Multidimensional Analogues Research Articles

Extension and Completion of Wynn's Theory on Convergence of Columns of the Epsilon Table

Let {Sn}∞n=0be such thatSn∽S+∑∞j=1ajλnjasn→∞, with 1>|λ1|>|λ2|>…, such that limj→∞λj=0. A well-known result by Wynn states that when the Shanks transformation or its equivalentε-algorithm is applied to {Sn}∞n=0, thenε(n)2k−S∽ak+1[∏ki=1(λk+1−λi)/(1−λi)]2λnk+1asn→∞. In the present work we extend this result (i) by allowing some of theλjto have the same modulus and (ii) by replacing the constantsajby some polynomialsPj(n) inn. Sequences {Sn}∞n=0with these characteristics arise frequently, e.g., in fixed point iterative solution of linear systems and in trapezoidal rule approximation of finite range integrals with logarithmic endpoint singularities and their multidimensional analogues. The results of this work are obtained by exploiting the connection between the Shanks transformation and Padé approximants and by using some recent results of the author on Padé approximants for meromorphic functions.

Optimal recovery using thin plate splines in finite volume methods for the numerical solution of hyperbolic conservation laws

The theory of optimal recovery is applied to finite volume methods for the numerical solution of conservation laws in multiple space dimensions. Classical polynomial ENO (essentially non-oscillatory) algorithms can be interpreted as trivial recovery operators for the point functional in the light of this theory. Thin plate splines are identified as optimal recovery functions in Beppo-Levi spaces and can be seen as multi-dimensional analogues of cubic splines. Recovery algorithms of ENO-type based on thin plate splines are developed and applied to test problems including the Euler equations of compressible gas dynamics.

Reduction formulas associated with series involving arbitrary coefficients

In the present investigation two different classes of reduction formulas are derived in terms of series involving arbitrary coefficients. Multidimensional analogues are also obtained. Applications of the main results include various examples on reduction formulas incorporating the generalized hypergeometric functions, the Laguerre polynomials and certain multivariable special functions.*

Multidimensional generalized superalgebras

The multidimensional analogues of the one-dimensional generalized superalgebras are considered in the framework of quantum mechanics.

A partial upwind difference scheme for nonlinear parabolic equations

In this paper, we present an explicit partial upwind difference scheme for the problem: u t − du xx + f( u)) x + g( x, u) = 0, d > 0, 0 ⩽ x ⩽ 1, t ⩾ 0, with u( x, 0), u(0, t) and u(1, t) all prescribed. Formulas for the upwind factors are provided. We use the method of lower and upper solutions and the theory of M-matrices. The present method is more accurate and allows larger time steps than the corresponding one-sided method. We obtain the order of convergence for our scheme. We also prove the convergence of the time evolving solutions to the unique steady-state solution as the time approaches infinity. The method can be generalized to multi-dimensional analogues. To support the theory, numerical results are given.

A weighted difference scheme and monotone iterative methods for quasilinear boundary value problems

We consider the boundary value problem – y xx + f( x, y) y x + g( x, y) = 0, 0< x <1, y(0) = b 0, y(1) = b 1. We present a weighted two sided difference scheme to approximate the differential equation. Monotone iterative methods are proposed to solve the resulting nonlinear system. The scheme applies equally well to multidimensional analogues. Numerical results are included.

On the gaussian approximation of convolutions under multidimensional analogues of S.N. Bernstein's inequality conditions

On etudie le taux d'approximation de distribution de sommes de vecteurs aleatoires independants par des distributions gaussiennes correspondantes

Multidimensional Hardy inequalities and the absence of positive eigenvalues for the Schr�dinger operator with a complex potential

The operator −(d/dx)2+ν where ν is a complex function, |J(x)(1+|x|1+e|>∞, has no positive eigenvalues. This follows from the existence of a triangular Green function for the operator −(d/dx)z−λ,gl>0, and from Hardy type inequalities. We give the multidimensional analogues of these inequalities and we prove the absence of positive eigenvalues for the operator −Δ+ν.

MULTIDIMENSIONAL TAUBERIAN COMPARISON THEOREMS FOR GENERALIZED FUNCTIONS IN CONES

This article deals with the proofs of some multidimensional Tauberian comparison theorems for generalized functions with supports in homogeneous cones, in particular, for measures and functions whose Laplace transforms have nonnegative imaginary parts. Admissible generalized functions, which can be regarded as multidimensional analogues of the so-called R-O functions of Karamata, serve as comparison functions in these theorems. For circular and n-faced cones a criterion is obtained for admissibility which generalizes the well-known Keldysh Tauberian condition to the multidimensional case.Bibliography: 9 titles.

Lebesgue constants of multiple de la Vallée-Poussin sums over polyhedra

One considers linear summation methods for the multiple Fourier series the multidimensional analogues of the de la Vallé-Poussin sums. The summation of the Fourier series is carried out over the homotheties of an m-dimensional starshaped polyhedron Λ. It is shown that if Λ has rational vertices, then the Lebesgue constants of the considered methods, with the accuracy of O((p+1)−1. logm−1 (n+2)) are equal to where is the Fourier transform of the function ϕ. The exact value of the principal term of the Lebesgue constant is computed in two particular cases: 1) Λ is obtained from an m-dimensional cube by means of a linear nonsingular transformation; 2) ρ=0. Λ is an m-dimensional simplex.

