a nonstandard construction of levy brownian motion a nonstandard construction of levy brownian motion on d is presented which extends r m anderson s nonstandard representation of brownian motion it involves a nonstandard construction of white noise and gives as a classical corollary a new white noise integral representation of levy brownian motion moreover a new invariance principle can be deduced in a similar way as donsker s invariance principles follows from anderson s construction

the zeros of derivatives of entire functions and the polya wiman conjecture an entire function f x which assumes only real values on the real axis is said to be a real entire function thus if f is a real entire function then its maclaurin coefficients are all real and consequently the zeros of f are symmetrically located with respect to the real axis one of the intuitive principles underlying the theory of distribution of zeros of successive derivatives of real entire functions was formulated by polya p4 in his celebrated survey article entitled on the zeros of the derivatives of a function and its analytic character in this work p4 p 181 polya states the real axis seems to exert an influence on the complex zeros of f n z it seems to attract these zeros when the order is less than 2 and it seems to repel them when the order is greater than 2 indeed the first confirmation of this principle was made by alander a2 see also p6lya p2 who in 1930 showed that the polya wiman conjecture is true if the order of f x is less than 2 3 in wil and wi2 wiman established the validity of this conjecture for functions f x of order at most 1 while in 1937 polya p3 proved it for functions of order less than 4 3 a general survey of these results and related conjectures may be found in p2 and p4 for more recent surveys of this and related areas of research see boas b2 and prather pr it seems that no progress has been made on the p6lya wiman conjecture since the publication of polya s 4 3 theorem in 1937 our proof of the conjecture is self contained and does not rely on the results cited above in section 2 we will 1 introduce some definitions and notations 2 state some of the properties of functions in the laguerre p6lya class and 3 prove the polya wiman conjecture theorem 3 in the proof of theorem 3 we

concentration in a thin euclidean shell for log concave measures abstract a weak version of a conjecture stated by kannan lovasz and simonovits claims that an isotropic log concave probability on r n should be concentrated in a thin euclidean shell in the following way 1 t 0 n x r n 1 t n x n 1 t n 1 c e c t where 1 2 and c and c are positive absolute constants for 1 10 02 this inequality has been established by klartag by combining different approaches introduced by klartag and by guedon paouris and the author we improve this result by showing that the inequality 1 holds with 1 8

a stability result for mean width of lp centroid bodies we give a different proof of a recent result of klartag b klartag a central limit theorem for convex sets invent math 168 1 2007 91 131 concerning the concentration of the volume of a convex body within a thin euclidean shell and proving a conjecture of anttila ball and perissinaki m anttila k ball i perissinaki the central limit problem for convex bodies trans amer math soc 355 12 2003 4723 4735 it is based on the study of the lp centroid bodies we prove an almost isometric reverse holder inequality for their mean width and a refined form of a stability result

reality of the zeros of an entire function and its derivatives in 1914 p6lya raised the problem of classifying the entire functions which together with all their derivatives have only real zeros in earlier work hellerstein and williamson settled this problem for entire functions which are real on the real axis we complete the classification in all cases and show that it is sufficient to consider the function and its first two derivatives introduction in 12 and 13 g polya posed the problem of classifying all entire functions which have along with all their derivatives only real zeros he conjectured that such a function must have one of the following forms 0 1 f z aebz 0 2 f z a eicz eid 0 3 f z azme az jii l1 ez an where a and b are eomplex constants a 0 b c d and the an are real i l a2 x and m is a nonnegative integer in 10 b ja levin and i v ostrovskii showed that if f is a real i e real on the real axis entire function with only real zeros satisfying the growth condition 0 4 limsup loglog m r f rlog r x then the zeros cn of f satisfy 0 5 2 i m llcn 0oc in particular f has infinitely many nonreal zeros polya s conjecture was completely affirmed for real entire functions by two of the present authors in 6 and 7 where the cases of functions of finite order and of infinite order growth slower than that of 0 4 were successfully treated these results coupled with those of levin and ostrovskii showed that if f f and f have only real zeros and f is real entire then f is in the laguerre p6lya class i e of the form 0 3 received by the editors august 14 1981 and in revised form december 31 1981 1980 mathematics subject classification primary 30d30 secondary 30d35 partially supported by nsf grant mcs 8003108 2part of research towards ph d degree university of wisconsin 1983 american mathematical society 0002 9947 82 0000 0099 04 00

