a wavelet based joint estimator of the parameters of long range dependence a joint estimator is presented for the two parameters that define the long range dependence phenomenon in the simplest case the estimator is based on the coefficients of a discrete wavelet decomposition improving a wavelet based estimator of the scaling parameter abry and veitch 1998 as well as extending it to include the associated power parameter an important feature is its conceptual and practical simplicity consisting essentially in measuring the slope and the intercept of a linear fit after a discrete wavelet transform is performed a very fast o n operation under well justified technical idealizations the estimator is shown to be unbiased and of minimum or close to minimum variance for the scale parameter and asymptotically unbiased and efficient for the second parameter through theoretical arguments and numerical simulations it is shown that in practice even for small data sets the bias is very small and the variance close to optimal for both parameters closed form expressions are given for the covariance matrix of the estimator as a function of data length and are shown by simulation to be very accurate even when the technical idealizations are not satisfied comparisons are made against two maximum likelihood estimators in terms of robustness and computational cost the wavelet estimator is found to be clearly superior and statistically its performance is comparable we apply the tool to the analysis of ethernet teletraffic data completing an earlier study on the scaling parameter alone

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

weighted and vector valued inequalities for potential operators on etablit des inegalites ponderees et a valeurs vectorielles pour quelques operateurs integraux de type potentiel qui appliquent les fonctions definies sur r n dans des fonctions sur r n 1 r n 0

weighted norm inequalities for potentials with applications to schrodinger operators fourier transforms and carleson measures on donne une nouvelle caracterisation de l inegalite de trace pour des operateurs potentiel et on l utilise pour affiner des resultats sur la distribution des valeurs propres des operateurs de schrodinger

identification of the multivariate fractional brownian motion this paper deals with the identification of the multivariate fractional brownian motion a recently developed extension of the fractional brownian motion to the multivariate case this process is a p multivariate self similar gaussian process parameterized by p different hurst exponents hi p scaling coefficients i of each component and also by p p 1 coefficients ij ij for i j 1 p with j i allowing two components to be more or less strongly correlated and allowing the process to be time reversible or not we investigate the use of discrete filtering techniques to estimate jointly or separately the different parameters and prove the efficiency of the methodology with a simulation study and the derivation of asymptotic results

laguerre 2d functions and their application in quantum optics a set of orthonormalized and complete functions of two real variables in complex representation involving laguerre polynomials as a substantial part is introduced and is referred to as the set of laguerre two dimensional 2d functions the properties of the set of laguerre 2d functions are discussed and the fourier and radon transforms of these functions are calculated the laguerre 2d functions form a basis for a realization of the five dimensional lie algebra to the heisenberg weyl group for a two mode system real representation of this set of functions by a sum over products of hermite functions involving jacobi polynomials at the zero argument as coefficients is derived and it leads to new connections between laguerre and hermite polynomials in both directions the set of laguerre 2d functions is the most appropriate set of functions for the fock state representation of quasi probabilities in quantum optics the wigner quasi probability in fock state representation is up to a factor and an argument scaling directly given for each matrix element by a corresponding laguerre 2d function the properties of orthonormality and completeness of the laguerre 2d functions provide the fock state matrix elements of the density operator directly from the quasi probabilities the perina mista representation of the glauber sudarshan quasi probability can be represented with advantage by the laguerre 2d functions the fock state matrix elements of the displacement operator and the scalar product of displaced fock states are closely related to laguerre 2d functions

fractal connectivity of long memory networks using the multivariate long memory lm model and taylor expansions we find the conditions for convergence of the wavelet correlations between two lm processes on an asymptotic value at low frequencies these mathematical results and a least squares estimator of lm parameters are validated in simulations and applied to neurophysiological human brain and financial market time series both brain and market systems had multivariate lm properties including a fractal connectivity regime of scales over which wavelet correlations were invariantly close to their asymptotic value this analysis provides efficient and unbiased estimation of long term correlations in diverse dynamic networks

asymptotic formulae for two variable hermite polynomials new asymptotic formulae for the two dimensional hermite polynomials with large values of indices are found the applications to the photon distribution functions in squeezed one mode mixed quantum states are considered