SUPREMA OF FOURIER COEFFICIENTS ON CLASSES OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS OF SEVERAL VARIABLES

Equalities that are multidimensional analogues of the familiar Lebesgue-Nikol'skiĭ-Efimov relations are obtained for upper bounds of Fourier coefficients on classes of continuous and differentiable functions of several variables. Bibliography: 6 titles.

Nonlinear Singular Perturbation Problems and One Sided Difference Schemes

We consider the boundary value problem $\varepsilon y'' - a(y)y' - b(x,y) = F(x), - 1 \leqq x \leqq 1$, for $y( - 1)$ and $y(1)$ given and $b(x,0) \equiv 0$, $b_y (x,y) \geqq 0$. We construct a simple difference scheme approximating $\varepsilon u_{xx} - a(u)u_x - b(x,u) - F(x) = u_t $ for $t \geqq 0, - 1 \leqq x \leqq 1$ with the same boundary conditions and an arbitrary initial guess. As the numerical artificial time approaches infinity, we prove convergence to a unique steady state solution. As the mesh size $\Delta x$ and the parameter $\varepsilon $ approach zero in any order, we prove convergence to the solution of the corresponding O.D.E. We use one-sided differences to approximate the transport term. The method applies equally well to multidimensional analogues. We present numerical results verifying the theory.

Multidimensional analogues of Newton's formulas for systems of nonlinear algebraic equations and some of their applications

Multidimensional analogues of Newton's formulas for systems of nonlinear algebraic equations and some of their applications

An Extension of S. N. Bernstein’s Inequalities to Multidimensional Distributions

Previous article Next article An Extension of S. N. Bernstein’s Inequalities to Multidimensional DistributionsYu. V. ProkhorovYu. V. Prokhorovhttps://doi.org/10.1137/1113029PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. N. Bernstein, On a modification of Chebyshev's inequality and on the error in Laplace's formulaCollected Works, Vol. 4, Izd-vo `Nauka', Moscow, 1964, 71–80, (In Russian.) Google Scholar[2] S. N. Bernstein, Probability Theory, GTTI, Moscow, 1964, (In Russian.) Google Scholar[3] A. Kolmogoroff, Über das Gesetz des iterierten Logarithmus, Math. Ann., 101 (1929), 126–135 10.1007/BF01454828 MR1512520 CrossrefGoogle Scholar[4] Yu. V. Prokhorov, An extremal problem in probability theory, Theor. Probability Appl., 4 (1959), 201–203 10.1137/1104017 MR0121857 (22:12587) 0093.15102 LinkGoogle Scholar[5] S. Bennett, Probability inequalities for a sum of independent random variables, J. Amer. Statist. Assoc., 59 (1962), 33–45 CrossrefGoogle Scholar[6] Wassily Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58 (1963), 13–30 MR0144363 (26:1908) 0127.10602 CrossrefGoogle Scholar[7] A. A. Borovkov, Some inequalities for sums of multidimensional random variables, Theory Prob. Applications, 13 (1968), 156–160 10.1137/1113013 0196.19501 LinkGoogle Scholar[8] Yu. V. Prokhorov, Convergence of stochastic processes and limit theorems in probability theory, Theory Prob. Applications, 1 (1956), 157–214 10.1137/1101016 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Evaluating probabilistic forecasts of extremes using continuous ranked probability score distributionsInternational Journal of Forecasting, Vol. 7 | 1 Aug 2022 Cross Ref BibliographyFundamental Aspects of Operational Risk and Insurance Analytics | 23 January 2015 Cross Ref Sharp Bounds for the Tails of Functionals of Markov ChainsP. Bertail and S. ClémençonTheory of Probability & Its Applications, Vol. 54, No. 3 | 26 August 2010AbstractPDF (214 KB)A large deviation inequality for vector functions on finite reversible Markov ChainsThe Annals of Applied Probability, Vol. 17, No. 4 | 1 Aug 2007 Cross Ref The Exact and Asymptotic Distributions of Cramér-Von Mises StatisticsJournal of the Royal Statistical Society: Series B (Methodological), Vol. 58, No. 1 | 5 December 2018 Cross Ref On Exponential Bounds for Probabilities of Large DeviationsV. V. YurinskiiTheory of Probability & Its Applications, Vol. 37, No. 1 | 28 July 2006PDF (208 KB)Exponential Bounds for the Distribution of the Maximum of a Non-Gaussian Random FieldE. I. OstrovskiiTheory of Probability & Its Applications, Vol. 35, No. 3 | 17 July 2006AbstractPDF (1196 KB)A refinement of S. N. Bernstein's inequalityMathematical Notes of the Academy of Sciences of the USSR, Vol. 47, No. 6 | 1 Jun 1990 Cross Ref The Empirical Characteristic Process When Parameters Are EstimatedContributions to Probability | 1 Jan 1981 Cross Ref Probability inequalities for sums of absolutely regular processes and their applicationsZeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol. 43, No. 4 | 1 Jan 1978 Cross Ref Summary of Papers Presented at Sessions of the Probability and Statistics Section of the Moscow Mathematical Society (February–October 1974)Theory of Probability & Its Applications, Vol. 20, No. 2 | 17 July 2006AbstractPDF (1740 KB)Multidimensional Limit Theorems for Large DeviationsL. V. OsipovTheory of Probability & Its Applications, Vol. 20, No. 1 | 17 July 2006AbstractPDF (1239 KB)On Convergence in the $L_2$-Norm of Probability Density EstimatesE. A. NadarayaTheory of Probability & Its Applications, Vol. 18, No. 4 | 28 July 2006AbstractPDF (311 KB)On the S. N. Bernstein InequalityS. L. Blyumin and B. D. KotlyarTheory of Probability & Its Applications, Vol. 16, No. 1 | 17 July 2006AbstractPDF (287 KB)On Multidimensional Analogues of S. N. Bernstein’s InequalitiesA. V. ProkhorovTheory of Probability & Its Applications, Vol. 13, No. 2 | 28 July 2006AbstractPDF (931 KB)Some Remarks on Multidimensional Bernstein–Kolmogorov-Type InequalitiesV. M. ZolotarevTheory of Probability & Its Applications, Vol. 13, No. 2 | 28 July 2006AbstractPDF (458 KB)S. N. Bernstein’s Inequalities in the Multidimensional CaseA. V. ProkhorovTheory of Probability & Its Applications, Vol. 13, No. 3 | 17 July 2006AbstractPDF (652 KB) Volume 13, Issue 2| 1968Theory of Probability & Its Applications197-358 History Submitted:30 January 1968Published online:28 July 2006 InformationCopyright © 1968 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1113029Article page range:pp. 260-267ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On Multidimensional Analogues of S. N. Bernstein’s Inequalities

