21. CWI Report MAS-R9828 41A58 (Series expansions (eg Taylor, Lidstone series, but not Fourier series)),41A60 (Asymptotic approximations, asymptotic expansions (steepest descent, etc http://www.cwi.nl/static/publications/reports/abs/MAS-R9828.html

22. Stochastic Expansions And Asymptotic Approximations. Home Journals List Econometric Theory Stochastic expansions and Asymptoticapproximations. Stochastic expansions and Asymptotic approximations. http://netec.mcc.ac.uk/BibEc/data/Articles/cupetheorv:8:y:1992:i:3:p:343-67.html

mirrored in Providing the latest research results since 1993 Search tips: title=fiscal or author=levine Working Papers Series Journals Authors JEL Classification ... Econometric Theory >> Stochastic Expansions and Asymptotic Approximations. Stochastic Expansions and Asymptotic Approximations. Magdalinos, Michael A Econometric Theory web site (RePEc:cup:etheor:v:8:y:1992:i:3:p:343-67) Pages: 343-67 Volume: 8 Month: September Year: 1992 Issue: 3 go top Information for authors: Are you an author of this paper? Please take the time and register at our new service. Read all about it at: http://netec.mcc.ac.uk/HoPEc/geminiabout.html . Note you do not need to register in order to use the search service!! Access statistics BibEc is a RePEc service managed by Jose Manuel Barrueco and Thomas Krichel contact us Last updated: 2003-03-16 17:59:26

23. DLMF: Asymptotic Approximations expansions (continued). §AS.11(6), Direct Numerical Transformations. References.Digital Library of Mathematical Functions, Chapter AS. Asymptotic approximations. http://dlmf.nist.gov/Contents/AS/outline.php

Chapter AS Asymptotic Approximations by Frank W. J. Olver and R. Wong Obsolete This page is obsolete. Draft Outline (about chapter status) Mathematical Properties Definitions and Elementary Properties Asymptotic and Order Symbols Integration and Differentiation Asymptotic Expansions Uniform Asymptotic Expansions Generalized Asymptotic Expansions Transcendental Equations Integrals of a Real Variable Integration by Parts Watson's Lemma Laplace's Method Method of Stationary Phase Coalescing Peak and Endpoint: Bleistein's Method Contour Integrals Watson's Lemma Inverse Laplace Transforms Laplace's Method Saddle Points Coalescing Saddle Points: Chester, Friedman and Ursell's Method Other Coalescing Critical Points Mellin Transformation Methods Distributional Methods Mathematical Properties continued Differential Equations Regular Singularities: Fuchs-Frobenius Theory Irregular Singularities of Rank 1 Liouville-Green (WKBJ) Approximation Numerically Satisfactory Solutions Differential Equations with a Parameter Classification of Cases Case I: No Transition Points Case II: Simple Turning Point Case III: Simple Pole Case IV: Multiple and Fractional Turning Points Case V: Coalescing Transition Points Difference Equations Distinct Characteristic Values Coincident Characteristic Values Sums and Sequences Euler-Maclaurin Formula Summation by Parts Asymptotic Expansions of Entire Functions Taylor and Laurent Coefficients: Darboux's Method Remainder Terms; Stokes Phenomenon

24. 41-XX 41XX approximations and expansions. 41A60 Asymptotic approximations,asymptotic expansions (steepest descent, etc.) See also 30E15; http://www.ams.org/mathweb/msc1991/41-XX.html

41-XX Approximations and expansions and ; for all trigonometric approximation and interpolation, see and ; for numerical approximation, see 41-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 41-01 Instructional exposition (textbooks, tutorial papers, etc.) 41-02 Research exposition (monographs, survey articles) 41-03 Historical (must be assigned at least one classification number from Section 01) 41-04 Explicit machine computation and programs (not the theory of computation or programming) 41-06 Proceedings, conferences, collections, etc. 41A05 Interpolation [See also and 41A15 Spline approximation 41A20 Approximation by rational functions 41A25 Rate of convergence, degree of approximation 41A27 Inverse theorems 41A28 Simultaneous approximation 41A29 Approximation with constraints 41A30 Approximation by other special function classes 41A35 Approximation by operators (in particular, by integral operators) 41A36 Approximation by positive operators 41A40 Saturation 41A44 Best constants 41A45 Approximation by arbitrary linear expressions 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy

