generalizations of laguerre polynomials it is shown that the polynomials ln m0 m1 mn x n 0 defined by ln m0m1 mn x k 0n 1ak dkln x for certain real coefficients ak k 0n 1 are orthogonal with respect to the inner product f g 1 1 0 x e x f x g x dx v 0nmv f v 0 g v 0 where 1 n n and mv 0 for all v 0 1 2 n for these new polynomials n m0 m1 mn x n 0 an orthogonality relation and a second order differential equation are derived further we obtain a representation as a n 2fn 2 hypergeometric series and a 2n 3 terms recurrence relation which gives rise to a christoffel darboux type formula

computing orthogonal polynomials in sobolev spaces numerical methods are considered for generating polynomials orthogonal with respect to an inner product of sobolev type i e one that involves derivatives up to some given order each having its own positive measure associated with it the principal objective is to compute the coefficients in the increasing order recurrence relation that these polynomials satisfy by virtue of them forming a sequence of monic polynomials with degrees increasing by 1 from one member to the next as a by product of this computation one gains access to the zeros of these polynomials via eigenvalues of an upper hessenberg matrix formed by the coefficients generated two methods are developed one is based on the modified moments of the constitutive measures and generalizes what for ordinary orthogonal polynomials is known as modified chebyshev algorithm the other a generalization of stieltjes s procedure expresses the desired coefficients in terms of a sobolev inner product involving the orthogonal polynomials in question whereby the inner product is evaluated by numerical quadrature and the polynomials involved are computed by means of the recurrence relation already generated up to that point the numerical characteristics of these methods are illustrated in the case of sobolev orthogonal polynomials of old as well as new types based on extensive numerical experimentation a number of conjectures are formulated with regard to the location and interlacing properties of the respective zeros

on the analysis and interpretation of inhomogeneous quadratic forms as receptive fields in this letter we introduce some mathematical and numerical tools to analyze and interpret inhomogeneous quadratic forms the resulting characterization is in some aspects similar to that given by experimental studies of cortical cells making it particularly suitable for application to second order approximations and theoretical models of physiological receptive fields we first discuss two ways of analyzing a quadratic form by visualizing the coefficients of its quadratic and linear term directly and by considering the eigenvectors of its quadratic term we then present an algorithm to compute the optimal excitatory and inhibitory stimuli those that maximize and minimize the considered quadratic form respectively given a fixed energy constraint the analysis of the optimal stimuli is completed by considering their invariances which are the transformations to which the quadratic form is most insensitive and by introducing a test to determine which of these are statistically significant next we propose a way to measure the relative contribution of the quadratic and linear term to the total output of the quadratic form furthermore we derive simpler versions of the above techniques in the special case of a quadratic form without linear term in the final part of the letter we show that for each quadratic form it is possible to build an equivalent two layer neural network which is compatible with but more general than related networks used in some recent articles and with the energy model of complex cells we show that the neural network is unique only up to an arbitrary orthogonal transformation of the excitatory and inhibitory subunits in the first layer

generation and evaluation of orthogonal polynomials in discrete sobolev spaces i algorithms in this paper we study theoretically the determination and evaluation of polynomials that are orthogonal with respect to a general discrete sobolev inner product that is an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives this sobolev inner product has the property that the orthogonal polynomials with respect to it satisfy a linear recurrence relation of fixed order we provide a complete set of formulas to compute the coefficients of this recurrence besides we study the determination of the fourier sobolev coefficients of a finite approximation of a function and the numerical evaluation of the resulting finite series at a general point

emergence of phase and shift invariant features by decomposition of natural images into independent feature subspaces olshausen and field 1996 applied the principle of independence maximization by sparse coding to extract features from natural images this leads to the emergence of oriented linear filters that have simultaneous localization in space and in frequency thus resembling gabor functions and simple cell receptive fields in this article we show that the same principle of independence maximization can explain the emergence of phase and shift invariant features similar to those found in complex cells this new kind of emergence is obtained by maximizing the independence between norms of projections on linear subspaces instead of the independence of simple linear filter outputs the norms of the projections on such independent feature subspaces then indicate the values of invariant features

the search for differential equations for certain sets of orthogonal polynomials abstract we look for differential equations of the form i 0ci x y i x 0 where the coefficients ci x i 0 are continuous functions on the real line and where ci x i 1 are independent of n for the generalized jacobi polynomials p m nn x n 0 and for generalized laguerre polynomials l m nn x n 0 which are orthogonal with respect to an inner product of sobolev type we use a method involving computer algebra packages like maple and mathematica and we will give some preliminary results

