hypergeometric transforms of functions with derivative in a half plane let a be the class of normalized analytic functions in the unit disk f a b c z and a c z denote respectively the gaussian and confluent hypergeometric functions let r lcub a r rcub such that re ei z 0 z rcub for a we define the hypergeometric transforms va b c and ua c by the convolution va b c zf a b c z z and ua c z a c z z respectively the main aim of this paper is to find conditions on 1 2 and the parameters a b c such that each of the operators va b c and ua c maps r 1 into r 2 we also find conditions such that the fu cab f a b c z 1 or ca a c z 1 is in r

fourier expansions of functions with bounded variation of several variables in the first part of the paper we establish the pointwise convergence as t for convolution operators rd t d k ty x y dy under the assumptions that y has integrable derivatives up to an order a and that k y c 1 y with d we also estimate the hausdorff dimension of the set where divergence may occur in particular when the kernel is the fourier transform of a bounded set in the plane we recover a two dimensional analog of the dirichlet theorem on the convergence of fourier series of functions with bounded variation in the second part of the paper we prove an equiconvergence result between fourier integrals on euclidean spaces and expansions in eigenfunctions of elliptic operators on manifolds which allows us to transfer some of the results proved for fourier integrals to eigenfunction expansions finally we present some examples of different behaviors between fourier integrals fourier series and spherical harmonic expansions

harmonic analysis as spectral theory of laplacians by writing the fourier inversion formula on euclidean space in polar coordinates we obtain f x 0 fgl x dx where fgl x 2fgl x which is the spectral theory of the laplacian how do properties of f relate to properties of the family of eigenfunctions fgl answers are provided for the following spaces l2 s s d in odd dimensions analogous results are obtained for harmonic analysis on hyperbolic space constant curvature semi riemannian spaces the heisenberg group and for differential forms on hyperbolic space for f l2 there is a plancherel formula f 22 limt o 1t bt z f x 2dx d where bt z denotes the ball of radius t about an arbitrary point z which is independent of the dimension and the identical formula holds in hyperbolic space if the parametrization of the eigenvalues is shifted for certain semi riemannian symmetric spaces we obtain a paley wiener theorem that explains the role of discrete series in producing functions of compact support and also involve certain non unitary representations that contain the discrete series representations for the heisenberg group the plancherel formula is of a different nature requiring that the eigenfunctions be almost periodic in one variable

starlike and convexity properties for hypergeometric functions abstract given the hypergeometric function f a b c z n 0 a n b n c n 1 n z n we place conditions on a b c to guarantee that zf a b c z will be in various subclasses of starlike and convex functions further constraints lead to coefficient characterizations of the families operators related to the hypergeometric function are also examined

integral transforms of functions to be in certain class defined by the combination of starlike and convex functions let p c b b 0 z d for some f r let m m a 0 m m which unifies s m and c m respectively the classes of analytic functions that map d onto the starlike and convex domain in this work we consider integral transforms of the form v l f z 0 1 l t f tz tdt the aim of this paper is to find conditions on l t so that the above transformation carry p c b into m m a as applications for specific values of l t it is found that several known integral operators carry p c b into m m a

pointwise fourier inversion on rank one symmetric spaces and related topics abstract this paper develops necessary and sufficient conditions for pointwise inversion of fourier transforms on rank one symmetric spaces of non compact type for functions in the piecewise smooth category this extends results of pinsky for isotropic riemannian manifolds of constant curvature methodologically a similar result on the jacobi transform and a transplantation scheme from even dimensional spaces to certain odd dimensional real hyperbolic spaces naturally enter the picture

on hypergeometric functions and function spaces the aim of this paper is to discuss the role of hypergeometric functions in function spaces and to prove some new results for these functions the first part of this paper proves results such as monotone convexity and concavity properties of sums of products of hypergeometric functions the second part of our results deals with the space a of all normalized analytic functions f f 0 0 f 0 1 in the unit disk and the subspace r f a a n r such that re ei f z 0 z for f a we consider integral transforms of the type v f 01 t f tz t dt where t is a real valued nonnegative weight function normalized so that 01 t 1 we obtain conditions on and the function such that v f takes each member of r into a starlike function of order 0 1 2 these results extend and improve the earlier known results in these directions we end the paper with an open problem