25. KLUWER Academic Publishers Approximations And Expansions Home » Browse by Subject » Mathematics » Analysis » Approximationsand expansions. Sort listing by AZ ZA Publication Date. Results http://www.baltzer.nl/home/topics/J/5/1/?type=Kluwer Looseleafs

26. Abstract Of: Uniform Approximations Of Bernoulli And Euler Polynomials In Terms In addition, the convergence of these improved expansions is stronger and also forreal argument the accuracy of these improved approximations is better in the http://db.cwi.nl/rapporten/abstract.php?abstractnr=792

27. Abstract Of: Uniform Asymptotic Expansions Of Integrals: A Selection Of Problems Abstract of Uniform asymptotic expansions of integrals a selection of problems ofasymptotic methods for integrals, in particular on uniform approximations. http://db.cwi.nl/rapporten/abstract.php?abstractnr=90

28. Calculus And Mathematica Courses and . Converting known expansions to others via change of variable.expansions for approximations. Science and math experience. http://www-cm.math.uiuc.edu/class/130syl.htm

Expansions and techiques of integration Authors: Bill Davis, Horacio Porta and Jerry Uhl Producer: Bruce Carpenter Publisher: Distr ibutor: 3.Approximation 3.01 Splines Mathematics. Remarkable plots explained by order of contact. Splining for smoothness at the knots. Science and math experience. 3.02 Expansions in Powers of x Mathematics. The expansion of a function in powers of as a file of polynomials with higher and higher orders of contact with at . The expansions every literate calculus person knows: , and Converting known expansions to others via change of variable. Expansions for approximations. Science and math experience. . Expansions by substitution. Expansions by differentiation. ;Expansions by integration. Recognition of expansions. Expansions that satisfy a priori error bounds. 3.03 Using Expansions Mathematics. The expansion of a function in powers of as a file of polynomials with higher and higher orders of contact with at Science and math experience.

29. Welcome To CourseSpace 3.03 Using expansions expansions in powers of (x b) and approximations basedon them, tangent lines and Newton's method, using expansions to help to http://www-cm.math.uiuc.edu/coursespace/ntu/351asyll.html

Please choose your CourseSpace from the list of Current Courses to the right. Contact an admin 351a: Calculus Refresher Feel of Mathematica an introduction to using mathematica, plotting functions with mathematica. 1.01 Growth rates of growth of linear functions/power functions/the exponential function, percentage growth rates, the global scale, interpolation of data sets. 1.02 Natural Logs and Exponentials the natural base e and the natural logarithm, percentage growth of exponential functions: doubling time and half life, unnatural bases, exponential models, e and exponential data analysis, e and finance. 1.03 Instantaneous Growth Rates instantaneous growth rates, instantaneous growth rate of x^k, sin[x], cos[x], Log[x] and exp[x]; average growth rate versus instantaneous growth rate. 1.04 Rules of the Derivative derivatives, instantaneous growth rates, the chain rule, general rules for taking derivatives, using the logarithm to calculational advantage, dominance in growth rates 1.05 Using the Tools

30. 41-XX 41XX approximations and expansions,. 41A60 Asymptotic approximations,asymptotic expansions (steepest descent, etc.), See also {30E15}; http://www.ma.hw.ac.uk/~chris/MR/41-XX.html

41-XX Approximations and expansions, and and 41-00 General reference works (handbooks, dictionaries, bibliographies, etc.) 41-01 Instructional exposition (textbooks, tutorial papers, etc.) 41-02 Research exposition (monographs, survey articles) 41-03 Historical (must be assigned at least one classification number from 01-XX 41-04 Explicit machine computation and programs (not the theory of computation or programming) 41-06 Proceedings, conferences, collections, etc. and 41A15 Spline approximation 41A17 Inequalities in approximation (Bernstein, Jackson, Nikolskiui type inequalities) 41A20 Approximation by rational functions 41A21 Pade approximation 41A25 Rate of convergence, degree of approximation 41A27 Inverse theorems 41A28 Simultaneous approximation 41A29 Approximation with constraints 41A30 Approximation by other special function classes 41A35 Approximation by operators (in particular, by integral operators) 41A36 Approximation by positive operators 41A40 Saturation 41A44 Best constants 41A45 Approximation by arbitrary linear expressions 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy

