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# Topic: Uniform convergence

###### In the News (Tue 21 May 13)

 Uniform convergence - Wikipedia, the free encyclopedia It is therefore plain that uniform convergence implies pointwise convergence. Counterexample to the converse of the uniform convergence theorem. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable, and even if it is differentiable, the derivative of the limit function need not be equal to the limit of the derivatives. en.wikipedia.org /wiki/Uniform_convergence   (860 words)

 Pointwise convergence - Wikipedia, the free encyclopedia This concept is often contrasted with uniform convergence. Uniform convergence, on the other hand, does not make sense for functions taking values in topological spaces generally, but makes sense for functions taking values in metric spaces, and, more generally, in uniform spaces. Pointwise convergence may also be formulated as convergence in the topology which arises from the seminorm given by en.wikipedia.org /wiki/Pointwise_convergence   (261 words)

 Learn more about Real analysis in the online encyclopedia.   (Site not responding. Last check: 2007-10-07) The concept of convergence, central to analysis, is introduced via limits of sequences. At this point, it is useful to study the notions of continuity and convergence in a more abstract setting, in order to later consider spaces of functions. The notion of uniform convergence is important in this context. www.onlineencyclopedia.org /r/re/real_analysis.html   (489 words)

 Uniforms To You   (Site not responding. Last check: 2007-10-07) Modern uniforms are worn by armed forces and paramilitary organisations such as police, emergency services, security guards, in some workplaces and schools, and by inmates in prisons. Uniform spaces are topological spaces with additional structure which is used to define uniform properties such as completeness, uniform continuity and uniform convergence. The conceptual difference between uniform and topological structures is that in a uniform space, you can formalize the idea that "''x'' is as close to ''a'' as ''y'' is to ''b''", while in a topological space you can only formalize "''x'' is as close to ''a'' as ''y'' is to ''a''". www.blownspeakers.com /pages3/91/uniforms-to-you.html   (1099 words)

 Uniform convergence   (Site not responding. Last check: 2007-10-07) Uniform convergence of martingales in the one-dimensional branching random walk.... Abstract of: Uniform convergence of curve estimators for ergodic diffusion proce... Uniform convergence of martingales in the branching random walk... www.scienceoxygen.com /math/381.html   (312 words)

 lab12.html Is the convergence uniform, pointwise, or in the norm? Pointwise convergence on [-1,1] is seen in the fact that the difference approaches zero for every point in the interval. Because there is not pointwise convergence, it must be the case that the sequence does not converge uniformly. www.math.sc.edu /~meade/math142-S03/labs/lab121.html   (1613 words)

 MATH 251 FINAL EXAM INFORMATION   (Site not responding. Last check: 2007-10-07) Convergence of a sequence of functions: definition of pointwise convergence, definition of uniform convergence. Theorem that the uniform limit of a sequence of continuous functions is continuous. Theorem that a uniformly Cauchy sequence of functions is guaranteed to converge uniformly to a function. www.math.wvu.edu /~sherm/m251/final.remarks.html   (421 words)

 Old Level 2   (Site not responding. Last check: 2007-10-07) Uniform limit of a sequence of continuous functions is continuous. The Cauchy principle for uniform convergence of sequences and the general uniform convergence principle for series. Uniform convergence of a sequence of functions on a closed bounded interval implies convergence of the corresponding integrals. www.qub.ac.uk /mp/pmt/courses/level21.htm   (1067 words)

 Edinburgh Mathematics Programme The uniform metric on C[a,b] and the notion of uniform convergence. The convergence of Fourier series (pointwise/uniform), uniform convergence of Fejer means for continuous functions, convergence in the mean-square metric d_2 of Fourier series of continuous functions and Parseval's Theorem; Weierstrass approximation theorem. In this module, metrics provide a language for discussing convergence and completeness, and so the emphasis is on the definition of a metric space and giving plenty of useful examples of metrics. www.maths.ed.ac.uk /~carbery/FoA.html   (932 words)

 Uniform Convergence   (Site not responding. Last check: 2007-10-07) We investigate this convergence graphically by looking at the maximum difference between the partial sums and the function over the entire interval as a function of n. Another way to visualize uniform convergence is to look at a strip of width epsilon around the function. Here we do not expect uniform convergence of the partial sums to the function and in fact, we see that the partial sums, while they do converge pointwise, do not converge uniformly to the function. amath.colorado.edu /courses/4350/2002fall/uniform.html   (384 words)

 Analysis WebNotes: Chapter 05, Class 20 We haven't yet had a formal definition of convergence for anything but sequences of real numbers, but the analogy with the real case will be pretty strong in these examples. We'll come to a formal definition of convergence which is broad enough to encompass all of these examples, in the next section. The next big step is to introduce a general mathematical setup for studying convergence and continuity which is broad enough to encompass the fairly diverse types of examples we have just been looking at, together with many more that will arise. www.math.unl.edu /~webnotes/classes/class20/class20.htm   (584 words)

 Uniform convergence of martingales in the branching random walk   (Site not responding. Last check: 2007-10-07) The convergence of these martingales uniformly in $\lambda$, for $\lambda$ lying in a suitable set, is the first main result of this paper. This will imply that, on that set, the martingale limit \Wl is actually an analytic function of \la. The uniform convergence results are used to obtain extensions of known results on the growth of $\Zn(nc+D)$ with $n$, for bounded intervals $D$ and fixed $c$. Finally similar results, both on martingale convergence and uniform local large deviations, are also obtained for continuous time models including branching Brownian motion. www.shef.ac.uk /~st1jdb/mart-www.html   (192 words)