weighted poincare and sobolev inequalities and estimates for weighted peano maximal functions 1 introduction in this paper we derive local poincare and sobolev inequalities involving two weight functions the estimates we obtain include those proved by fabes kenig and serapioni in 8 for the one weight function case we also consider the notion of weighted peano derivatives and show that the peano derivative formed with a given weight function can be estimated in terms of a maximal function involving a second weight function in the special case of ordinary lebesgue measure results of this kind lead to the theorems of a p calderon and a zygmund about local differentiability of functions in sobolev spaces and of functions which are of bounded variation in the sense of tonelli see 3 and 4 all our results are based on variants of the basic estimate in 5 but we do not presuppose any facts from 5 let v x be a weight function on r i e let v x be nonnegative and locally integrable with respect to lebesgue measure we shall use the notation v e se v x dx ordinary lebesgue measure will be denoted ie l if bh x is the open ball with center x and radius h we say v satisfies the doubling condition with respect to lebesgue measure if

incomplete 2d hermite polynomials properties and applications incomplete forms of two variable two index hermite polynomials are introduced their link with laguerre polynomials is discussed and it is shown that they are a useful tool to study quantum mechanical harmonic oscillator entangled states the possibility of developing the theory of complete 2d hermite polynomials from the point of view of the incomplete forms is analyzed too the orthogonality properties of the associated harmonic oscillator functions are finally discussed

an improvement of dimension free sobolev imbeddings in r i spaces abstract we prove a dimension invariant imbedding estimate for sobolev spaces of first order into a small lebesgue space and we establish the optimality of its fundamental function namely for any 1 p the inequality f y p 0 1 c p f l p b n f w 0 1 p b n n p where y p 0 1 is a rearrangement invariant banach function space independent of the dimension n b n is the ball in r n of measure 1 and c p is a constant independent of n is satisfied by the small lebesgue space l p p 2 0 1 moreover we show that the smallest space y p 0 1 in the sense of the continuous imbedding such that is true has the fundamental function equivalent to that of l p p 2 0 1 as a byproduct of our results we get that the space l p log l p 2 is optimal in the framework of the orlicz spaces satisfying

fractional integrals on weighted and spaces we characterize the pairs of doubling weights u v on rn such that iiwiihj ch hj 0 0 is the fractional integral operator we also consider the behavior of an associated maximal function applications of the results to sobolev inequalities in weighted lp spaces are given 1 introduction and notation in this paper we will characterize the pairs of doubling weights it v on rn such that pafwm 0 is the fractional integral operator we shall also give applications of these results to sobolev inequalities in weighted lp spaces a weight function u is said to belong to ds p 1 if u tq 1 and every cube q c rn where u q denotes the u measure of q and tq is the cube with the same center as q but with t times the edgelength we write dx j idu analogously u g rdv v 0 if u tq ctnuu q for every t 1 and every cube q it is not hard to see that if u g d then u g rdv for some v 0 we write u rhr r 1 if 1 1 r 1 rprr u x t dx i x dx 0 and e s then we denote as usual f x t f 0t x where t x t n x t and n f x sup f y t vfithta x y t yern t 0 x y 0

on limit relations between some families of bivariate hypergeometric orthogonal polynomials in this paper we deal with limit relations between bivariate hypergeometric polynomials we analyze the limit relation from trinomial distribution to bivariate gaussian distribution obtaining the limit transition from the second order partial difference equation satisfied by bivariate hypergeometric kravchuk polynomials to the second order partial differential equation verified by bivariate hypergeometric hermite polynomials as a consequence the limit relation between both families of orthogonal polynomials is established a similar analysis between bivariate hahn and bivariate appell orthogonal polynomials is also presented

