on the multifractal nature of fully developed turbulence and chaotic systems it is generally argued that the energy dissipation of three dimensional turbulent flow is concentrated on a set with non integer hausdorff dimension recently in order to explain experimental data it has been proposed that this set does not possess a global dilatation invariance it can be considered to be a multifractal set the authors review the concept of multifractal sets in both turbulent flows and dynamical systems using a generalisation of the beta model

legendre multi galerkin methods for fredholm integral equations with weakly singular kernel and the corresponding eigenvalue problem abstract in this paper we consider legendre multi galerkin methods to solve fredholm integral equations of the second kind with weakly singular kernel and the corresponding eigenvalue problem we obtain the convergence rates for the approximated solution and iterated solution in weakly singular fredholm integral equations of the second kind in both l 2 and infinity norm we also establish error bounds of approximated eigenelements with exact eigenelements in the eigenvalue problem of a compact integral operator with weakly singular kernel in both infinity and l 2 norm numerical examples are presented to prove the theoretical estimates

lacunarity and multifractal analysis of the large dla mass distribution abstract we show the methodology used to analyze fractal and mass multifractal properties of very large diffusion limited aggregation dla clusters with a maximum of 10 9 particles for 2d aggregates and 10 8 particles for 3d clusters to support our main result the scaling behavior obtained by our experimental results corresponds to the expected performance of monofractal objects in order to estimate lacunarity measures for large dla clusters we develop a variant of the gliding box algorithm which reduces the computer time needed to obtain experimental results we show how our mass multifractal data have a tendency to present monofractal behavior for the mass distribution of the cases presented in this paper in the limit of very large clusters lacunarity analysis shows provided we study small clusters mass distributions data which might be interpreted as two different values of fractal dimensions while the cluster grows however this effect tends to vanish when the cluster size increases further in such a way that monofractality is achieved the outcomes of this paper lead us to conclude that the previously reported mass multifractality behavior vicsek et al 1990 13 detected for dla clusters is a consequence of finite size effects and floating point precision limitations and not an intrinsic feature of the phenomena since the scaling behavior of our dla clusters space corresponds to monofractal objects being this situation remarkably noticeable in the limit of very large clusters

product integration in the presence of a singularity this paper considers the approximate evaluation of int a b k x f x dx where k is a fixed lebesgue integrable function by quadrature rules of the form sigma i 0 m n w in f x in normally rules of this kind are used only for smooth functions f and any singularities are incorporated into k here however we allow f to have a singularity either at an endpoint or in the interior general convergence properties are established for a wide class of product integration rules more detailed results are established for the class of rules which are exact if f belongs to a specified family of piecewise polynomials

multifractality of space filling bearings and apollonian packings we study the structure of space filling bearings sfb and apollonian packings ap by determining the scaling behavior of the density distribution of points where the circles touch each other application of the sand box method to the average of theqth power of the number of touching points reveals the multifractal aspects of the structure of sfb and ap

on the convergence of spline product quadratures for cauchy principal value integrals abstract in a recent paper this journal 1990 the authors proposed product integration formulas for the numerical evaluation of the cauchy principal value integral f1 1u x f x x dx based on cubic spline interpolation of f and obtained convergence results for functions f ck 1 1 k 1 2 or 3 in this report the same rules are considered and their convergence is investigated for a larger class of functions f an error bound and some uniform convergence results are established in the case of equally spaced quadrature nodes for functions f satisfying a holder condition of order on 1 1 0

sequency ordered complex hadamard transform properties computational complexity and applications in this paper the generation of sequency ordered complex hadamard transform scht based on the complex rademacher matrices is presented the exponential form of scht is also derived and the proof for the unitary property of scht is given using the sparse matrix factorization the fast and efficient algorithm to compute the scht transform is developed and its computation load is described certain properties of the scht matrices are derived and analyzed with the discussion of scht applications in spectrum analysis and image watermarking relations of scht with fast fourier transform fft and unified complex hadamard transform ucht are discussed