a probabilistic approach to the geometry of the lᵨⁿ ball this article investigates by probabilistic methods various geometric questions on b n p the unit ball of ln p we propose realizations in terms of independent random variables of several distributions on b n p including the normalized volume measure these representations allow us to unify and extend the known results of the sub independence of coordinate slabs in b n p as another application we compute moments of linear functionals on b n p which gives sharp constants in khinchine s inequalities on b n p and determines the 2 constant of all directions on b n p we also study the extremal values of several gaussian averages on sections of b n p including mean width and l norm and derive several monotonicity results as p varies applications to balancing vectors in l 2 and to covering numbers of polyhedra complete the exposition

hyperfinite methods applied to the critical branching diffusion we apply the hyperfinite methods of re to the construction of a version of the critical branching diffusion studied by dawson et al several new sample path properties are derived from this construction

reverse faber krahn and mahler inequalities for the cheeger constant we prove a reverse faber krahn inequality for the cheeger constant stating that every convex body in 2 has an affine image such that the product between its cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle an analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square we also prove a mahler type inequality for the cheeger constant stating that every axisymmetric convex body in 2 has a linear image such that the product between its cheeger constant and the cheeger constant of its polar body is not larger than the same quantity for the square

one dimensional stochastic partial differential equations and the branching measure diffusion an existence theorem in the spirit of keisler ke is proved for the simple one dimensional diffusion equation driven by white noise modulated by a non linear function of the solution this is used to obtain a density and a stochastic partial differential equation in one dimension for the critical branching diffusion studied by dawson et al

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

the hurwitz type theorem for the regular coulomb wave function via hankel determinants abstract we derive a closed formula for the determinant of the hankel matrix whose entries are given by sums of negative powers of the zeros of the regular coulomb wave function this new identity applied together with results of grommer and chebotarev allows us to prove a hurwitz type theorem about the zeros of the regular coulomb wave function as a particular case we obtain a new proof of the classical hurwitz s theorem from the theory of bessel functions that is based on algebraic arguments in addition several hankel determinants with entries given by the rayleigh function and bernoulli numbers are also evaluated

extremal slabs in the cube and the laplace transform abstract we study the volume of symmetric slabs in the unit cube we show that for t 3 4 the slab parallel to a face has the minimal volume among all symmetric slabs with width t for large width we prove the asymptotic extremality of the slab orthogonal to the main diagonal the proof is based on certain concavity properties of the laplace transform and on several limit theorems from probability the central limit theorem and classical principles of moderate and large deviations finally we extend some of the results to more general classes of bodies

on the nonlinear brascamp lieb inequality we prove a nonlinear variant of the general brascamp lieb inequality our proof consists of running an efficient or tight induction on scales argument which uses the existence of gaussian near extremizers to the underlying linear brascamp lieb inequality lieb s theorem in a fundamental way a key ingredient is an effective version of lieb s theorem which we establish via a careful analysis of near minimizers of weighted sums of exponential functions instances of this inequality are quite prevalent in mathematics and we illustrate this with some applications in harmonic analysis

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

a berry esseen type inequality for convex bodies with an unconditional basis suppose x x1 xn is a random vector distributed uniformly in a convex body k subset mathbb r n we assume the normalization mathbb e x i 2 1 for i 1 n the body k is further required to be invariant under coordinate reflections that is we assume that x1 xn has the same distribution as x1 xn for any choice of signs then we show that mathbb e left x sqrt n right 2 leq c 2 where c 4 is a positive universal constant and is the standard euclidean norm in mathbb r n the estimate is tight up to the value of the constant it leads to a berry esseen type bound in the central limit theorem for unconditional convex bodies

power law estimates for the central limit theorem for convex sets abstract we investigate the rate of convergence in the central limit theorem for convex sets established in b klartag a central limit theorem for convex sets invent math in press 8 we obtain bounds with a power law dependence on the dimension these bounds are asymptotically better than the logarithmic estimates which follow from the original proof of the central limit theorem for convex sets

volume inequalities of convex bodies from cosine transforms on grassmann manifolds abstract the l p cosine transform on grassmann manifolds naturally induces finite dimensional banach norms whose unit balls are origin symmetric convex bodies in r n reverse isoperimetric type volume inequalities for these bodies are established which extend results from the sphere to grassmann manifolds