fock expansion of multimode pure gaussian states the fock expansion of multimode pure gaussian states is derived starting from their representation as displaced and squeezed multimode vacuum states the approach is new and appears to be simpler and more general than previous ones starting from the phase space representation given by the characteristic or wigner function fock expansion is performed in terms of easily evaluable two variable hermite kampe de feriet polynomials a relatively simple and compact expression for the joint statistical distribution of the photon numbers in the different modes is obtained in particular this result enables one to give a simple characterization of separable and entangled states as shown for two mode and three mode gaussian states

general hermite and laguerre two dimensional polynomials general hermite and laguerre two dimensional 2d polynomials which form a complex three parameter unification of the special hermite and laguerre 2d polynomials are defined and investigated the general hermite 2d polynomials are related to the two variable hermite polynomials but are not the same the advantage of the newly introduced hermite and laguerre 2d polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the two variable hermite polynomials two different sets of such polynomials are introduced which are biorthonormal to each other the matrix which plays a role in the new definition of hermite and laguerre 2d polynomials is in a considered sense the square root of the matrix which plays a role in the definition of two variable hermite polynomials two essentially different explicit representations of the hermite and laguerre 2d polynomials are derived where the first involves jacobi polynomials as coefficients in superpositions of special hermite or laguerre 2d polynomials and the second is a superposition of products of two hermite polynomials with decreasing indices and with coefficients related to the special laguerre 2d polynomials generating functions are derived for the hermite and laguerre 2d polynomials

pointwise symmetrization inequalities for sobolev functions and applications we develop a technique to obtain new symmetrization inequalities that provide a unified framework to study sobolev inequalities concentration inequalities and sharp integrability of solutions of elliptic equations

a characterization of two weight norm inequalities for fractional and poisson integrals caracterisation de deux inegalites de normes ponderees pour les integrales fractionnaires et de poisson applications aux operateurs differentiels elliptiques degeneres

an invariant class of wave packets for the wigner transform abstract generalised hagedorn wave packets appear as exact solutions of schrodinger equations with quadratic possibly complex potential and are given by a polynomial times a gaussian we show that the wigner transform of generalised hagedorn wave packets is a wave packet of the same type in phase space the proofs build on a parametrisation via lagrangian frames and a detailed analysis of the polynomial prefactors including a novel laguerre connection our findings directly imply the recently found tensor product structure of the wigner transform of hagedorn wave packets

implicit summation formulae for hermite and related polynomials in this article we derive some implicit summation formulae for hermite and related polynomials by using different analytical means on their respective generating functions

transformations of laguerre 2d polynomials with applications to quasiprobabilities laguerre 2d polynomials are defined and their properties are investigated the laguerre 2d functions introduced in 1 2 are related to the laguerre 2d polynomials in such a way that they also include the weight function for the orthonormalization of the laguerre 2d polynomials a one parameter group of transformations applicable to certain classes of polynomials and discrete sets of functions is investigated and applied in particular to hermite polynomials and to laguerre 2d polynomials these transformations allow us to represent the polynomials of the corresponding classes by superpositions of the same kind of polynomials with stretched arguments they contain limiting cases with delta functions and their derivatives and lead to regularized representations of the delta functions and their derivatives as demonstrated for hermite and laguerre 2d polynomials applications of the laguerre 2d polynomials and 2d functions and their transformations to problems of quantum optics as the representation of quasiprobabilities in the fock state basis and by normally and otherwise ordered moments are considered the inversion of these representations is obtained in all cases a restricted design of quasiprobabilities should become possible

a two weight weak type inequality for fractional integrals for 1 p q oo 0 a n and w x v x nonnegative weight functions on r 1 we show that the weak type inequality f xw x dx a q if x l v x dx holds for allf 0 if and only if f ta xqw x v x i dx p 1b aw cc for all cubes q in rn here t7 denotes the fractional integral of order a ta f x f i x y i f y dy more generally we can replace ta by any suitable convolution operator with radial kernel decreasing in i x i