on the rate distortion performance and computational efficiency of the karhunen loeve transform for lossy data compression we examine the rate distortion performance and computational complexity of linear transforms for lossy data compression the goal is to better understand the performance complexity tradeoffs associated with using the karhunen loeve transform klt and its fast approximations since the optimal transform for transform coding is unknown in general we investigate the performance penalties associated with using the klt by examining cases where the klt fails developing a new transform that corrects the klt s failures in those examples and then empirically testing the performance difference between this new transform and the klt experiments demonstrate that while the worst klt can yield transform coding performance at least 3 db worse than that of alternative block transforms the performance penalty associated with using the klt on real data sets seems to be significantly smaller giving at most 0 5 db difference in our experiments the klt and its fast variations studied here range in complexity requirements from o n sup 2 to o n log n in coding vectors of dimension n we empirically investigate the rate distortion performance tradeoffs associated with traversing this range of options for example an algorithm with complexity o n sup 3 2 and memory o n gives 0 4 db performance loss relative to the full klt in our image compression experiments

optimality of klt for high rate transform coding of gaussian vector scale mixtures application to reconstruction estimation and classification the karhunen loeacuteve transform klt is known to be optimal for high rate transform coding of gaussian vectors for both fixed rate and variable rate encoding the klt is also known to be suboptimal for some non gaussian models this paper proves high rate optimality of the klt for variable rate encoding of a broad class of non gaussian vectors gaussian vector scale mixtures gvsm which extend the gaussian scale mixture gsm model of natural signals a key concavity property of the scalar gsm same as the scalar gvsm is derived to complete the proof optimality holds under a broad class of quadratic criteria which include mean squared error mse as well as generalized f divergence loss in estimation and binary classification systems finally the theory is illustrated using two applications signal estimation in multiplicative noise and joint optimization of classification reconstruction systems

hybrid collocation methods for eigenvalue problem of a compact integral operator with weakly singular kernel abstract in this paper we consider the hybrid collocation methods to solve the eigenvalue problem of a compact integral operator with weakly singular kernels of algebraic and logarithmic type we obtain the global convergence rates for eigenvalues the gap between the spectral subspaces and iterated eigenvectors the numerical examples are presented to verify the theoretical estimates and also shown that this method is computationally useful in comparison to other methods

remarks on some properties of geometrical multifractals abstract in a porous medium the local density often exhibits spatial variations these variations can be characterized by a multifractal spectrum as long as suitable scaling characteristics are present we derive some of the properties of geometrical multifractals and show with examples how such objects can be constructed

interpolary product integration in the presence of singularities l 2 theory we study interpolatory product integration rules based on gauss radau and lobatto points with respect to a generalized jacobi weight function as applied to the integration of functions with singularities we give sufficient conditions for convergence of such rules for both endpoint and interior singularities

a lagrangian formulation of zador s entropy constrained quantization theorem zador s 1963 1966 classic result for the asymptotic high rate behavior of entropy constrained vector quantization is recast in a lagrangian form which better matches the lloyd algorithm used to optimize such quantizers the equivalence of the two formulations is shown and the result is proved for source distributions that are absolutely continuous with respect to the lebesgue measure which satisfy an entropy condition thereby generalizing the conditions stated by zador under which the result holds

numerical integration in the presence of an interior singularity abstract the convergence of sequences of integration rules applied to an improperly integrable function with an interior singularity is investigated the rules studied include the gauss jacobi rules and the interpolatory integration rules based on the zeros of 1 x 2 m p n 2m x m 0 1 for certain values of and the results are then applied to the study of the convergence of hunter s method for cauchy principal value integrals

asymptotics for the remainder of a class of positive quadratures for integrands with an interior singularity for functions with an interior singularity the errors of a class of positive quadrature formulae with high algebraic degree are reduced to those of the much simpler euler maclaurin type formulae applying this method to certain classes of functions such as for example f x h x x u where 1 with a sufficiently smooth functionh we obtain the main term of the error expansion for quadrature rules of ultraspherical type

gauss type quadratures for weakly singular integrals and their application to fredholm integral equations of the second kind in this paper we establish gauss type quadrature formulas for weakly singular integrals an application of the quadrature scheme is given to obtain numerical solutions of the weakly singular fredholm integral equation of the second kind we call this method a discrete product integration method since the weights involved in the standard product integration method are computed numerically

transform coding with backward adaptive updates the karhunen loeve transform klt is optimal for transform coding of a gaussian source this is established for all scale invariant quantizers generalizing previous results a backward adaptive technique for combating the data dependence of the klt is proposed and analyzed when the adapted transform converges to a klt the scheme is universal among transform coders a variety of convergence results are proven

numerical integration based on a multilevel quartic quasi interpolation operator quasi interpolation is very important in the study of the scattered data approximation numerical integration and numerical solutions of differential equations in this paper we proposed a multilevel quasi interpolation scheme by using b spline basis functions on quartic spline space moreover the proposed scheme is applied to the numerical integration of two dimensional singular integrals some numerical results with contrast to the existing methods are provided