fourier series of radial functions in several variables abstract we prove that the spherical partial sums of the fourier series of the indicator function of a ball inside the cube of width 2 converge at the center of the ball if and only if the dimension is strictly less than three for more general radial functions in three dimensions we give a necessary and sufficient condition for convergence related questions of non localization and convergence of fourier transforms of radial functions in three dimensions are examined

operators on radial functions a general theory is provided that allows to write multi variate fourier transforms or convolutions of radial functions as very simple univariate operations as a byproduct an interesting group of operators i x r with i i i i i is defined it contains the classical derivatives as i1 d dr and is intimately connected to the fourier transform applications to the construction of new positive definite radial and to new identities for special functions are included

duality techniques for certain integral transforms to be starlike abstract for p denote the class of all normalized analytic functions f such that re e i 1 f z z f z 0 z for some r let s 0 s f re zf z f z z in this paper we find conditions so that the integral transform v f z 0 1 t f tz t dt carries p into s we provide a number of applications for certain special choices of t these results extend previously known results by a number of authors

classical locally finite tits chamber systems of rank 3 a chamber system v 9 j zi is over some index set z is a set of chambers together with partitions in z of v if ze z and c e v then bj is the join of all partitions fijzi e z and d c is the element of fij containing c d c is again a chamber system over j with fij restricted to z we say q is connected if d c furthermore 111 is the rank of v the most important examples of chamber systems are those obtained from groups if g is a group b a subgroup of g and xi ie z are subgroups of g containing b then 5 g b xi icl has as chambers the cosets bg g e g two such cosets bg bh being i adjacent if and only if xi g xih then g acts chamber transitively on with kernel b n bg and is connected if and only if g x 1 i e z furthermore each chamber system with transitive automorphism group g can be obtained in this way by setting b g and xi gd cr a rank 2 chamber system u hi fi is called a classical generalized ma gon mq 3 if is isomorphic to g b pi p where g is an essentially simple rank 2 group of lie type in the sense of cl b is a bore1 subgroup and p pj are the two maximal parabolic subgroups of g containing b here md is given by the relation vu wj wiw g 1 defined by the weyl group of g a rank 2 chamber system is a generalized digon if is isomorphic to v g pin p pi p where pi p are different proper subgroups of g satisfying g pipj pjpi a connected chamber system over z is called a classical tits chamber system if for each ce and i j c z di j c is either a generalized digon or a classical generalized m gon for some mi e n is focally finite if all d c are finite for motivation to consider such classical locally finite tits chamber systems see 15 and 16 if is a classical tits chamber system then the diagram d z is defined in 9 0021 8693 89 3 00

on the positivity of certain trigonometric sums and their applications in this paper we find conditions on the coefficients b k k 1 n such that the corresponding trigonometric cosine and sine sums given respectively by k 1nb email protected 0 and k 1nb email protected 0 for all email protected n are positive using these results we find that the functions f that are in the class of analytic functions a are starlike of certain order in the unit disc d by means of conditions on the taylor coefficients of f as an application we also find conditions such that the cesaro means of order b of f z are close to convex and starlike in d

on the order of starlikeness of the shifted gauss hypergeometric function abstract we determine for several ranges of real parameters the order of starlikeness of the shifted gauss hypergeometric function and we give some consequences of our results in particular some mapping properties of the carlson shaffer convolution operator