non linear wavelet regression and branch bound optimization for the full identification of bivariate operator fractional brownian motion self similarity is widely considered the reference framework for modeling the scaling properties of real world data however most theoretical studies and their practical use have remained univariate operator fractional brownian motion ofbm was recently proposed as a multivariate model for self similarity yet it has remained seldom used in applications because of serious issues that appear in the joint estimation of its numerous parameters while the univariate fractional brownian motion requires the estimation of two parameters only its mere bivariate extension already involves seven parameters that are very different in nature the present contribution proposes a method for the full identification of bivariate ofbm i e the joint estimation of all parameters through an original formulation as a non linear wavelet regression coupled with a custom made branch bound numerical scheme the estimation performance consistency and asymptotic normality is mathematically established and numerically assessed by means of monte carlo experiments the impact of the parameters defining ofbm on the estimation performance as well as the associated computational costs are also thoroughly investigated

on dimension free sobolev imbeddings i abstract we prove dimension invariant imbedding theorems for sobolev spaces using the gross logarithmic inequality

modes in n dimensional first order systems properties of the extended hermite polynomials are studied it is proved apparently for the first time that these functions times gaussian functions are the modes of an n dimensional first order system thus generalizing an earlier result due to hardy sheng and siegman for one dimension misalignment nonlosslessness and the exact normalization constant are explicitedly included in the treatment phase conjugators are also analyzed

generalized zernike or disc polynomials we investigate generalized zernike or disc polynomials p m n z z which are orthogonal 2d polynomials in the unit disc 0 zz 1 is a free parameter these polynomials can be expressed by jacobi polynomials of transformed arguments in connection with a simple angle dependence a limiting procedure leads to laguerre 2d polynomials lm n z z furthermore we introduce the corresponding orthonormalized disc functions the disc polynomials and disc functions obey two differential equations a first order and a second order one with a certain degree of freedom and the operators of lowering and raising of the indices are found these operators can be closed to a lie algebra su 1 1 su 1 1 new generating functions are derived from an operational representation which is alternative to the rodrigues type representation the one dimensional analogue of the disc polynomials which are orthogonal polynomials in the interval 0 r 1 with weight factors 1 r2 x are ultraspherical or gegenbauer polynomials in a new standardization the lowering and raising operators to the corresponding orthonormalized functions form a simple su 1 1 lie algebra this is given in the appendix in sketched form

carleson measures and multipliers of dirichlet type spaces a function p from 0 1 onto itself is a dirichlet weight if it is increasing p 0 and limr o x p x 0 the corresponding dirichlettype space dp consists of those bounded holomorphic functions on u z e c izl 1 such that if z 12p 1 izi is integrable with respect to lebesgue measure on u we characterize in terms of a carleson type maximal operator the functions in the set of pointwise multipliers of dp m dp g u c gf e dp vf e dp

new relations for two dimensional hermite polynomials the effective formulas reducing the two dimensional hermite polynomials to the classical one dimensional orthogonal polynomials by jacobi gegenbauer legendre laguerre and hermite are given new one parameter generating functions for the diagonal multidimensional hermite polynomials are derived the factorial moments and cumulants of the distribution functions related to the hermite polynomials of two variables with equal indices are expressed in terms of the legendre and chebyshev polynomials asymptotical formulas for the two dimensional polynomials with large values of indices and zero arguments are found the applications to the squeezed one mode states and to the time dependent quantum harmonic oscillator are considered

bivariate raising and lowering differential operators for eigenfunctions of a 2d fourier transform we define a two dimensional 2d fourier transform that self reproduces a one parameter family of bivariate hermite functions these are eigenfunctions of a hamiltonian differential operator of second order whose exponential is that transform we find explicit forms of the bivariate raising and lowering partial differential operators of first degree for the eigenfunctions of this 2d fourier transform

hermite and laguerre 2d polynomials abstract we define hermite 2d polynomials hm n u x y and laguerre 2d polynomials l m n u z z as functions of two variables with an arbitrary 2d matrix u as parameter and discuss their properties and their explicit representation recursion relations and generating functions for these polynomials are derived the advantage of the introduced hermite and laguerre 2d polynomials in comparison to the related usual two variable hermite polynomials is that they satisfy orthogonality relations in a direct way whereas for the purpose of orthonormalization of the last one has to introduce two different kinds of such polynomials which are biorthogonal to each other