sensitivity analysis for szegő polynomials szeg polynomials are orthogonal with respect to an inner product on the unit circle numerical methods for weighted least squares approximation by trigonometric polynomials conveniently can be derived and expressed with the aid of szeg polynomials this paper discusses the conditioning of several mappings involving szeg polynomials and thereby sheds light on the sensitivity of some approximation problems involving trigonometric polynomials

the finite volume element method for diffusion equations on general triangulations this paper develops discretization error estimates for the finite volume element method on general triangulations of a polygonal domain in mathcal r 2 using a special type of control volume the theory applies to diffusion equations of the form begin gathered nabla a nabla u f quad text in omega hfill u 0 quad text on partial omega hfill end gathered under fairly general conditions the theory establishes o h estimates of the error in a discrete mathcal h 1 seminorm under an additional assumption concerning local uniformity of the triangulation the estimate is improved to o h 2

slow feature analysis unsupervised learning of invariances invariant features of temporally varying signals are useful for analysis and classification slow feature analysis sfa is a new method for learning invariant or slowly varying features from a vectorial input signal it is based on a nonlinear expansion of the input signal and application of principal component analysis to this expanded signal and its time derivative it is guaranteed to find the optimal solution within a family of functions directly and can learn to extract a large number of decorrelated features which are ordered by their degree of invariance sfa can be applied hierarchically to process high dimensional input signals and extract complex features sfa is applied first to complex cell tuning properties based on simple cell output including disparity and motion then more complicated input output functions are learned by repeated application of sfa finally a hierarchical network of sfa modules is presented as a simple model of the visual system the same unstructured network can learn translation size rotation contrast or to a lesser degree illumination invariance for one dimensional objects depending on only the training stimulus surprisingly only a few training objects suffice to achieve good generalization to new objects the generated representation is suitable for object recognition performance degrades if the network is trained to learn multiple invariances simultaneously

inner products involving q differences the little q laguerre sobolev polynomials in this paper polynomials which are orthogonal with respect to the inner product p r s k 0 p qk r qk aq k aq q q q k k 0 dqp qk dqr qk aq k aq q q q k where dq is the q difference operator 0 0 q 1 and 0 aq 1 are studied for these polynomials algebraic properties and q difference equations are obtained as well as their relation with the monic little q laguerre polynomials some properties about the zeros of these polynomials are also deduced finally the relative asymptotics qn x pn x a q n on compact subsets of c 0 1 is given where qn x is the nth degree monic orthogonal polynomial with respect to the above inner product and pn x a q denotes the monic little q laguerre polynomial of degree n

on computational aspects of discrete sobolev inner products on the unit circle in this paper we show how to compute in o n 2 steps the fourier coefficients associated with the gelfand levitan approach for discrete sobolev orthogonal polynomials on the unit circle when the support of the discrete component involving derivatives is located outside the closed unit disk as a consequence we deduce the outer relative asymptotics of these polynomials in terms of those associated with the original orthogonality measure moreover we show how to recover the discrete part of our sobolev inner product

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

orthogonal polynomials on sobolev spaces old and new directions abstract during the last years orthogonal polynomials on sobolev spaces have attracted considerable attention algebraic properties distribution of their zeros and fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case in this paper we present an introductory survey about the subject the origin of the problems and their development show the interest and the promising future of this field

generation and evaluation of orthogonal polynomials in discrete sobolev spaces ii numerical stability in this paper we concern ourselves with the determination and evaluation of polynomials that are orthogonal with respect to a general discrete sobolev inner product that is an ordinary inner product on the real line plus a finite sum of atomic inner products involving a finite number of derivatives in a previous paper we provided a complete set of formulas to compute the coefficients of this recurrence here we study the numerical stability of these algorithms for the generation and evaluation of a finite series of sobolev orthogonal polynomials besides we propose several techniques for reducing and controlling the rounding errors via theoretical running error bounds and a carefully chosen recurrence

on real and complex zeros of orthogonal polynomials in a discrete sobolev space abstract let sn x c n denote a set of polynomials orthogonal with respect to the discrete sobolev inner product f g f x g x d x nf c g c where n 0 c r for n 0 put kn x sn x 0 then sn x c n has at least n 2 different real zeros their position with respect to the zeros of kn can be determined using the tangent to the graph of y kn x in c kn c on the other hand if n 3 then c can be chosen such that sn x c n has two complex zeros if n is sufficiently large