sections of the unit ball of ipn let bnp xi rn i li n xi p 1 1 p and let ek be a k dimensional subspace of rn it is proved that if p 2 resp then vol bpn ek vol bpk resp we give some applications to linear forms

on the number of nonreal zeros of real entire functions and the fourier polya conjecture this paper is concerned with a general theorem on the number of nonreal zeros of transcendental functions j fourier formulated the theorem in his work analyse des equations determineesin 1831 but he did not give a proof roughly speaking the theorem states that if a real entire function f x can be expressed as a product of linear factors then we can count the nonreal zeros of f x by observing the behavior of the derivatives of f x on the real axis alone as we shall see in the sequel this theorem completely justifies his former argument by which he tried to prove that the function j0 2 x has only real zeros it seems that no complete proof of the theorem is known and no general theorem has been published that justifies the argument later in 1930 g polya published a paper entitled some problems connected with fourier s work on transcendental equations p3 in this paper polya conjectured two hypothetical theorems that are closely related to fourier s unproved theorem in fact he conjectured three but he proved that two of them are equivalent to each other the first hypothetical theorem is a modernized formulation of the theorem and it justifies fourier s argument completely the second conjecture was proved in 1990 but it is impossible to justify the argument using the conjecture alone in the present paper we prove polya s formulation of the theorem his first conjecture as well as its extensions give a very simple and direct proof of the second conjecture mentioned above and exhibit some applications of the results in particular we completely justify fourier s argument by our general theorems acknowledgments professor fefferman has encouraged and helped us to publish this paper the authors truly thank him for this the authors also thank professor csordas for his kind interest in the results and his valuable suggestions on the proof of the polya wiman conjecture 1 historical introduction in this section we briefly explain our results as well as their background a real entire function is an entire function that assumes only real values on the real axis fourier s unproven theorem asserts that we can know the number of nonreal zeros of such real entire functions by counting their critical points which are defined as follows let f x be a real analytic function defined in an open

shadows of convex bodies it is proved that if c is a convex body in rn then c has an affine image c of nonzero volume so that if p is any 1 codimensional orthogonal projection pca 1 cl n l n it is also shown that there is a pathological body k all of whose orthogonal projections have volume about vx times as large as iki n 1 n 0 introduction the problems discussed in this paper concern the areas of shadows orthogonal projections of convex bodies and to a lesser extent the surface areas of such bodies if c is a convex body in irn and 0 a unit vector p c will denote the orthogonal projection of c onto the 1 codimensional space perpendicular to 0 volumes and areas of convex bodies and their surfaces will be denoted with i 1 the relationship between shadows and surface areas of convex bodies is expressed in cauchy s well known formula for each n e n let vn be the volume of the n dimensional euclidean unit ball and let a an be the rotationally invariant probability on the unit sphere sn cauchy s formula states that if c is a convex body in rn then its surface area is ac i s p6ci dcr o vni jn i the classical isoperimetric inequality in rn states that any body has surface area at least as large as a euclidean ball of the same volume the first section of this paper is devoted to the proof of a local isoperimetric inequality showing that all bodies have large shadows rather than merely large surface area or average shadow the principal motivation for this result is its relationship to a conjecture of vaaler and the important problems surrounding it this theorem and its connection with vaaler s conjecture are described at the beginning of 1 an important role is played in the theory of convex bodies by the so called projection body of a convex body it is easily seen by considering polytopes received by the editors september 12 1989 1980 mathematics subject classification 1985 revision primary 52a20 52a40 the author was supported in part by nsf dms 8807243 1991 american mathematical society 0002 9947 91 1 00 25 per page

on the convergence of the extremal eigenvalues of empirical covariance matrices with dependence consider a sample of a centered random vector with unit covariance matrix we show that under certain regularity assumptions and up to a natural scaling the smallest and the largest eigenvalues of the empirical covariance matrix converge when the dimension and the sample size both tend to infinity to the left and right edges of the marchenko pastur distribution the assumptions are related to tails of norms of orthogonal projections they cover isotropic log concave random vectors as well as random vectors with i i d coordinates with almost optimal moment conditions the method is a refinement of the rank one update approach used by srivastava and vershynin to produce non asymptotic quantitative estimates in other words we provide a new proof of the bai and yin theorem using basic tools from probability theory and linear algebra together with a new extension of this theorem to random matrices with dependent entries

