Tchebychev and uniform approximation I

The aim of this note is to introduce another way of defining the almost sure uniform convergence, which is necessary when studying some mathematical results on the existence of price bubbles in certain scenarios of trading securities. This mode of convergence of random variables' sequences is intermediate between the uniform and the almost sure ones, and, more specifically, between the uniform and the complete convergences. In this way, this paper presents some mathematical characterizations of both almost sure uniform and complete convergences, and shows that the almost sure uniform convergence is a particular case of complete convergence, when the number of summands in the series defining this mode of convergence is finite. Finally, this paper presents the relation of almost surely uniform convergence with convergence in mean when the random variable limit is integrable. Moreover, almost surely convergence and local boundedness of the sequence of random variables minus its limit are sufficient to derive convergence in mean.

Comparative study of singularly perturbed two-point BVPs via: Fitted-mesh finite difference method, B-spline collocation method and finite element method

Noncompact uniform universal approximation

Let Ω be an open set in R n and E be a relatively closed subset of Ω. Which pairs (Ω,E) have the property that functions harmonic on (some neighbourhood of) E can be uniformly approximated by functions harmonic on Ω? Which pairs permit similar approximation of functions continuous on E.ind harmonic on E°? This paper reviews recently-obtained solutions to the above problems, and adds some new results concerning better - than - uniform approximation.

We investigate the convergence of phase fields for the Willmore problem away from the support of a limiting measure $$\mu $$ . For this purpose, we introduce a suitable notion of essentially uniform convergence. This mode of convergence is a natural generalisation of uniform convergence that precisely describes the convergence of phase fields in three dimensions. More in detail, we show that, in three space dimensions, points close to which the phase fields stay bounded away from a pure phase lie either in the support of the limiting mass measure $$\mu $$ or contribute a positive amount to the limiting Willmore energy. Thus there can only be finitely many such points. As an application, we investigate the Hausdorff limit of level sets of sequences of phase fields with bounded energy. We also obtain results on boundedness and $$L^p$$ -convergence of phase fields and convergence from outside the interval between the wells of a double-well potential. For minimisers of suitable energy functionals, we deduce uniform convergence of the phase fields from essentially uniform convergence.

Repetitive Play in Finite Statistical Games with Unknown Distributions

Let (X, p) and (Y, d) be metric spaces with at least two points. It is usual for introductory courses in topology to study the set Yx of all functions mapping X to Y with the pointwise, compact-open, uniform convergence, and uniform convergence on compacta topologies. Some care is taken to show sufficient conditions for these topologies to be equivalent [1, 2]. However, the question of necessary conditions are dismissed with examples showing that the topologies are not in general equivalent.

Previous article Next article Uniform Generalized Weight Function Polynomial Approximation with InterpolationH. L. Loeb, D. G. Moursund, L. L. Schumaker, and G. D. TaylorH. L. Loeb, D. G. Moursund, L. L. Schumaker, and G. D. Taylorhttps://doi.org/10.1137/0706026PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. Paszkowski, On approximation with nodes, Rozprawy Mat., 14 (1957), 63– MR0092888 0083.29002 Google Scholar[2] Frank Deutsch, On uniform approximation with interpolatory constraints, J. Math. Anal. Appl., 24 (1968), 62–79 10.1016/0022-247X(68)90049-8 MR0230017 0174.35501 CrossrefISIGoogle Scholar[3] E. W. Cheney and , H. L. Loeb, Generalized rational approximation, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 1 (1964), 11–25 MR0178288 0138.04403 LinkGoogle Scholar[4] Samuel Karlin and , William J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, Vol. XV, Interscience Publishers John Wiley & Sons, New York-London-Sydney, 1966xviii+586 MR0204922 0153.38902 Google Scholar[5] David G. Moursund, Chebyshev approximation using a generalized weight function, SIAM J. Numer. Anal., 3 (1966), 435–450 10.1137/0703038 MR0204936 0163.07302 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A general framework for the optimal approximation of circular arcs by parametric polynomial curvesJournal of Computational and Applied Mathematics, Vol. 345 | 1 Jan 2019 Cross Ref Strong unicity of order 2 in C(T)Journal of Approximation Theory, Vol. 56, No. 3 | 1 Mar 1989 Cross Ref The Best Interpolating Approximation is a Limit of Best Weighted ApproximationsCanadian Mathematical Bulletin, Vol. 25, No. 4 | 20 November 2018 Cross Ref The best interpolating approximation and weighted approximationJournal of Approximation Theory, Vol. 35, No. 1 | 1 Oct 1982 Cross Ref Remez algorithm for Chebyshev approximation with interpolationComputing, Vol. 28, No. 1 | 1 Mar 1982 Cross Ref Discrete chebyshev approximation with interpolationInternational Journal of Computer Mathematics, Vol. 11, No. 3-4 | 19 March 2007 Cross Ref Approximation with interpolatory constraintsJournal of Mathematical Analysis and Applications, Vol. 64, No. 2 | 1 Jun 1978 Cross Ref Approximation with constraints in normed linear spacesJournal of Approximation Theory, Vol. 21, No. 3 | 1 Nov 1977 Cross Ref An alternation theory for copositive approximationJournal of Approximation Theory, Vol. 19, No. 2 | 1 Feb 1977 Cross Ref The Remez exchange algorithm for approximation with linear restrictionsTransactions of the American Mathematical Society, Vol. 223, No. 0 | 1 January 1976 Cross Ref Best uniform approximation with Hermite-Birkhoff interpolatory side conditionsJournal of Approximation Theory, Vol. 15, No. 2 | 1 Oct 1975 Cross Ref Optimal rational starting approximationsJournal of Approximation Theory, Vol. 12, No. 2 | 1 Oct 1974 Cross Ref Optimal starting approximations for iterative schemesJournal of Approximation Theory, Vol. 9, No. 1 | 1 Sep 1973 Cross Ref Approximation with Convex ConstraintsJames T. LewisSIAM Review, Vol. 15, No. 1 | 18 July 2006AbstractPDF (2399 KB)A generalization of the varisolvency and unisolvency propertiesJournal of Approximation Theory, Vol. 7, No. 1 | 1 Jan 1973 Cross Ref Nonlinear Tchebycheff approximation with constraintsJournal of Approximation Theory, Vol. 6, No. 3 | 1 Oct 1972 Cross Ref A unified approach to uniform real approximation by polynomials with linear restrictionsTransactions of the American Mathematical Society, Vol. 166, No. 0 | 1 January 1972 Cross Ref Kolmogoroff's criterion for constrained rational approximationJournal of Approximation Theory, Vol. 4, No. 2 | 1 Jun 1971 Cross Ref Uniform rational approximation with osculatory interpolationJournal of Computer and System Sciences, Vol. 4, No. 6 | 1 Dec 1970 Cross Ref Approximation by functions having restricted ranges: Equality caseNumerische Mathematik, Vol. 14, No. 1 | 1 Dec 1969 Cross Ref Volume 6, Issue 2| 1969SIAM Journal on Numerical Analysis161-315 History Submitted:30 September 1968Published online:14 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0706026Article page range:pp. 284-293ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

