe this behavior without any concrete calculation of z a p a 0 a paley wiener schwartz theorem for a nonconvex case which is a consequence of the result is given

the ring structure on the cohomology of coordinate subspace arrangements every simplicial complex delta subset2 n on the vertex set n 1 ldots n defines a real resp complex arrangement of coordinate subspaces in mathbb r n resp mathbb c n via the correspondence delta ni sigma mapsto rm span e i i in sigma the linear structure of the cohomology of the complement of such an arrangement is explicitly given in terms of the combinatorics of delta and its links by the goresky macpherson formula here we derive by combinatorial means the ring structure on the integral cohomology in terms of data of delta we provide a non trivial example of different cohomology rings in the real and complex case furthermore we give an example of a coordinate arrangement that yields non trivial multiplication of torsion elements

a sampling theorem on homogeneous manifolds we consider a generalization of entire functions of spherical exponential type and lagrangian splines on manifolds an analog of the paleywiener theorem is given we also show that every spectral entire functionl on a manifold is uniquely determined by its values on some discrete sets of points the main result of the paper is a formula for reconstruction of spectral entire functions from their values on discrete sets using lagrangian splines

real paley wiener theorems and roe s theorem associated with the opdam cherednik transform abstract we reconsider roe s theorem associated with jacobi cherednik operators we first prove two new real paley wiener theorems for the opdam cherednik transform characterizing those functions whose transform has support inside or outside intervals as a corollary we get a characterization of those functions whose transform has support at the endpoints which we use to prove our version of roe s theorem

on the gegenbauer transformation the gegenbauer transformation g k is defined for 1 2 k 0 1 2 by where if being the gegenbauer polynomial of index and degree k and l 0 k is the tchebichef polynomial of degree k the transformation is studied on the spaces l p denned by the norm and its boundedness and range on these spaces is determined and inversion formulae are found

on real paley wiener theorems for certain integral transforms we prove real paley wiener theorems for the inverse jacobi transform characterising the space of l2 functions whose image under the jacobi transform are smooth functions with compact support

paley wiener theorems for the dunkl transform we conjecture a geometrical form of the paley wiener theorem for the dunkl transform and prove three instances thereof by using a reduction to the one dimensional even case shift operators and a limit transition from opdam s results for the graded hecke algebra respectively these paley wiener theorems are used to extend dunkl s intertwining operator to arbitrary smooth functions furthermore the connection between dunkl operators and the cartan mo tion group is established it is shown how the algebra of radial parts of in variant differential operators can be described explicitly in terms of dunkl operators this description implies that the generalized bessel functions coin cide with the spherical functions in this context of the cartan motion group the restriction of dunkl s intertwining operator to the invariants can be inter preted in terms of the abel transform we also show that for certain values of the multiplicities of the restricted roots the abel transform is essentially inverted by a differential operator 1 introduction and overview in recent times the study of special functions associated with root systems has developed to a considerable degree starting with a number of conjectures by mac donald and the work of heckman and opdam on multivariable hypergeometric functions in the late 1980 s the development of the theory was greatly enhanced by the introduction of rational dunkl operators by dunkl 6 through various intermediate steps of generalization these operators can even be said to have ulti mately provided crucial building blocks for cherednik s work on double affine hecke algebras and the q macdonald conjectures originally before the introduction of dunkl operators the idea when studying special functions related to root systems was to consider root multiplicities in the theory of spherical functions on lie groups as parameters and then to develop a theory for weyl group invariant objects in this more general situation without the aid of the presence of the group it was this point of view which underlay the macdonald conjectures and which led heckman and opdam to the development of their theory of hypergeometric functions in higher dimension one of the main technical problems in this context is the description of the generalized radial parts of invariant differential operators apart from an explicit formula for the generalized radial part of the laplacian an expression which was in fact the starting point for heckman and opdam the other operators remain somewhat intangible