non real zeros of real differential polynomials the main results of the paper determine all real meromorphic functions f of finite lower order in the plane such that f has finitely many zeros and non real poles and certain combinations of derivatives of f have few non real zeros msc 2000 30d20 30d35

sudakov s typical marginals random linear functionals and a conditional central limit theorem v n sudakov sud78 proved that the one dimensional marginals of a high dimensional second order measure are close to each other in most directions extending this and a related result in the context of projection pursuit of p diaconis and d freedman dia84 we give for a probability measure p and a random a s linear functional f on a hilbert space simple sufficient conditions under which most of the one dimensional images of p under f are close to their canonical mixture which turns out to be almost a mixed normal distribution using the concept of approximate conditioning we deduce a conditional central limit theorem theorem 3 for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions

poincare inequality in mean value for gaussian polytopes let k n g 1 g n be the absolute convex hull of n independent standard gaussian random points in mathbb r n with n n we prove that for any 1 lipschitz function f mathbb r n rightarrow mathbb r the polytope k n satisfies the following poincare inequality in mean value mathbb e omega int limits k n omega left f x frac 1 textup vol n left k n omega right int limits k n omega f y dy right 2 dx leq frac c n mathbb e omega int limits k n omega x 2dx where c 0 is an absolute constant this poincare inequality is the one suggested by a conjecture of kannan lovasz and simonovits for general convex bodies moreover we prove in mean value that the volume of the polytope k n is concentrated in a subexponential way within a thin euclidean shell with the optimal dependence of the dimension n an important tool of the proofs is a representation of the law of g 1 g n conditioned by the event that the convex hull of g 1 g n is a n 1 face of k n as an application we also get an estimate of the number of n 1 faces of the polytope k n valid for every n n

high dimensional random sections of isotropic convex bodies we study two properties of random high dimensional sections of convex bodies in the first part of the paper we estimate the central section function k f n k1 k for random f gn k and k rn a centrally symmetric isotropic convex body this partially answers a question raised by v d milman and a pajor see v d milman a pajor isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n dimensional space in lecture notes in math vol 1376 springer 1989 p 88 in the second part we show that every symmetric convex body has random high dimensional sections f gn k with outer volume ratio bounded by ovr k f cnn klog 1 nn k

pointwise estimates for marginals of convex bodies we prove a pointwise version of the multi dimensional central limit theorem for convex bodies namely let be an isotropic log concave probability measure on rn for a typical subspace e rn of dimension nc consider the probability density of the projection of onto e we show that the ratio between this probability density and the standard gaussian density in e is very close to 1 in large parts of e here c 0 is a universal constant this complements a recent result by the second named author where the total variation metric between the densities was considered

the borell ehrhard game a precise description of the convexity of gaussian measures is provided by sharp brunn minkowski type inequalities due to ehrhard and borell we show that these are manifestations of a game theoretic mechanism a minimax variational principle for brownian motion as an application we obtain a gaussian improvement of barthe s reverse brascamp lieb inequality

the central limit problem for convex bodies it is shown that every symmetric convex body which satisfies a kind of weak law of large numbers has the property that almost all its marginal distributions are approximately gaussian several quite broad classes of bodies are shown to satisfy the condition

proof of a conjecture of polya on the zeros of successive derivatives of real entire functions we prove polya s conjecture of 1943 for a real entire function of order greater than 2 with finitely many non real zeros the number of non real zeros of the nth derivative tends to infinity as n to infty we use the saddle point method and potential theory combined with the theory of analytic functions with positive imaginary part in the upper half plane

differentiation evens out zero spacings if f is a polynomial with all of its roots on the real line then the roots of the derivative f are more evenly spaced than the roots of f the same holds for a real entire function of order 1 with all its zeros on a line in particular we show that if f is entire of order 1 and has sufficient regularity in its zero spacing then under repeated differentiation the function approaches after normalization the cosine function we also study polynomials with all their zeros on a circle and we find a close analogy between the two situations this sheds light on the spacing between zeros of the riemann zeta function and its connection to random matrix polynomials

on a reverse form of the brascamp lieb inequality we prove a reverse form of the multidimensional brascamp lieb inequality our method also gives a new way to derive the brascamp lieb inequality and is rather convenient for the study of equality cases