 PlanetMath: compact-open topology   (Site not responding. Last check: 2007-10-07) is a metric space), then this is the topology of uniform convergence on compact sets. is a compact space, then this is the topology of uniform convergence. Cross-references: converges, sequence, compact sets, uniform convergence, metric space, uniform space, subbasis, open set, subspace, compact, continuous maps, topological spaces planetmath.org /encyclopedia/CompactOpenTopology.html   (87 words)

 Uniform Convergence   (Site not responding. Last check: 2007-10-07) Local uniform convergence and convergence of Julia sets... A uniform convergence theorem for the numerical solving of the nonlinear filteri... UNIFORM CONVERGENCE ESTIMATES FOR A COLLOCATION METHOD FOR THE CAUCHY SINGULAR I... www.scienceoxygen.com /math/356.html   (297 words)

 Mathematical Analysis by Gordon H. Fullerton - Apronus.com Uniform convergence of sequences of functions is defined and related to the completeness of C(X). Chapter 3 deals with further results on the uniform convergence sequences and series of functions. In Chapter 4, the Lebesgue integral is defined, using Daniell's approach, as a linear functional on a certain class of functions. www.apronus.com /math/fullerton_analysis.htm   (174 words)

 [No title] The corresponding question for uniform limit spaces has been solved by Wyler [19], whereas completions in the realm of filter spaces have been constructed by Kent and Rath [11] as well as by Császár [6]. The category Conv of convergence spaces (and uniformly continuous maps) is isomorphic to the category KConvs of symmetric Kent convergence spaces (and continuous maps); it is a bicoreflective subcategory of SUConv (cf. [2] Behling, A.: 1992, Einbettung uniformer Räume in topologische Universen, Thesis, Free University, Berlin. www.mathematik.uni-osnabrueck.de /projects/carmen/AP11/test/file14.html   (3798 words)

 [No title]   (Site not responding. Last check: 2007-10-07) Since the text book includes some more difficult results about the topic of uniform convergence, and which we do not expect you to know yet, let me state explicitly what we DO expect you to know at this stage about uniform convergence of Fourier series. You should already know (also from earlier courses) the definition of uniform convergence of a given series of functions on a given interval. If a Fourier series converges uniformly on some interval, then the function which is the sum of that series must be continuous on that same interval. www.math.technion.ac.il /~mcwikel/fsit27apr01.txt   (477 words)

 Nat' Academies Press, Large-Scale Structures in Acoustics and Electromagnetics: Proceedings of a Symposium (1996) The Galerkin convergence curves are the same ones that were presented in Figure 6.5, and are included here for comparison with GLS convergence. We consider first the GLS convergence with uniform refinement, and hereafter refer to it as GLS/uniform (analogous notation is used for the other cases). Figure 6.8 repeats the convergence study of Figure 6.7, this time showing the estimated error as a function of mesh refinement. www.nap.edu /books/0309053374/html/137.html   (843 words)

 Abstract of: Uniform convergence of curve estimators for ergodic diffusion processes   (Site not responding. Last check: 2007-10-07) Using empirical process theory for martingales, we first prove a theorem regarding the uniform weak convergence of the empirical density. This result is then used to derive uniform weak convergence for the kernel estimator of the invariant density. For kernel estimators of the derivatives of the invariant density and for a nonparametric drift estimator that was proposed by Banon, we give bounds for the rate at which the uniform distance between the estimator and the true curve vanishes. db.cwi.nl /rapporten/abstract.php?abstractnr=927   (179 words)

 NeuroCOLT: Neural Networks and Computational Learning Theory   (Site not responding. Last check: 2007-10-07) Learnability in Valiant's PAC learning model has been shown to be strongly related to the existence of uniform laws of large numbers. These laws define a distribution-free convergence property of means to expectations uniformly over classes of random variables. In this paper we prove, through a generalization of Sauer's lemma that may be interesting in its own right, a new characterization of uniform Glivenko-Cantelli classes. www.neurocolt.com /abs/1996/abs96009.html   (171 words)

 [No title] Knowledge of key results of analysis such as a sequence of real numbers converges if and only if it is a Cauchy sequence and the uniform convergence of a sequence of continuous functions gives a continuous limit. Cauchy sequences and the Cauchy criterion for convergence. Pointwise and uniform convergence of a sequence of functions. www.brunel.ac.uk /~icstmkw/syllabus/ma2034a.htm   (464 words)

 Gerhard Preuss   (Site not responding. Last check: 2007-10-07) Recently semiuniform convergence spaces have been studied by the author as a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces) as well as of uniform limit spaces (and thus of uniform spaces) with many convenient properties such as cartesian closedness, hereditariness and the fact that products of quotients are quotients. They form the suitable framework for studying continuity, Cauchy continuity and uniform continuity as well as convergence structures in function spaces, namely simple convergence, continuous convergence and uniform convergence. Furthermore, the localization of compactness (when considered in the realm of semiuniform convergence spaces) leads to a cartesian closed topological category and, in contrast to the situation for topological spaces, the locally compact spaces are exactly the compactly generated spaces (even without the Hausdorff axiom!). www.utm.edu /staff/jschomme/topology/c/a/a/i/88.htm   (239 words)

 Edinburgh Mathematics Programme To cover basic ideas of convergence and completeness in R. To introduce the idea of uniform convergence. Sequences and series, uniform convergence, the Riemann integral, convergence of Fourier series, applications, functions and distributions, transform techniques, existence and uniqueness for PDEs, fundamental solutions and Green's functions. Uniform convergence [5] Definition and illustrations; uniform limit of continuous is continuous; uniform convergent series and the M-test; term-by-term integration and differentiation; examples and applications. www.maths.ed.ac.uk /~derek/Syll/PAA.html   (528 words)

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