directed compact percolation cluster size and hyperscaling exact recurrence relations are obtained for the length and size distributions of compact directed percolation clusters on the square lattice the corresponding relation for the moment generating function of the length distribution is obtained in closed form whereas in the case of the size distribution only the first three moments are obtained the work is carried out for clusters grown from a seed of arbitrary width on an anisotropic lattice a duality property is shown to exist which relates the moment generating functions on the two sides of the critical curve the moments of both distributions have critical exponents which satisfy a constant gap hypothesis with gap exponents nu 2 and delta 3 corresponding to the scaling length and scaling size respectively the usual hyperscaling relation for directed percolation is found to be invalid for compact clusters and is replaced by delta nu d nu perpendicular to where nu perpendicular to is the exponent corresponding to the cluster width and d is the number of transverse dimensions 1 for the square lattice

basic sturm liouville problems this paper is devoted to studying a q analogue of sturm liouville eigenvalue problems we formulate a self adjoint q difference operator in a hilbert space some of the properties of the eigenvalues and the eigenfunctions are discussed green s function is constructed and the problem in question is inverted into a q type fredholm integral operator with a symmetric kernel the set of eigenfunctions is shown to be a complete orthogonal set in the hilbert space examples involving basic trigonometric functions are involved

the zeros of basic bessel functions the functions jv ax x and associated orthogonal polynomials abstract the modified lommel polynomials are generalized and their orthogonality relation is obtained as a byproduct we prove that the non zero roots of jv ax x are real simple and interlace with those of jv ax 1 x the q lommel polynomials are shown to play in the theory of basic bessel function the role played by the lommel polynomials in the bessel function theory the q lommel polynomials are proved to be orthogonal with respect to a purely discrete measure with bounded support this is then used to prove that the prositive zeros of x j 2 v x q are real simple and interlace with the zeros of x 1j 2 v 1 x q when 1 we also establish the complete monotonicity of 1x 1 2 j v 1 2 i x q j v 2 i x q

real paley wiener theorems and local spectral radius formulas we systematically develop real paley wiener theory for the fourier transform on d for schwartz functions l p functions and distributions in an elementary treatment based on the inversion theorem as an application we show how versions of classical paley wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use for other transforms of fourier type as well an explanation is also given as to why the easily applied classical paley wiener theorems are unlikely to be able to yield information about the support of a function or distribution which is more precise than giving its convex hull whereas real paley wiener theorems can be used to reconstruct the support precisely albeit at the cost of combinatorial complexity we indicate a possible application of real paley wiener theory to partial differential equations in this vein and furthermore we give evidence that a number of real paley wiener results can be expected to have an interpretation as local spectral radius formulas a comprehensive overview of the literature on real paley wiener theory is included

on a q paley wiener theorem abstract after the analysis of the q even translation and the q cosine fourier transform by a fitouhi and f bouzeffour in q cosine fourier transform and q heat equation ramanujan j in press it is natural to look for the q analogue of some well known classical theorems in this paper we purpose to give a q version of the paley wiener theorem related to this fourier transform using standard methods of the q calculus

paley wiener subspace of vectors in a hilbert space with applications to integral transforms article history the goal of this article is to introduce an analogue of the paley wiener space of bandlimited functions pw in hilbert spaces and then apply the general result to more specific examples guided by the role that the differentiation operator plays in some of the characterizations of the paley wiener space we construct a space of vectors using a self adjoint operator d in a hilbert space h and denote this space by pw d t he article can be virtually divided into two parts in the first part we show that the space pw d has similar properties to those of the space pw including an analogue of the bernstein inequality and the riesz interpolation formula we also develop a new characterization of the abstract paley wiener space in terms of solutions of cauchy problems associated with abstract schrodinger equations finally we prove two sampling theorems for vectors in pw d one of which uses the notion of hilbert frames and the other is based on the notion of variational splines in h in the second part of the paper we apply our abstract results to integral transforms associated with singular sturm liouville problems in particular we obtain two new sampling formulas related to one dimensional schrodinger

on the inversion of integral transforms associated with sturm liouville problems consider the sturm liouville boundary value problem 1 1 y q x y t2y a x b 2 2 y a cos y a sin 0 3 3 y b cos y b sin 0 where q x is continuous on a b let x t be a solution of either the initial value problem 1 and 2 or 1 and 3 in this paper we develop two techniques to invert the integral f t abf x x t dx where f x l2 a b one technique is based on the construction of some biorthogonal sequence of functions and the other is based on poisson s summation formula