discrepancy chaining and subgaussian processes the research leading to the results presented here has received funding from the european research council under the european community s seventh frame work programme fp7 2007 2013 erc grant agreement 203134 from the israel science foundation grant 666 06 and from the australian research council grant dp0986563

diagonalizing properties of the discrete cosine transforms since its introduction in 1974 by ahmed et al the discrete cosine transform dct has become a significant tool in many areas of digital signal processing especially in signal compression there exist eight types of discrete cosine transforms dcts we obtain the eight types of dcts as the complete orthonormal set of eigenvectors generated by a general form of matrices in the same way as the discrete fourier transform dft can be obtained as the eigenvectors of an arbitrary circulant matrix these matrices can be decomposed as the sum of a symmetric toeplitz matrix plus a hankel or close to hankel matrix scaled by some constant factors we also show that all the previously proposed generating matrices for the dcts are simply particular cases of these general matrix forms using these matrices we obtain for each dct a class of stationary processes verifying certain conditions with respect to which the corresponding dct has a good asymptotic behavior in the sense that it approaches karhunen loeve transform performance as the block size n tends to infinity as a particular result we prove that the eight types of dcts are asymptotically optimal for all finite order markov processes we finally study the decorrelating power of the dcts obtaining expressions that show the decorrelating behavior of each dct with respect to any stationary processes

symmetric convolution using unitary transform matrices we demonstrate relationships between the convolutional and unitary forms of certain trigonometric transform matrices we then use the new relationships to derive forms of the symmetric convolution multiplication property based on unitary rather than convolutional forms of the transform matrices

symmetric convolution and the discrete sine and cosine transforms this paper discusses the use of symmetric convolution and the discrete sine and cosine transforms dsts and dcts for general digital signal processing the operation of symmetric convolution is a formalized approach to convolving symmetrically extended sequences the result is the same as that obtained by taking an inverse discrete trigonometric transform dtt of the product of the forward dtts of those two sequences there are 16 members in the family of dtts each provides a representation for a corresponding distinct type of symmetric periodic sequence the author defines symmetric convolution relates the dsts and dcts to symmetric periodic sequences and then use these principles to develop simple but powerful convolution multiplication properties for the entire family of dsts and dcts symmetric convolution can be used for discrete linear filtering when the filter is symmetric or antisymmetric the filtering will be efficient because fast algorithms exist for all versions of the dtts conventional linear convolution is possible if one first zero pad the input data symmetric convolution and its fast implementation using dtts are now an alternative to circular convolution and the dft

the fox wright functions and laguerre multiplier sequences abstract linear operators which 1 preserve the reality of zeros of polynomials having only real zeros and 2 map stable polynomials into stable polynomials are investigated using recently established results concerning the zeros of certain fox wright functions and generalized mittag leffler functions the paper includes several open problems and questions

moment inequalities and central limit properties of isotropic convex bodies the object of our investigations are isotropic convex bodies k subseteq mathbb r n centred at the origin and normed to volume one in arbitrary dimensions we show that a certain subset of these bodies specified by bounds on the second and fourth moments is invariant under forming expanded joinsrsquo considering a body k as above as a probability space and taking u in s n 1 we define random variables x k u x cdot u on k it is known that for subclasses of isotropic convex bodies satisfying a concentration of mass property the distributions of these random variables are close to gaussian distributions for high dimensions n and most directions u in s n 1 we show that this central limit property which is known to hold with respect to convergence in law is also true with respect to l 1 convergence and l infty convergence of the corresponding densities

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

invariances in variance estimates we provide variants and improvements of the brascamp lieb variance inequality which takes into account the invariance properties of the underlying measure this is applied to spectral gap estimates for log concave measures with many symmetries and to non interacting conservative spin systems

a central limit theorem for convex sets we show that there exists a sequence varepsilon n searrow0 for which the following holds let k n be a compact convex set with a non empty interior let x be a random vector that is distributed uniformly in k then there exist a unit vector in n t0 and 0 such that sup a subset mathbb r left textit prob langle x theta rangle in a frac 1 sqrt 2 pi sigma int ae frac t t 0 2 2 sigma 2 dt right leq varepsilon n qquad ast where the supremum runs over all measurable sets a and where denotes the usual scalar product in n furthermore under the additional assumptions that the expectation of x is zero and that the covariance matrix of x is the identity matrix we may assert that most unit vectors satisfy with t0 0 and 1 corresponding principles also hold for multi dimensional marginal distributions of convex sets