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

saturated fusion systems with parabolic families abstract let g be a group a finite p subgroup s of g is a sylow p subgroup if every finite p subgroup of g is conjugate to a subgroup of s in this paper we examine the relations between the fusion system over s which is given by conjugation in g and a certain chamber system c on which g acts chamber transitively with chamber stabilizer n g s next we introduce the notion of a fusion system with a parabolic family and we show that a chamber system can be associated to such a fusion system we determine some conditions the chamber system has to fulfill in order to assure the saturation of the underlying fusion system we give an application to fusion systems with parabolic families of classical type

generalized hypergeometric functions associated with k uniformly convex functions abstract for a certain linear operator which is defined by means of the hadamard product or convolution with a generalized hypergeometric function the authors aim at investigating various mapping and inclusion properties involving such subclasses of analytic and univalent functions as for example k uniformly convex functions and k starlike functions relevant connections of the definitions and results presented in this paper with those in several earlier works on the subject are also pointed out

generalized hardy identity and relations to gibbs wilbraham and pinsky phenomena abstract we consider the fourier series of the indicator functions of several dimensional balls for the spherical partial sum of the fourier series we extract the gibbs wilbraham or gibbs pinsky and the third phenomena as an extension of hardy s identity the third phenomenon has been shown by kuratsubo recently we prove the gibbs wilbraham phenomenon for all dimensions and give another proof of the pinsky phenomenon pinsky gave the order of the divergence for the fourier inversion at the origin we give the order of the divergence of the fourier series at the origin and show that both orders coincide we also investigate the uniform convergence for the fourier series and the fourier inversion

precise coefficient estimates for close to convex harmonic univalent mappings the class sh consists of harmonic univalent and sense preserving functions f in the open unit disk u z z 1 such that f h g where h z z n 2 anzn and g z n 1 a nzn let s0h ch and c0h denote the subclass of sh with a 1 0 the subclass of sh with f being a close to convex mapping and the intersection of s0h and ch respectively in this paper for f c0h and f ch we prove that the harmonic analogue of the bieberbach conjecture and the generalization of the bieberbach conjecture are true

sufficient conditions for hypergeometric functions to be in a certain class of analytic functions the author aims at finding certain conditions on the parameters a b and c such that the normalized gaussian hypergeometric function zf a b c z given by f a b c z email protected n 0 a n b n c n 1 nz n z 1 is in a certain class of analytic functions various results for the particular values of these parameters are deduced

harmonic univalent functions with negative coefficients we give sufficient coefficient conditions for normalized harmonic functions to map onto either starlike or convex regions these conditions are also shown to be necessary when the coefficients are negative this leads to distortion bounds and extreme points

radii of starlikeness and convexity of some q bessel functions abstract geometric properties of the jackson and hahn exton q bessel functions are studied for each of them three different normalizations are applied in such a way that the resulting functions are analytic in the unit disk of the complex plane for each of the six functions we determine the radii of starlikeness and convexity precisely by using their hadamard factorization these are q generalizations of some known results for bessel functions of the first kind the characterization of entire functions from the laguerre polya class via hyperbolic polynomials plays an important role in this paper moreover the interlacing property of the zeros of jackson and hahn exton q bessel functions and their derivatives is also useful in the proof of the main results we also deduce a necessary and sufficient condition for the close to convexity of a normalized jackson q bessel function and its derivatives some open problems are proposed at the end of the paper

parabolic systems the gf 3 case parabolic systems defined over gf q have been classified by timmesfeld for q 4 and by stroth for q 2 see references we deal with the case q 3

norm estimates for convolution transforms of certain classes of analytic functions abstract for normalized analytic functions f in the unit disk we consider the class f f re 1 z f z f z 3 2 z where 2 3 1 and s a b f z f z f z 1 a z 1 b z z where 1 b a 1 and denotes the subordination set k a b f z f s a b for 0 b c define j b c f by j b c f z z f 1 b c z f z where f a b c z is the gauss hypergeometric function and denotes the convolution or hadamard product for a locally univalent analytic function f defined on we consider the pre schwarzian norm by f sup z 1 z 2 f z f z in this paper we find sharp norm estimates for functions in f and in k a b in addition we also present sharp norm estimates for j b c f z when f ranges over the classes f and k a b respectively

norm estimates for the alexander transforms of convex functions of order alpha abstract the hyperbolic sup norm of the pre schwarzian derivative of a locally univalent function on the unit disk measures the deviation of the function from similarities we present sharp norm estimates for the alexander transforms of convex functions of order 0 1