Previous article Next article On Multidimensional Analogues of S. N. Bernstein’s InequalitiesA. V. ProkhorovA. V. Prokhorovhttps://doi.org/10.1137/1113030PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Yu. V. Prokhorov, An extension of S. N. Bernstein's inequalities to multidimensional distributions, Theory Prob. Applications, 13 (1968), 260–267 10.1137/1113029 0167.46903 LinkGoogle Scholar[2] W. Hoeffding, On a theorem of V. M. Zolotarev, Theory Prob. Applications, 9 (1964), 89–91 10.1137/1109008 LinkGoogle Scholar[3] Wassily Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58 (1963), 13–30 MR0144363 (26:1908) 0127.10602 CrossrefGoogle Scholar[4] A. V. Prokhorov, On a probability inequality, Mat. Zametki, 3 (1968), , (In Russian.) Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails On Exponential Bounds for Probabilities of Large DeviationsV. V. Yurinskii28 July 2006 | Theory of Probability & Its Applications, Vol. 37, No. 1PDF (208 KB)Exponential Bounds for the Distribution of the Maximum of a Non-Gaussian Random FieldE. I. Ostrovskii17 July 2006 | Theory of Probability & Its Applications, Vol. 35, No. 3AbstractPDF (1196 KB)The Rate of Convergence of the Distribution of the Smirnov-von Mises StatisticA. I. Orlov28 July 2006 | Theory of Probability & Its Applications, Vol. 19, No. 4AbstractPDF (1852 KB)Exponential Bounds for Large DeviationsV. V. Jurinskii28 July 2006 | Theory of Probability & Its Applications, Vol. 19, No. 1PDF (187 KB)On the S. N. Bernstein InequalityS. L. Blyumin and B. D. Kotlyar17 July 2006 | Theory of Probability & Its Applications, Vol. 16, No. 1AbstractPDF (287 KB)S. N. Bernstein’s Inequalities in the Multidimensional CaseA. V. Prokhorov17 July 2006 | Theory of Probability & Its Applications, Vol. 13, No. 3AbstractPDF (652 KB) Volume 13, Issue 2| 1968Theory of Probability & Its Applications History Submitted:08 February 1968Published online:28 July 2006 InformationCopyright © 1968 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1113030Article page range:pp. 268-280ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Regularization of difference schemes

Regularization of difference schemes

Interpolative Solution of Systems of Nonlinear Equations

This paper presents an iterative method for numerical solution of a system of n nonlinear functional equations in n unknowns, by repeated linear interpolation. It is a variation of Newton’s method, in which the partial derivatives are replaced by the multidimensional analogues of difference quotients. The only computations required are evaluation of the given functions and simple matrix operations. The method is a generalization of an idea of C. F. Gauss [6], and can also be regarded as generalizing the secant method, or regula falsi, as used for solving functions of one variable [4]. It produces superlinear convergence (of order $\frac{1}{2}(1 + \sqrt 5 )$, or approximately 1.62) to a simple zero, provided the given functions are twice continuously differentiable in a neighborhood of the zero, and the initial approximations are close enough. A convergence analysis for this method is given, and some numerical results are presented.