31. Distance Calculus At Suffolk University • Courses Offered 0. The expansions every literate calculus person knows (1/(1 x), ex,sinx and cosx). expansions for approximations. The expansion http://www.distancecalculus.com/new/courses/calcIII.php

contact us enroll now Information: about the course the classroom courses offered where ... student page visit our sister program at The Ohio State University visit our Computer Algebra in Mathematics Education courses At Suffolk University Calculus III (Series) Detailed Syllabus Mathematics Remarkable plots explained by order of contact. Splining for smoothness at the knots. The expansion of a function f[x] in powers of x as a file of polynomials with higher and higher orders of contact with f[x] at x = 0. The expansions every literate calculus person knows (1/(1 - x), ex, sin[x] and cos[x]). Expansions for approximations. The expansion of a function f[x] in powers of (x - b) as a file of polynomials with higher and higher orders of contact with f[x] at x = b. Newton's method. Multiplying and dividing expansions. Using expansions to help to calculate limits at a point. Expansions and the complex exponential function. Using expansions to help to get precise estimates of some integrals. Science and Math Experience Experiments geared at discovering that the smoother the transition from one curve to another at a knot, the better both curves approximate each other near the knot.

32. Calculus&Mathematica: 153 Course Description person knows 1/(1 x); e x; sinx; cosx. expansions for approximations.Science and math experience. Experiments geared toward http://socrates.math.ohio-state.edu/about/coursedesc/describe153.php3

Go to.... About our Program For Students For Schools The People Contact Us Links Mathematica Bill Davis, Horacio Porta and Jerry Uhl Technical Crew: Don Brown, Gary Binyamin, Alan DeGuzman, Justin Gallivan, Corey Mutter, David Taubenheim, Jennifer Welch and David Wiltz The rights to all modifications are assigned to Addison-Wesley Publishing Company, Inc. Developed with support from the National Science Foundation at the University of Illinois at Urbana-Champaign and the Ohio State University. Mathematica Approximations Book 3 3.01 Splines Mathematics Remarkable plots explained by order of contact. Splining for smoothness at the knots. Science and math experience. Experiments geared at discovering that the smoother the transition from one curve to another at a knot, the better both curves approximate each other near the knot. Splining functions and polynomials. Splines in road design. Landing an airplane. The natural cubic spline. Order of contact for derivatives and integrals. 3.02 Expansions Mathematics The expansion of a function f[x] in powers of x as a file of polynomials with higher and higher orders of contact with f[x] at x = 0. The expansions every literate calculus person knows:

33. RR-3427 : Analytic Expansions Of (max,+) Lyapunov Exponents Translate this page order, together with an error estimate for finite order Taylor approximations. Severalextensions of this are discussed, including expansions of multinomial http://www.inria.fr/rrrt/rr-3427.html

RR-3427 - Analytic Expansions of (max,+) Lyapunov Exponents Hong, Dohy Les rapports de cet auteur Rapport de recherche de l'INRIA- Sophia Antipolis Page d'accueil de l'unité de recherche Fichier PostScript / PostScript file Fichier postscript du document : 382 Ko Fichier PDF / PDF file Fichier PDF du document : 595 Ko Projet : MISTRAL - 51 pages - Mai 1998 - Document en anglais Page d'accueil du projet KEY-WORDS : TAYLOR SERIES / LYAPUNOV EXPONENTS / (MAX / +) SEMIRING / STRONG COUPLING / RENOVATING EVENTS / STATIONARY STATE VARIABLES / ANALYTICITY / VECTORIAL RECURRENCE RELATION / NETWORK MODELING / STOCHASTIC PETRI NETS

34. 130 C M Syllabus orders of contact with fx at x = 0. The expansions every literate calculus personknows (1/(1 x), e^x sinx and cosx). expansions for approximations. http://www.math.uiuc.edu/UndergraduateProgram/engineering/syl130mma.html

Syllabi are provided for students and advisors only as an indication of the nature of the course and its contents. Individual instructors may choose a different approach to the material. Some instructors may post special materials concerning their section on a WWW page. (See the Classes listing on the Mathematics Department Webpage Calculus and Mathematica consists of five chapters of interactive computer notebooks. This course is composed of the following selections from the notebooks. 4 Approximation 4.01 Splines Mathematics Remarkable plots explained by order of contact. Splining for smoothness at the knots.. Science and math experience. Experiments geared at discovering that the smoother the tranistion from one curve to another at a knot, the better both curves approximate each other near the knot. Splining functions and polynomials. Splines in road design. Landing an airplane. The natural cubic spline. Order of contact for derivatives and integrals. 4.02 Expansions Mathematics The expansion of a function f[x] in powers of x as a file of polynomials with higher and higher orders of contact with f[x] at x = 0. The expansions every literate calculus person knows (1/(1 - x), e^x sin[x] and cos[x]). Expansions for approximations.