a pushing up result and some consequences for the embedding of 2 constrained subgroups let m m1 be maximal 2 locai subgroups of some finite group g with o g 1 satisfying qi o mi f mi for i 0 1 is it possible in this situation that q1 qo the bulk of the proof of l consists of showing that there exists no k group g of characteristic 2 type satisfying these and certain non generational conditions for mi which by results of g seitz strongly restrict the structure of mr if one tries to consider this problem more generally one realizes that it splits naturally into two cases

subordination results for classes of analytic functions related to conic domains defined by a fractional operator abstract let a fractional operator d n n n 0 0 1 2 0 1 0 be defined by d 0 0 f z d 1 f z 1 f z z f z d f z d 2 f z d d 1 f z d n f z d d n 1 f z where f z 2 z d z f z and d z is the known fractional derivative in this paper several interesting subordination results are derived for certain classes of analytic functions related to conic domains defined by the operator d n which yield sharp distortion rotation theorems and koebe domain these results extended corresponding previously known results

connections between various subclasses of planar harmonic mappings involving hypergeometric functions the main purpose of this paper is to establish connections between various subclasses of harmonic mappings in the plane by applying certain convolution operators involving hypergeometric functions to be more precise we investigate such connections with harmonic k uniformly starlike and harmonic k uniformly convex mappings in the plane

pointwise fourier inversion and related eigenfunction expansions necessary and sufficient conditions are found for the convergence at a pre assigned point of the spherical partial sums resp integrals of the fourier series resp integral in the class of piecewise smooth functions on euclidean space these results carry over unchanged to spherical harmonic expansions fourier transforms on hyperbolic space and dirichlet eigenfunction expansions with respect to the laplace operator on a class of riemannian manifolds 1994 john wiley sons inc

inclusion theorems of convolution operators associated with normalized hypergeometric functions for the normalized gaussian hypergeometric function zf a b c z given by f a b c z n 0 a n b n c n l n zn z 0 r z a convolution result for the class p is also given analogous results for the confluent hypergeometric functions are also discussed

starlikeness and convexity results for hypergeometric functions the objective of the present paper is to give some characterizations for a gaussian hypergeometric function to be in various subclasses of starlike and convex functions of order a type b we also consider an integral operator related to the hypergeometric function

the radius of uniform convexity of bessel functions abstract in this paper we determine the radius of uniform convexity for three kinds of normalized bessel functions of the first kind in the cases considered the normalized bessel functions are uniformly convex on the determined disks moreover necessary and sufficient conditions are given for the parameters of the three normalized functions such that they to be uniformly convex in the open unit disk the basic tool of this study is bessel functions in series

close to convexity properties of gaussian hypergeometric functions abstract let a be the class of normalized analytic functions in the unit disk let z be either zf a b c z or c ab f a b c z 1 where f a b c z denotes the classical hypergeometric function the purpose of this paper is to study close to convexity and hence univalency of z in the unit disc more generally we find conditions on a b c and such that satisfies re e i 1 z z 0 for all z and for some real 1 2 1 2

inequalities for zero balanced hypergeometric functions the authors study certain monotoneity and convexity properties of the gaussian hypergeometric function and those of the euler gamma function

univalency of weighted integral transforms of certain functions for 0 z for some r for f p we consider the integral transform v f z 01 t f tz t dt where t is a real valued nonnegative weight function so that 01 t dt 1 the main aim of this paper is to find conditions such that v f p1 whenever f p for 1 we also obtain conditions such that v f p1 whenever f p0 for various choices of t as a useful consequence we find conditions for certain hypergeometric functions to be univalent

the rate of convergence of fourier expansions in the plane a geometric viewpoint let a be an appropriate planar domain and let f be a piecewise smooth function on mathbb r 2 we discuss the rate of convergence of s lambda f x int lambda a widehat f xi exp 2 pi i xi cdot x d xi in terms of the interaction between the geometry of a and the geometry of the singularities of f the most subtle case is when x belongs to the singular set of f