linear decision trees subspace arrangements and mobius functions let p be a set in rn given a point x we want to test if x e p our model of computation is a linear decision tree a rooted ternary tree t where with each interior node v is associated a linear function lv x l j ajxj b and the three edges connecting v to its descendants are labelled starting from the root we move down the tree at each internal node v we check whether iv x 0 and follow the appropriately labelled edge leaves are labelled yes and no and arriving at a leaf we read off the answer to the question is x e p linear decision trees are sometimes surprisingly powerful devices we mention here the result that some np complete problems like the knapsack problem have polynomial depth linear decision trees mh lower bound results on the size or depth of linear decision trees usually depend on counting the number of connected components of p or r n p this method was introduced by dobkin and lipton dl and further developed by steele and yao sy ben or bo and others in this paper we describe topological methods that can be applied to obtain lower bounds also if these sets are connected the basic idea of this paper was first explored in bly the euler characteristic of p or r n p was used to estimate the number of leaves in a linear decision tree for p here we improve and extend the topological methods of bly in particular we show that the euler characteristic can be replaced by the sum of the betti numbers of r n p in bl y the method was applied to the case when p is a subspace arrangement i e a union of a finite number of affine subspaces the euler characteristic of the union is essentially the mobius function of the intersection lattice of these subspaces in this paper using a formula of goresky and macpherson gm we work with the betti numbers of the complement of this union and thereby obtain better bounds the motivating problem in bly was the k equal problem given n real numbers xl xn decide if some k of them are equal it was shown that ev ry linear decision tree for the k equal problem has size at least 2njk 0 n and consequently depth at least q nlog 2njk which is best possible we

a generalized weber transform and its inverse formula a generalized weber transform is introduced and its inverse formula is proved a well with constant output production in an infinite fractal reservoir of fluid mechanics in porous medium has been considered as an application of this generalized weber transform

directed compact percolation near a wall iii exact results for the mean length and number of contacts existing exact results for the percolation probability and mean cluster size for compact percolation near a dry wall are extended to the mean cluster length and the mean number of contacts with the wall the results are derived from our previous work on vesicles near an attractive wall and involve elliptic integrals as opposed to the simple rational forms found for the percolation probability and cluster size below the results for the cluster length satisfy previously conjectured differential equations a closed expression is conjectured for the mean size above in terms of a hypergeometric function

on q analogues of the fourier and hankel transforms for h exton s q analogue of the bessel function going back to w hahn in a special case but different from f h jackson s q bessel functions we derive hansen lommel type orthogonality relations which by a symmetry turn out to be equivalent to orthogonality relations which are q analogues of the hankel integral transform pair these results are implicit in the context of quantum groups in a paper by vaksman and korogodskii as a specialization we get q cosines and q sines which admit q analogues of the fourier cosine and fourier sine transforms

paley wiener type theorems for a class of integral transforms abstract a characterization of weighted l 2 i spaces in terms of their images under various integral transformations is derived where i is an interval finite or infinite this characterization is then used to derive paley wiener type theorems for these spaces unlike the classical paley wiener theorem our theorems use real variable techniques and do not require analytic continuation to the complex plane the class of integral transformations considered is related to singular sturm liouville boundary value problems on a half line and on the whole line

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

critical exponents from nonlinear functional equations for partially directed cluster models we present a method for the derivation of the generating function and computation of critical exponents for several cluster models staircase bar graph and directed column convex polygons as well as partially directed self avoiding walks starting with nonlinear functional equations for the generating function by linearizing these equations we first give a derivation of the generating functions the nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz alternatively taking the continuum limit leads to nonlinear differential equations from which one can extract the scaling function we find that all the above models are in the same universality class with exponents y 1 2 7 1 3 and b 2 3 all models have as their scaling function the logarithmic derivative of the airy function