a new class of high order finite volume methods for second order elliptic equations in the numerical simulation of many practical problems in physics and engineering finite volume methods are an important and popular class of discretization methods due to the local conservation and the capability of discretizing domains with complex geometry however they are limited by low order approximation since most existing finite volume methods use piecewise constant or linear function space to approximate the solution in this paper a new class of high order finite volume methods for second order elliptic equations is developed by combining high order finite element methods and linear finite volume methods optimal convergence rate in h 1 norm for our new quadratic finite volume methods over two dimensional triangular or rectangular grids is obtained

sobolev orthogonal polynomials and m n coherent pairs of measures we introduce the notion of m n coherent pair of measures as a generalization of the concept of coherent pair of measures introduced by iserles et al a iserles p e koch s p norsett j m sanz serna on polynomials orthogonal with respect to certain sobolev inner products j approx theory 65 2 1991 151 175 and subsequently generalized by several authors a pair of measures email protected 0 email protected 1 is called m n coherent if the corresponding orthogonal polynomial sequences p n n and q n n resp satisfy a non zero structure relation such as i 0nr i np n i x email protected i 0ms i nq n i 1 x for all n 0 1 2 where m and n are fixed non negative integer numbers and r i n and s i n are given real parameters satisfying some natural conditions we prove that the regular moment linear functionals associated to an m n coherent pair are semiclassical and they are related by a rational modification in the usual sense of distribution theory we also discuss the converse statement under the assumption that email protected 0 email protected 1 form an m n coherent pair we study the sequence s n l n of the monic orthogonal polynomials with respect to the sobolev inner product l email protected email protected email protected f g email protected 1 defined in the space of all polynomials with real coefficients where l 0 an efficient algorithm is stated to compute the coefficients in the fourier sobolev type series f x email protected n 0 c n ls n l x with respect to l for suitable smooth functions f such that email protected l m 0 2 r and f l m 1 2 r finally some illustrative computational examples are presented

zero distribution of orthogonal polynomials in a certain discrete sobolev space abstract we study the zero distribution of the polynomials snn which are orthogonal with respect to the discrete sobolev inner product f g 0 x g x d x nf r 0 g r 0 where is a distribution function n 0 r 1 snn has n real simple zeros at most one of them is outside 0 the location of these zeros is given in relation to the position of the zeros of some classical polynomials i e polynomials with respect to an inner product with n 0

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

global properties of zeros for sobolev type orthogonal polynomials perez t e and m a pifiar global properties of zeros for sobolev type orthogonal polynomials journal of computational and applied mathematics 49 1993 22 5 232 in this paper we analyze some properties concerning the zeros of orthogonal polynomials q x associated to the inner product f g f xlg x dp x1 t mf c g c nf c g c where i is a not necessarily bounded real interval p is a positive measure on i c e w and m n 2 0 in particular some properties concerning the localization and separation for the roots of q x are obtained

orthogonal polynomials in matlab a suite of matlab programs has been developed as part of the book orthogonal polynomials computation and approximation oxford university press oxford 2004 by gautschi the package contains routines for generating orthogonal polynomials as well as routines dealing with applications in this paper a brief review of the first part of the package is given dealing with procedures for generating the three term recurrence relation for orthogonal polynomials and more general recurrence relations for sobolev orthogonal polynomials moment based methods and discretization methods and their implementation in matlab are among the principal topics discussed

a tool for locating zeros of orthogonal polynomials in sobolev inner product spaces abstract in the theory of polynomials orthogonal with respect to an inner product of the form f 0f x g x d x mk 1akf ik 0 g ik 0 one is confronted with the following situation for certain values of the parameters the orthogonal polynomial of degree n does not have all its zeros inside the support of the distribution function d this paper gives a method to investigate the zero distribution by looking at a type of limiting polynomial for the case m 2 it is shown that there are exactly two zeros outside the true interval of orthogonality for a1 a2 large moreover it is proved that these zeros are nonreal complex conjugates in the case i1 1 i2 also several examples are given

ratio and plancherel rotach asymptotics for meixner sobolev orthogonal polynomials abstract we study the analytic properties of the monic meixner sobolev polynomials qn orthogonal with respect to the inner product involving differences p q s i 0 p i q i p i q i i i i 0 0 where 0 is the forward difference operator f x f x 1 f x and n denotes the pochhammer symbol relative asymptotics for meixner sobolev polynomials with respect to meixner polynomials is obtained this relative asymptotics is also given for the scaled polynomials moreover a zero distribution for the scaled meixner sobolev polynomials and plancherel rotach asymptotics for qn are deduced