properties of interpolatory product integration rules this paper is concerned with the approximate evaluation of int 1 1 k x f x d x where k is lebesgue integrable and f is at least riemann integrable and preferably smooth the integral is approximated by a rule of the form sum nolimits i 1 n w ni k f x ni where the points x ni are chosen in some prescribed way and the weights w ni k are such that the rule is exact if f is any polynomial of degree n for suitable choices of the points and suitable functions k the rule is shown to converge to the exact integral and the companion rule sum w ni k f x ni to int k x f x dx for all riemann integrable functions f the companion property ensures for example that the weights have the asymptotic positivity property if k is positive error bounds are given that guarantee rapid convergence of the rule if f is smooth finally by establishing a connection with mean convergence of lagrangian interpolation some negative theorems are proved in pa

geometrical multifractality of growing structures the authors study the mass distribution of fractal objects growing by subsequent addition of units to the structure it is demonstrated that in the limit of very large sizes such structures generally exhibit multifractal properties without defining a singular measure on the fractal support using the mass as a measure on the fractal for a few simple deterministic processes they give the spectrum of corresponding fractal dimensions exactly according to their results such geometrical multifractals should be very common and consist of parts which have a varying local fractal scaling of mass different from that of the whole object they demonstrate the variation of multifractal properties as a function of the increasing size in this way establishing for the calculation of multifractality an approach common in the studies of fractal properties of growing clusters

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

asymptotic expansions for trapezoidal type product integration rules this paper is concerned with the numerical integration of functions by piecewise polynomial product integration rules followed by application of extrapolation procedures the studied rules can be considered as generalizations of the conventional trapezoidal rule euler maclaurin type asymptotic expansions are obtained with only even powers furthermore numerical examples are given in order to show the effectiveness of these methods and a comparison with rules of similar characteristics is also made

rates of convergence of gaussian quadrature for singular integrands the authors obtain the rates of convergence or divergence of gaussian quadrature on functions with an algebraic or logarithmic singularity inside or at an endpoint of the interval of integration a typical result is the following for a bounded smooth weight function on 1 1 the error in n point gaussian quadrature of f x ix y1 1 is o n 2 28 if y 1 and o n if y e 1 1 provided we avoid the singularity if we ignore the singularityy the error is o n1 23 log n 8 loglog n 8t 1 for almost all choices of y these assertions are sharp with respect to order

a product quadrature algorithm by hermite interpolation this paper is concerned with numerical integration of 1 1f x k x dx by product integration rules based on hermite interpolation special attention is given to the kernel k x ei x with a view to providing high precision rules for oscillatory integrals convergence results and error estimates are obtained in the case where the points of integration are zeros of pn w x or of 1 x2 pn 2 w x where pn w x n 0 1 2 are the orthonormal polynomials associated with a generalized jacobi weight w further examples are given that test the performance of the algorithm for oscillatory weight functions

multiresolution graph fourier transform for compression of piecewise smooth images piecewise smooth pws images e g depth maps or animation images contain unique signal characteristics such as sharp object boundaries and slowly varying interior surfaces leveraging on recent advances in graph signal processing in this paper we propose to compress the pws images using suitable graph fourier transforms gfts to minimize the total signal representation cost of each pixel block considering both the sparsity of the signal s transform coefficients and the compactness of transform description unlike fixed transforms such as the discrete cosine transform we can adapt gft to a particular class of pixel blocks in particular we select one among a defined search space of gfts to minimize total representation cost via our proposed algorithms leveraging on graph optimization techniques such as spectral clustering and minimum graph cuts furthermore for practical implementation of gft we introduce two techniques to reduce computation complexity first at the encoder we low pass filter and downsample a high resolution hr pixel block to obtain a low resolution lr one so that a lr gft can be employed at the decoder upsampling and interpolation are performed adaptively along hr boundaries coded using arithmetic edge coding so that sharp object boundaries can be well preserved second instead of computing gft from a graph in real time via eigen decomposition the most popular lr gfts are pre computed and stored in a table for lookup during encoding and decoding using depth maps and computer graphics images as examples of the pws images experimental results show that our proposed multiresolution gft scheme outperforms h 264 intra by 6 8 db on average in peak signal to noise ratio at the same bit rate