mean and almost everywhere convergence of fourier neumann series abstract let j denote the bessel function of order the functions x 2 2 1 2j 2n 1 x1 2 n 0 1 2 form an orthogonal system in l2 0 x dx when 1 in this paper we analyze the range of p and for which the fourier series with respect to this system converges in the lp 0 x dx norm also we describe the space in which the span of the system is dense and we show some of its properties finally we study the almost everywhere convergence of the fourier series for functions in such spaces

combinatorial geometries convex polyhedra and schubert cells this paper is a continuation of gm which was published in the same journal we will explore a remarkable connection between the geometry of the schubert cells in the grassmann manifold the theory of convex polyhedra and the theory of combinatorial geometries in the sense of crapo and rota cr the results in this paper were obtained simultaneously and independently by gelfand and serganova in moscow and by goresky and macpherson at the i h e s in paris as part of larger programs with different purposes see below the geometry of this simple example is so beautiful that we decided to publish it independently of the applications we believe that combinatorial methods will play an increasing role in the future of geometry and topology we consider the grassmann manifold gi of all n k dimensional subspaces of c by fixing the standard basis in c we obtain an action of the torus h i on g pk which is induced from stretching the coordinate axes in c see also sect 1 we will describe not only the trajectories but also the strata of a new and interesting decomposition of the 301 oool 8708 87 7 50

on an exactly solvable compact cluster growth model on a square lattice we consider a cluster growth model on a square lattice that is in the compact directed percolation universality class the model is exactly solvable as regards its static cluster properties and expressions are given for various quantities of interest such as the mean perimeter length and the mean cluster size the results also provide new information on area perimeter generating functions for a class of self avoiding polygons on square lattices

directed compact percolation near a wall i biased growth the directed compact percolation cluster model of domany and kinzel 1984 is considered in the presence of a wall which is parallel to the growth direction and hence restricts the lateral growth of the cluster in one direction the critical exponents are found to depend on whether the wall is wet or dry in the former case the model is solved exactly for all the standard percolation functions and the critical behaviour is found to be the same as that for cluster growth with no wall present with this boundary condition the cluster is completely attached to the wall and the model may also be viewed as one of symmetric compact cluster growth in the case of a dry wall the cluster may repeatedly leave and return to the wail as it grows and in this case the percolation probability has been derived exactly by lin 1992 and found to have a critical exponent different from that of the bulk lin s result is rederived and an exact formula for the percolation probability is found for a more general model in which the cluster growth is biased either towards or away from the wall it is found that the unbiased case is special in that any bias away from the wall recovers the bulk critical exponent and a bias towards the wall produces a problem in the same class as the wet wall model

the normalized acoustic power of a thin clamped annular plate vibrating in a flat screen around a perpendicular semi infinite circular cylinder abstract the neumann boundary value problem is presented for the time harmonic excited steady state vibrations of a thin clamped annular plate embedded in a flat rigid baffle around a perpendicular circular cylindrical rigid semi infinite baffle the system of two coupled differential equations is solved for this purpose one of them being the wave equation of the fluid filling the space between the two baffles and the other being the equation of motion of the plate for its excited vibrations first the acoustic pressure is presented in its spectral form using the weber inverse transform then the time averaged acoustic power is presented using the plate s modal impedance coefficients and the coupling factors the modal coefficients are presented as the integrals together with the corresponding asymptotic formulations the influence of the cylindrical baffle on the acoustic power is analyzed numerically the fluid loading and the internal attenuation of the plate s vibrations are both included whereas the attenuation of the acoustic waves in fluid is neglected

homotopy types of subspace arrangements via diagrams of spaces we prove combinatorial formulas for the homotopy type of the union of the subspaces in an affine compactified affine spherical or projective subspace arrangement from these formulas we derive results of goresky macpherson on the homology of the arrangement and the cohomology of its complement the union of an arrangement can be interpreted as the direct limit of a diagram of spaces over the intersection poset a closely related space is obtained by taking the homotopy direct limit of this diagram our method consists in constructing a combinatorial model diagram over the same poset whose homotopy limit can be compared to the original one by usual homotopy comparison results for diagrams of spaces