on the de bruijn newman constant abstract if 0 denotes the infimum of the set of real numbers such that the entire function represented by t 0 e 4 log x 2 i t 2 log x x 5 4 n 1 2 n 4 2 x 3 n 2 e n 2 x d x x has only real zeros then the de bruijn newman constant is defined as 4 0 the riemann hypothesis is equivalent to the inequality 0 the fact that the non trivial zeros of the riemann zeta function lie in the strip s 0 re s 1 and a theorem of de bruijn imply that 1 2 in this paper we prove that all but a finite number of zeros of are real and simple for each 0 and consequently that 1 2

on the polya wiman properties of differential operators abstract let x n x n be a formal power series with real coefficients and let d denote differentiation it is shown that for every real polynomial f there is a positive integer m 0 such that d m f has only real zeros whenever m m 0 if and only if 0 0 or 2 0 2 1 2 0 and that if does not represent a laguerre polya function then there is a laguerre polya function f of genus 0 such that for every positive integer m d m f represents a real entire function having infinitely many nonreal zeros

generating matrices for the discrete sine transforms in this correspondence we obtain a general form for the generating matrices of the eight types of discrete sine transforms these matrix forms can be used to study the performance of the different discrete sine transforms as substitutes of the karhunen loeve transform and will allow us to show in a very straightforward way that the discrete sine transforms as the discrete cosine transforms are asymptotically optimal for any finite order markov process

symmetric convolution of asymmetric multidimensional sequences using discrete trigonometric transforms this paper uses the fact that the discrete fourier transform diagonalizes a circulant matrix to provide an alternate derivation of the symmetric convolution multiplication property for discrete trigonometric transforms derived in this manner the symmetric convolution multiplication property extends easily to multiple dimensions using the notion of block circulant matrices and generalizes to multidimensional asymmetric sequences the symmetric convolution of multidimensional asymmetric sequences can then be accomplished by taking the product of the trigonometric transforms of the sequences and then applying an inverse trigonometric transform to the result an example is given of how this theory can be used for applying a two dimensional 2 d finite impulse response fir filter with nonlinear phase which models atmospheric turbulence

on the volume of non central sections of a cube abstract let q n be the cube of side length one centered at the origin in r n and let f be an affine n d dimensional subspace of r n having distance to the origin less than or equal to 1 2 where 0 d n we show that the n d dimensional volume of the section q n f is bounded below by a value c d depending only on the codimension d but not on the ambient dimension n or a particular subspace f in the case of hyperplanes d 1 we show that c 1 1 17 is a possible choice we also consider a complex analogue of this problem for a hyperplane section of the polydisc

the surface measure and cone measure on the sphere of ell_p n we prove a concentration inequality for the l n p norm on the l n p sphere for p q 0 this inequality which generalizes results of schechtman and zinn 2000 is used to study the distance between the cone measure and surface measure on the sphere of l n p in particular we obtain a significant strengthening of the inequality derived by naor and romik 2003 and calculate the precise dependence of the constants that appeared there on p

on the zeros of the derivatives of real entire functions and wiman s conjecture let f be an entire function of finite order real on the real axis and possessing only real zeros a classical problem proposed by g p6lya 9 10 and a wiman 1 is to determine from the hadamard canonical representation of the function the number of nonreal zeros of the higher derivatives of f in this paper we shall prove the following conjecture of wiman which has been open since 1915

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

l1 bounds in normal approximation the zero bias distribution w of w defined though the characterizing equation ewf w 2 ef 0 w for all smooth functions f exists for all w with mean zero and finite variance 2 for w and w defined on the same probability space the l 1 distance between f the distribution function of w with ew 0 and var w 1 and the cumulative standard normal has the simple upper bound f 1 2e w w this inequality is used to provide explicit l 1 bounds with moderate sized constants for independent sums projections of cone measure on the sphere s p simple random sampling and combinatorial central limit theorems

analysis of the structured perturbation for the bsccb linear system abstract in this paper based on block style spectral decomposition of the block skew circulant with circulant blocks bsccb matrix the structure perturbation is discussed which includes the condition number and relative error of the bsccb linear system then the optimal backward perturbation bound of the bsccb linear system is analyzed simultaneously the algorithm for the optimal backward perturbation bound is presented at the end of the paper a numerical example is provided to verify the effectiveness of the algorithm