numerical integration based on approximating splines abstract convergence results are proved for product integration rules based on approximating splines these results are both for bounded and unbounded integrands pointwise and uniform convergence results are proved for sequences of cauchy principal values of these approximating splines

on the evaluation of cauchy principal value integrals by rules based on quasi interpolating splines abstract in this paper we consider the numerical evaluation of one dimensional cauchy principal value integrals of the form 1 1 k x f x x d x 1 by rules obtained by subtracting out the singularity and then applying quadrature formulas based on quasi interpolating splines

robust quantization of memoryless sources using dispersive fir filters an approach to quantizing discrete time memoryless sources is presented an important feature is that its performance is largely insensitive to errors in modeling the input pdf the method involves changing the amplitude distribution of the source to be approximately gaussian by all pass filtering then applying a lloyd max quantizer designed for a gaussian source after quantization the samples are passed through another all pass filter which is an approximate inverse of the first filter the mean square error mse for the overall process is roughly equal to the quantization mse for the intermediate gaussian signal independent of the source statistics for some sources this is actually an improvement over direct correct model lloyd max quantization the cost of this technique is some delay due to filtering

directed recursion models for fractal growth fractals constructed by recursive processes are introduced to model growth phenomena these fractals are simultaneously directed and self similar in analogy with patterns growing under diffusion limited conditions the multifractal nature of the harmonic measure associated with laplacian interfaces is qualitatively interpreted using the models calculation of the largest singularity exponent allows one to make conclusions about the behaviour of diffusion limited aggregates

generalized triangular decomposition in transform coding a general family of optimal transform coders tcs is introduced here based on the generalized triangular decomposition gtd developed by jiang this family includes the karhunen loeve transform klt and the generalized version of the prediction based lower triangular transform plt introduced by phoong and lin as special cases the coding gain of the entire family with optimal bit allocation is equal to that of the klt and the plt even though the original plt introduced by phoong is not applicable for vectors that are not blocked versions of scalar wide sense stationary processes the gtd based family includes members that are natural extensions of the plt and therefore also enjoy the so called minlab structure of the plt which has the unit noise gain property other special cases of the gtd tc are the geometric mean decomposition gmd and the bidiagonal decomposition bid transform coders the gmd tc in particular has the property that the optimum bit allocation is a uniform allocation this is because all its transform domain coefficients have the same variance implying thereby that the dynamic ranges of the coefficients to be quantized are identical

product integration of singular integrands using quasi interpolatory splines we consider product integrals of the form kf 11k x f x dx where kf l1 but f is unbounded in the integration interval we shall approximate f by a sequence of quasi interpolatory splines qnf involving only function values and study the convergence of i kqnf to i kf it turns out that we can ignore an endpoint singularity but must in general avoid an interior singularity

product integration of piecewise continuous integrands based on cubic splineinterpolation at equally spaced nodes in this paper we consider the approximate evaluation of int limits a b k x f x dx wherek x is a fixed lebesgue integrable function by product formulas of the form sum limits i 0 n w i f x i based on cubic spline interpolation of the functionf generally whenever it is possible product quadratures incorporate the bad behaviour of the integrand in the kernelk here however we allowf to have a finite number of jump discontinuities in a b convergence results are established and some numerical applications are given for a logarithmic singularity structure in the kernel

a theoretical high rate analysis of causal versus unitary online transform coding backward adaptive or online transform coding tc of gaussian sources is investigated the karhunen loe spl grave ve transform klt unitary approach and the causal transform ct causal approach are compared in this context when the covariance matrix r sub x of the source is used in the tc scheme klt and ct present similar coding gains at high rates the aim of this study is to model analytically the behavior of these two coding structures when the ideal tc scheme gets perturbed that is when only a perturbed value r sub x spl delta r is known at the encoder in the online tc schemes considered here this estimate is used to compute both the transform and the bit assignment spl delta r is caused by two noise sources estimation noise finite set of available data at the encoder and quantization noise quantized data at the decoder furthermore not only does the transformation itself get perturbed but also the bit assignment does as well in this framework theoretical expressions for the coding gains in both the unitary and the causal cases are derived under the high rate assumption

on random walks in percolation models random walks on percolation clusters of any dimension consisting of both directed and undirected bonds are studied results conjectured to be exact and supported by monte carlo simulation are deduced for the critical parameters of the problem it is shown that in the undirected case of the model the probabilities of finding the particle in the sites of a finite cluster have a uniform limit distribution