on integral transforms whose kernels are solutions of singular sturm liouville problems on etudie des transformees integrales du type f x 0 f s x s d ou x s est la solution du probleme de sturm liouville y s 2 q x y 0 0 x avec y 0 cos y 0 sin 0 y x est bornee a l infini et d est la mesure spectrale

real paley wiener theorems in spaces of ultradifferentiable functions abstract we develop real paley wiener theorems for classes s of ultradifferentiable functions and related l p spaces in the spirit of bang and andersen for the schwartz class we introduce results of this type for the so called gabor transform and give a full characterization in terms of fourier and wigner transforms for several variables of a paley wiener theorem in this general setting which is new in the literature we also analyze this type of results when the support of the function is not compact using polynomials some examples are given

a pair of generalized hankel clifford transformations and their applications abstract two variants of the hankel transform which will be called hankel clifford transformations are simultaneously investigated in certain spaces of distributions by the procedure of the adjoint operator for this purpose we exploit the fact that a parseval equation involving both transforms can be constructed so we propose to define the first generalized transformation as the adjoint operator of the second one and conversely finally these transforms are used in the solution of some partial differential equations of kepinski type

on the range of the hankel and extended hankel transforms abstract the range of the hankel and extended hankel transforms on some spaces of functions is described the paley wiener theorem for the hankel transform is also obtained

k parabolic subspace arrangements in this paper we study k parabolic arrangements a generalization of the k equal arrangement for any finite real reflection group when k 2 these arrangements correspond to the well studied coxeter arrangements brieskorn 1971 showed that the fundamental group of the complement of the type w coxeter arrangement over mathbb c is isomorphic to the pure artin group of type w khovanov 1996 gave an algebraic description for the fundamental group of the complement of the 3 equal arrangement over mathbb r we generalize khovanov s result to obtain an algebraic description of the fundamental group of the complement of the 3 parabolic arrangement for arbitrary finite reflection group our description is a real analogue to brieskorn s description

a finite and an infinite whittaker integral transform abstract this paper is concerned with the construction and study of specific finite and infinite integral transforms arising from a singular sturm liouville problem on a half line which is connected to the hydrogen atom equation the paper goes further to describe fully the image of some spaces of square integrable functions with respect to some measure under the finite and infinite integral transforms

real paley wiener theorems for fourier series abstract consider an integrable function f on the torus t m with fourier coefficients f n n z m for functions with support in a neighbourhood of the origin we describe the support by means of l p growth properties of difference operators acting on the fourier coefficients f n we then generalize the result to compact lie groups using a certain weyl group invariant difference operator we also consider l 2 bernstein inequalities for the difference operators

compact directed percolation with modified boundary rules a forest fire model a modified version of compact directed percolation on a square lattice is examined in the context of a model problem for the spread of forest fires the modification relates to conditioning the extent of the fire spread along the forest boundaries exact expressions are given for the mean perimeter length and the mean size of the damaged forest under such conditioning

directed compact percolation near a wall ii cluster length and size for paper i see ibid vol 27 p 3743 50 1994 the mean cluster size and length for the unbiassed growth of compact clusters near a dry wall are considered in the case of the cluster size below pc an exact expression is obtained for a seed of arbitrary width and distance from the surface it is found that the critical exponent gamma 1 for any finite distance from the surface crossover to the bulk value gamma 2 as the distance from the surface tends to infinity is observed this extends an existing result for the exponent beta of the percolation probability which changes from a value of 2 in the presence of a surface to 1 in the bulk limit the value delta 3 of the scaling size exponent is unchanged by the introduction of the surface the cluster size above pc and the mean cluster length are investigated using differential approximants from which we conjecture that these functions satisfy second order differential equations accepting this conjecture gives a mean size exponent the same as below pc and a logarithmic divergence of the mean length from both sides of the critical point the latter result together with scaling theory predicts that the exponent v sub has the value 2 the same as for the bulk problem

sampling for the fourth order sturm liouville differential operator we extend the sampling method to compute the eigenvalues of a fourth order differential operator we show that the shannon sampling theorem is not applicable due to the growth of the solutions on the real line the inverse spectral theory as developed by mclaughlin and kramer s theorem allow us to express the characteristic function explicitly by a new sampling formula