35. Citations: Adaptive Nonlinear Approximations - Davis (ResearchIndex) the problem of finding M vector optimal approximations is NP hard. Because of thenumerical intractability of computing optimal expansions, in section 3 we http://citeseer.nj.nec.com/context/21384/185946

13 citations found. Retrieving documents... G. Davis, Adaptive Nonlinear Approximations .PhD thesis, New York University, Sept. 1994. Home/Search Document Details and Download Summary Related Articles ... Check This paper is cited in the following contexts: Fractal Video Coding By Matching Pursuit - Gharavi-Alkhansari, Huang (Correct) ....one may be interested in finding the smallest number of vectors in the dictionary whose linear combination approximates the given vector within a given error threshold. This problem, in the general case, is a di#cult combinatorial optimization problem, and has recently been proven to be NP hard However, an e#cient suboptimal greedy solution to this problem has been discovered by di#erent researchers in di#erent contexts but with basically the same underlying mathematics. In statistics, this greedy algorithm was found and named projection pursuit [10] It was used for computation of .... G. Davis, Adaptive Nonlinear Approximations .PhD thesis, New York University, Sept. 1994. Matching Pursuit and Atomic Signal Models Based on Recursive .. - Goodwin, Vetterli

36. Heavy-Traffic Asymptotic Expansions For The Asymptotic Decay Rates In The Bmap/g Related documents from cocitation More All 3 Exponential approximations forTail GL Choudhury and W. Whitt, Heavy-Traffic Asymptotic expansions for the http://citeseer.nj.nec.com/468898.html

Heavy-Traffic Asymptotic Expansions For The Asymptotic Decay Rates In The Bmap/G/1 Queue (Make Corrections) (2 citations) Gagan L. Choudhury Att Bell Laboratories Holmdel, Nj 07733-3030 Ward Whitt... Home/Search Context Related View or download: att.com/~wow/decay.ps Cached: PS.gz PS PDF DjVu ... Help From: att.com/~wow/choudhury (more) Homepages: G.Choudhury W.Whitt HPSearch (Update Links) Rate this article: (best) Comment on this article (Enter summary) Abstract: In great generality, the basic steady-state distributions in the BMAP/G/1 queue have asymptotically exponential tails. Here we develop asymptotic expansions for the asymptotic decay rates of these tail probabilities in powers of one minus the traffic intensity. The first term coincides with the decay rate of the exponential distribution arising in the standard heavy-traffic limit. The coefficients of these heavy-traffic expansions depend on the moments of the service-time distribution and the... (Update) Context of citations to this paper: More representation of the versatile Markovian point process or Neuts process which is a powerful model with many applications [23] For example, since superpositions of independent BMAPs are BMAPs, they are useful for studying statistical multiplexing. We have...

37. K10.4-Fourier.html Even expansions. Let call S(n,x), the Fourier series approximations withn coefficients S=(m,x) 1/2+sum(Ak*cos(k*Pi/2*x),k=1..m);. http://www.mapleapps.com/categories/engineering/engmathematics/html/K10.4-Fourie

Even expansions ge:=x*(Heaviside(x)-Heaviside(x-1))+(2-x)*(Heaviside(x-1)-Heaviside(x-2))+(-x)*(Heaviside(x+1)-Heaviside(x))+(2+x)*(Heaviside(x+2)-Heaviside(x+1)); plot(ge,x=-2.01..2.01); The coefficient a[0]:=1/4*Int('f',x=-2..2)=2*(1/4*(int(f1,x=0..1)+int(f2,x=1..2))); The coefficients a[n]:=2*1/2*Int('f'*cos(n*Pi/2*x),x=-2..2)=simplify(2*1/2*(int(f1*cos(n*Pi/2*x),x=0..1)+int(f2*cos(n*Pi/2*x),x=1..2))); The coefficient are all zero A[k]:=subs(n=k,simplify(2*1/2*(int(f1*cos(n*Pi/2*x),x=0..1)+int(f2*cos(n*Pi/2*x),x=1..2)))); Let call S(n,x), the Fourier series approximations with n coefficients: Now, we look at different approximations of S as m increases: plot([ge,S(3,x),S(7,x)],x=-2..2,title="Higher approximations: n=3 (Green),n=7 (Blue)",color=[red,green,blue],numpoints=100); The approximation gets better., Let's try some higher order approximations plot([S(100,x)],x=-6..6,title="Higher approximations: n=100 (Blue)",color=[red,blue],numpoints=150);