spline product quadrature rules for cauchy singular integrals abstract in this paper product quadrature rules based on cubic spline interpolation are obtained for the numerical evaluation of cauchy singular integrals and an error bound is proposed some numerical examples that test the performance of the rules are given

on the convergence of closed interpolatory integration rules based on the zeros of gegenbauer polynomials abstract the gegenbauer polynomials c n x are orthogonal with respect to the weight function 1 x2 1 2 it is known that the interpolatory integration rules based on the zeros of 1 x2 c n 2 x converge for all riemann integrable functions for 1 2 4 this is shown to hold also for 1 2 1 2

numerical evaluation of cauchy principal value integrals with singular integrands convergence results are proved for sequences of interpolatory integration rules for cauchy principal value integrals of the form fk x f x x a dx 1 a 1 when f x is singular at a point a 7 i and the singularity is ignored or avoided

a generalized lloyd type algorithm for adaptive transform coder design in this paper we establish a probabilistic framework for adaptive transform coding that leads to a generalized lloyd type algorithm for transform coder design transform coders are often constructed by concatenating an ad hoc choice of transform with suboptimal bit allocation and quantizer design instead we start from a probabilistic latent variable model in the form of a mixture of constrained gaussian mixtures from this model we derive an transform coder design algorithm which integrates optimization of all transform coder parameters an essential part this algorithm is our introduction of a new transform basis the coding optimal transform which unlike commonly used transforms minimizes compression distortion adaptive transform coders can be effective for compressing databases of related imagery since the high overhead associated with these coders can be amortized over the entire database for this work we performed compression experiments on a database of synthetic aperture radar images our results show that adaptive coders improve compressed signal to noise ratio snr by approximately 0 5 db compared with global coders coders that incorporated the coding optimal transform had the best snrs on the images used to develop the coder however coders that incorporated the discrete cosine transform generalized better to new images

deterministic models of fractal and multifractal growth abstract four deterministic models are presented in order to get more insight into the geometry and multifractal behavior associated with fractal growth phenomena these models allow for exact treatment and are used to demonstrate such properties as directed self affinity and self similarity and multifractal growth probability and mass distribution it is pointed out that in some cases the various fractal dimension definitions may lead to inconsistent results

piecewise polynomial quadratures for cauchy singular integrals in this paper we propose piecewise polynomial methods for the approximation of cauchy principal value integrals and develop a simple efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise polynomial quadratures we present two examples to illustrate the advantages of these quadratures versus the gauss jacobi quadratures

entropy encoding hilbert space and karhunen loeve transforms by introducing hilbert space and operators we show how probabilities approximations and entropy encoding from signal and image processing allow precise formulas and quantitative estimates our main results yield orthogonal bases which optimize distinct measures of data encoding

a new b spline collocation method for singular integro differential equations of higher orders abstract two numerical schemes based on b spline collocation method are presented for approximate solution of the systems of singular integro differential equations of higher orders appearing in fracture problems with steigmann ogden surface energy it is shown that the resulting systems of linear equations are well conditioned and the numerical examples are in good agreement with known solutions although this paper is concerned with the numerical solution to the specific equations the proposed method can be easily generalized to a large class of singular integro differential equations

convergence of product integration rules for functions with interior and end point singularities over bounded and unbounded intervals the convergence of product integration rules based on gaussian quadrature points is investigated for functions with interior and endpoint singularities over bounded and unbounded intervals the investigation is based on a new convergence result for lagrangian interpolation and gaussian quadrature of singular integrands

product integration of singular integrands based on cubic spline interpolation at equally spaced nodes in this paper the convergence of product integration rules based on cubic spline interpolation at equally spaced nodes with not a knot end condition is investigated for integrand functions with a interior or endpoint singularity in the integration interval

fracture propagation governed by the laplace equation we propose a model of fracture propagation when it is a two dimensional system and has displacement restricted to be perpendicular to the plane its displacement obeys the laplace equation because the diffusion limited aggregation dla is also governed by the laplace equation we can make use of many methods developed for dla to study our model monte carlo simulations are done and df of the model turns out to be about 1 55 0 05 it differs from that of dla moreover nagatani s renormalization is used to investigate the model and the multifractalities of both the growth probability and the mass distributions are found the generalized dimension dm q of the mass distribution is dm df dm 0 dm a fractal pattern is proposed to show the dependence of dm q upon q