on integral transformations with g function kernels integral transformations t of the formal form where g is a meijer g function are studied on the spaces r of functions f such that results are obtained on the boundedness representation and range of the transformations and on when they have an inverse of the same formal form

the cohomology rings of complements of subspace arrangements the ring structure of the integral cohomology of complements of real linear subspace arrangements is considered while the additive structure of the cohomology is given in terms of the intersection poset and dimension function by a theorem of goresky and macpherson we describe the multiplicative structure in terms of the intersection poset the dimension function and orientations of the participating subspaces for the class of arrangements without pairs of intersections of codimension one in particular this yields a description of the integral cohomology ring of complex arrangements conjectured by yuzvinsky for general real arrangements a weaker result is obtained the approach is geometric and the methods are elementary

real paley wiener theorems for the koornwinder swarttouw q hankel transform abstract we derive two real paley wiener theorems in the setting of quantum calculus the first uses techniques due to tuan and zayed v k tuan a i zayed paley wiener type theorems for a class of integral transforms j math anal appl 266 1 2002 200 226 in order to describe the image of the space l q 2 0 r under koornwinder and swarttouw q hankel transform t h koornwinder r f swarttouw on q analogues of the fourier and hankel transforms trans amer math soc 333 1 1992 445 461 and contains as a special case a description of the domain of the q sampling theorem associated with the q hankel transform l d abreu a q sampling theorem related to the q hankel transform proc amer math soc 133 4 2005 1197 1203 the second characterizes the image of compactly supported q smooth functions under a rescaled version of the q hankel transform and is a q analogue of a recent result due to andersen n b andersen real paley wiener theorems for the hankel transform j fourier anal appl 12 1 2006 17 25

staircase polygons scaling functions and asymmetric compact directed percolation the scaling function for compact directed percolation on a square lattice is investigated for the asymmetric case where two parameters control the critical behaviour a simple representation for the area perimeter generating function for staircase polygons is found which can be recast as a non linear functional equation from this the exact scaling function is extracted in the process the most concise derivations to date are given for the exact low order cluster moments

exact solution of the staircase and row convex polygon perimeter and area generating function an explicit expression is obtained for the perimeter and area generating function g y z sigma n or 2 sigma m or 1 cn mynzm where cn m is the number of row convex polygons with area m and perimeter n a similar expression is obtained for the area perimeter generating function for staircase polygons both expressions contain q series

combinatorial stratification of complex arrangements we present a method for discretizing complex hyperplane arrange ments by encoding their topology into a finite partially ordered set of sign vectors this is used in the following ways 1 a general method is given for constructing regular cell complexes having the homotopy type of the complement of the arrangement 2 for the case of complexified arrangements this specializes to the con struction of salvetti s we study the combinatorial structure of complexified arrangements and the salvetti complex in some detail 3 this general method simultaneously produces cell decompositions of the singularity link this link is shown to have the homotopy type of a wedge of d spheres for arrangements in cd d 4 4 the homology of the link and the cohomology of the complement are computed in terms of explicit bases which are matched by alexander duality this gives a new more elementary and more generally valid proof for the brieskorn orlik solomon theorem and some related results 5 our setup leads to a more general notion of 2 pseudoarrangements which can be thought of as topologically deformed complex arrangements re taining only the essential topological and combinatorial structure we show that all of the above remains true in this generality except for the sign patterns of the orlik solomon relations department of mathematics royal institute of technology s10044 stockholm sweden e mail address andersb nada kth se institut fur mathematik universitat augsburg d 8900 augsburg germany current address institut mittag leffler auravagen 17 s 18262 djursholm sweden address after 1 april 1992 konrad zuse zentrum fur informationstechnik berlin heilbronner str 10 w 1000 berlin 31 germany e mail addresses ziegler ml kva se ziegler zib berlin de this content downloaded from 157 55 39 211 on fri 24 jun 2016 06 44 45 utc all use subject to http about jstor org terms

the continuous α β jacobi transform and its inverse when α β 1 is a positive integer definition de la transformation de jacobi continue determination de la transformation inverse application a une variante du theoreme d echantillonnage