38. K10.4-Fourier.html plot(S(100,x),x=6..6,title= Higher approximations n=100 (Blue) ,color=red,blue,numpoints=150);.Comparison between odd and even expansions. http://www.mapleapps.com/categories/engineering/engmathematics/html/K10.4-Fourie

Section 1: Even vs Odd expansions Consider the triangle: g:=x*(Heaviside(x)-Heaviside(x-1))+(2-x)*(Heaviside(x-1)-Heaviside(x-2)); plot(g,x=0..2,title="The triangle"); f1:=x;f2:=(2-x); Even expansions Odd expansions go:=x*(Heaviside(x)-Heaviside(x-1))+(2-x)*(Heaviside(x-1)-Heaviside(x-2))+(x)*(Heaviside(x+1)-Heaviside(x))+(-2-x)*(Heaviside(x+2)-Heaviside(x+1)); plot(go,x=-2.01..2.01); The coefficients are all zero (including n=0) b[n]:=2*1/2*Int('f'*sin(n*Pi/2*x),x=-2..2)=simplify(2*1/2*(int(f1*sin(n*Pi/2*x),x=0..1)+int(f2*sin(n*Pi/2*x),x=1..2))); The coefficient are all zero B[k]:=subs(n=k,simplify(2*1/2*(int(f1*sin(n*Pi/2*x),x=0..1)+int(f2*sin(n*Pi/2*x),x=1..2)))); Let call S(n,x), the Fourier series approximations with n coefficients: Now, we look at different approximations of S as m increases: plot([go,S(3,x),S(7,x)],x=-2..2,title="Higher approximations: n=3 (Green),n=7 (Blue)",color=[red,green,blue],numpoints=100); The approximation gets better., Let's try some higher order approximations plot([S(100,x)],x=-6..6,title="Higher approximations: n=100 (Blue)",color=[red,blue],numpoints=150); Comparison between odd and even expansions plot([S(j,1),Seven(j,1)],j=2..51,title="Convergence of the even expansion (blue) and odd expansion (red) at one point x=1",color=[red,blue],numpoints=50);

39. 4 Fourier Expansions PDF (15) for (dashed line), and its EdgeworthPetrov approximations with 2 and4 terms in the expansion (solid). next previous Up expansions for nearly http://aanda.u-strasbg.fr:2002/articles/astro/full/1998/10/h0596/node4.html

Up: Expansions for nearly Gaussian 4 Fourier expansions In order better to understand the poor convergence properties of the Gram-Charlier series, let us first discuss how it is related to the Fourier expansion. A Fourier expansion ( Suetin 1979 ) for any function f x ) in the set of orthogonal polynomials P n is given by with Here h n is the squared norm and Now we can see that the Gram-Charlier series ( ) is just the Fourier expansion ( ) of f x p x Z x ) in the set of Chebyshev-Hermite polynomials with c n n a n The properties of the Gram-Charlier approximations of p x ) in Figs. and are to be considered in the general context of the convergence of Fourier expansions. The source of the divergence lies in the sensitivity of the Gram-Charlier series to the behavior of p x ) at infinity - the latter must fall to zero faster than for the series to converge ( Kendall 1952 ). This is often too restrictive for practical applications. Our example of the distribution in ( ), with its exponential behavior at infinity, clearly demonstrates this. The Fourier expansion of p x Z x ) in another set of Hermite polynomials H n x ) (not in Chebyshev-Hermite polynomials ), as for the Gram-Charlier series) is sometimes used:

40. EEVL | Full Record approximation theory, harmonic, multilevel methods, large scale eigenvalue problems,matrix, Fourier, approximations, expansions, acceleration, probabilistic http://www.eevl.ac.uk/show_full.htm?rec=980259601-16213

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