rates of convergence of gauss lobattom and radau integration rules for singular integrands rates of convergence or divergence are obtained in the application of gauss lobatto and radau integration rules to functions with an algebraic or logarithmic singularity inside or at an endpoint of the interval of integration a typical result is the following for a generalized jacobi weight function on 1 1 the error in applying an n point rule to f x ix vi 6 is o n 2 28 if y 1 and o n 1 6 if y e 1 1 providedwe avoid the singularity if we ignore the singularity y the error is o n 1 28 log n 6 log log n 1 e for almost all choices of y these assertions are sharp with respect to order 1 introduction this paper is a sequel to that of lubinsky and myself 2 on the rates of convergence of gauss integration rules for singular integrands in this paper we extend the results in 2 in several directions on the one hand we extend the results for gauss integration rules to results for lobatto and radau rules this extension follows easily from the generalized markov stieltjes inequality for lobatto and radau rules given in lemma 3 2 and from a representation of the coefficients or weights or cotes numbers in the lobatto and radau rules in terms of the coefficients in the gauss rule with respect to a related weight function a second extension is to the special case of gauss jacobi integration rules as well as lobatto jacobi and radau jacobi rules for certain values of the parameters defining the jacobi weight function

suboptimality of the karhunen loeve transform for transform coding we examine the performance of the karhunen loeve transform klt for transform coding applications the klt has long been viewed as the best available block transform for a system that orthogonally transforms a vector source scalar quantizes the components of the transformed vector using optimal bit allocation and then inverse transforms the vector this paper treats fixed rate and variable rate transform codes of non gaussian sources the fixed rate approach uses an optimal fixed rate scalar quantizer to describe the transform coefficients the variable rate approach uses a uniform scalar quantizer followed by an optimal entropy code and each quantized component is encoded separately earlier work shows that for the variable rate case there exist sources on which the klt is not unique and the optimal quantization and coding stage matched to a worst klt yields performance as much as 1 5 db worse than the optimal quantization and coding stage matched to a best klt in this paper we strengthen that result to show that in both the fixed rate and the variable rate coding frameworks there exist sources for which the performance penalty for using a worst klt can be made arbitrarily large further we demonstrate in both frameworks that there exist sources for which even a best klt gives suboptimal performance finally we show that even for vector sources where the klt yields independent coefficients the klt can be suboptimal for fixed rate coding

convergence results for piecewise linear quadratures for cauchy principal value integrals etablissement des conditions de convergence simple et uniforme des quadratures lineaires par morceaux pour les integrales a valeur principale de cauchy

determination of fractal dimensions for geometrical multifractals abstract two independent approaches the box counting and the sand box methods are used for the determination of the generalized dimensions d q associated with the geometrical structure of growing deterministic fractals we find that the multifractal nature of the geometry results in an unusually slow convergence of the numerically calculated d q s to their true values our study demonstrates that the above mentioned two methods are equivalent only if the sand box method is applied with an averaging over randomly selected centres in this case the latter approach provides better estimates of the generalized dimensions

product integration of logarithmic singular integrands based on cubic splines abstract this paper is concerned with the practical evaluation of the product integral 1 1 f x k x d x for the case when k x in x 1 1 and f is bounded in 1 1 the approximation is a quadrature rule where the weights w n n i are chosen to be exact when f is given by a linear combination of a chosen set of functions n j in this paper the functions n j are chosen to be cubic b splines an error bound for product quadrature rules based on cubic splines is provided examples that test the performance of the product quadrature rules for different choices of the function are given a comparison is made with product quadrature rules based on first kind chebyshev polynomials

mean convergence of lagrange interpolation for freud s weights with application to product integration rules the connection between convergence of product integration rules and mean convergence of lagrange interpolation in l 1 p 00 has been thoroughly analysed by sloan and smith 37 motivated by this connection we investigate mean convergence of lagrange interpolation at the zeros of orthogonal polynomials associated with freud weights on r our results apply to the weights exp x 2 m 2 4 6 and for the hermite weight m 2 extend results of nevai 28 and bonan 2 in at least one direction the results are sharp in l 1 p g 2 as a consequence we can improve results of smith sloan and opie 38 on convergence of product integration rules based on the zeros of the orthogonal polynomials associated with the hermite weight in the process we prove a new markov stieltjes inequality for gauss quadrature sums and solve a problem of nevai on how to estimate certain quadrature sums

uniform convergence results for cauchy principal value integrals a general uniform convergence theorem for numerical integration of cauchy principal value integrals is proved seven special instances of this theorem are given as corollaries