In this paper, the problem of best uniform polynomial approximation to a continuous function on a compact set X X is approached through the partitioning of X X and the definition of norms corresponding to the partition and each of the standard L p {L_p} norms 1 ≦ p > ∞ 1 \leqq p > \infty . For computational convenience, a pseudo norm is defined corresponding to each partition. When the partition is chosen appropriately, the corresponding best approximations (using both the norms and the pseudo norm) are arbitrarily close to a best uniform approximation. A chracterization theorem for best pseudo norm approximation is presented, along with an alternation theorem for best pseudo norm approximation to a univariate function.

The main result proved in the paper is: iff is absolutely continuous in (−∞, ∞) andf' is in the real Hardy space ReH1, then\(R_n (f) \leqslant C \cdot n^{ - 1} \left\| {f'} \right\|_{\operatorname{Re} H^1 }\) for everyn≥1, whereRn(f) is the best uniform approximation off by rational functions of degreen. This estimate together with the corresponding inverse estimate of V. Russak [15] provides a characterization of uniform rational approximation.

Semiclassical periodic-orbit theory and closed-orbit theory represent a quantum spectrum as a superposition of contributions from individual classical orbits. Close to a bifurcation, these contributions diverge and have to be replaced with a uniform approximation. Its construction requires a normal form that provides a local description of the bifurcation scenario. Usually, the normal form is constructed in flat space. We present an example taken from the hydrogen atom in an electric field where the normal form must be chosen to be defined on a sphere instead of a Euclidean plane. In the example, the necessity to base the normal form on a topologically non-trivial configuration space reveals a subtle interplay between local and global aspects of the phase space structure. We show that a uniform approximation for a bifurcation scenario with non-trivial topology can be constructed using the established uniformization techniques. Semiclassical photo-absorption spectra of the hydrogen atom in an electric field are significantly improved when based on the extended uniform approximations.

A closed subsetE of a Riemann surfaceS is called a set of uniform meromorphic approximation if every functionf continuous onE and holomorphic onE0 can be approximated uniformly onE by meromorphic functions onS. We show that ifE is a set of uniform meromorphic approximation, then so is\(E \cap \bar D\) for every compact parametric diskD. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A. G. Vitushkin.

Uniform polynomial approximation

: Existence, uniqueness and characterization of best uniform approximations of continuous functions by (Tchebycheffian) spline functions with free knots in (a,b) are investigated. (Author)

In this paper, we first give two uniform asymptotic approximations of the eigenfunctions of the weighted finite Fourier transform operator, defined by ${\displaystyle \mathcal F_c^{(\alpha)} f(x)=\int_{-1}^1 e^{icxy} f(y)\,(1-y^2)^{\alpha}\, dy,\,}$ where $ c >0, \alpha > -1$ are two fixed real numbers. The first uniform approximation is given in terms of a Bessel function, whereas the second one is given in terms of a normalized Jacobi polynomial. These eigenfunctions are called generalized prolate spheroidal wave functions (GPSWFs). By using the uniform asymptotic approximations of the GPSWFs, we prove the super-exponential decay rate of the eigenvalues of the operator $\mathcal F_c^{(\alpha)}$ in the case where $0<\alpha < 3/2.$ Finally, by computing the trace and an estimate of the norm of the operator ${\displaystyle \mathcal Q_c^{\alpha}=\frac{c}{2\pi} \mathcal F_c^{{\alpha}^*} \mathcal F_c^{\alpha},}$ we give a lower and an upper bound for the counting number of the eigenvalues of $Q_c^{\alpha},$ when $c>>1.$