Tchebycheff Approximation Research Articles

On existence theorems for some class of non-linear Tchebycheff approximations

On existence theorems for some class of non-linear Tchebycheff approximations

Approximate solutions to H2 optimization problems with H2 and H infinity constraints

In this paper, a multiobjective H2 optimal control problem is considered in which the controllers are constrained to satisfy prescribed upper bounds on the H2 or H infinity norms of several closed-loop transfer functions. A sequence of H2-cost/ H2-constraint problems (i.e. H2 problems with a single H2 constraint) is presented which aims at generating approximate solutions to the original problem this may be viewed as a generalization of the so-called Lawson algorithm for Tchebycheff approximation. Each such H2/H2 problem has a unique rational solution which can be computed through line search and spectral factorization. Under appropriate assumptions, it is shown that the sequence of optimal cost values for the H2/H2 problems is increasing and bounded from above (hence, it is convergent) and that, whenever the corresponding sequence of solutions converges, it does so to the solution of the original problem. In the special case in which there are only H2 constraints, convergence is established for the sequence of solutions of the auxiliary H2-cost/H2- constraint problems.

Phase and delay approximation for 1-D digital IIR filter in the L∞norm

The phase response of a digital all-pass filter is often used to approximate a prescribed phase or delay. Based on the nonlinear Tchebycheff approximation theory, the phase and delay functions are investigated. A number of necessary and sufficient (NS) conditions that characterize the best approximation are obtained. Numerical design methods using the phase or delay functions of a digital all-pass filter yield the minimal unique global solution. A trigonometric rational function is derived for optimal approximation of a desired group delay.

A TWO-DIMENSIONAL TCHEBYCHEFF COLLOCATION METHOD FOR THE STUDY OF THE VIBRATION OF A BAFFLED FLUID-LOADED RECTANGULAR PLATE

The vibration and the acoustic radiation of a baffled rectangular plate in contact with a dense fluid is considered. By using Green's representation theorem, the original system of differential equations governing both the displacement and the acoustic sound pressure is transformed into a system of coupled boundary integral equations. The unknown functions (the displacement of the plate, the pressure jump at the surface of the plate and the boundary sources) are expanded as series of Tchebycheff polynomials. The new unknowns (the coefficients of the series) are calculated by using a collocation method. Some results, both theoretical and numerical, are given concerning the resolution properties of the Tchebycheff approximation. The numerical difficulties are presented, and solutions of these are proposed. Numerical examples are given.

ALGORITHME DE PERTURBATION DIRECTRICE POUR LE PROBLÈME D'AJUSTEMENT DES DONNEES

In this paper we propose an algorithm, called Directrive Perturbation (PD-Algorithm), which decrease the maximum value of the absolute errors of a fitting formula obtained by a Least–Squares Method. Applying the algorithm we can increase the accuracy of the uniform approximation of the fitting formula with the original data (xi, f(xi)) (i=1, 2, …, m). Further, after applying a finite number of times of the perturbation, we can obtain the formuala of Tchebycheff approximation.

A stochastic Remes algorithm

We treat the linear Tchebycheff approximation problem for a regression function ƒ ϵ C 2[0, 1] . A stochastic Remes algorithm which uses only estimates of first derivatives is proposed and investigated for almost sure convergence and rate of convergence.

Tchebycheff approximation of continuous functions by harmonic polynomials on conic sections

The problem of finding a best Tchebycheff approximation to a given continuous function f, defined on a compact portion of a plane conic section, from the set of harmonic polynomials of degree n or less is studied. It is shown that the Haar condition is fulfilled by such harmonic polynomials. Interesting relationships which exist between this problem and certain classical approximation problems are explored. Numerical examples are given to illustrate the theory.

Optimality conditions in nondifferentiable programming and their applications to best approximations

This paper studies constrained optimization problems in Banach spaces without usual differentiability and convexity assumptions on the functionals involved in the data. The aim is to give optimality conditions for the problems from which one can derive the characterization of best approximations. The objective and the inequality constraint functionals are assumed to have one-sided directional derivatives. First-order necessary conditions are given in terms of subdifferentials of the directional derivatives. The notion of “max-pseudoconvexity” weaker than pseudoconvexity is introduced for sufficiency. The optimality conditions are applied to linear and nonlinear Tchebycheff approximation problems to derive the characterization of best approximations.

A FREQUENCY DOMAIN MAPPING APPROXIMATION OF MOVEOUT FILTERS WITH APPLICATIONS*

AbstractWavenumber aliasing is the main limitation of conventional optimum least‐squares linear moveout filters: it prevents adequate reject domain weighting for efficient coherent noise rejection. A general frequency domain multichannel filter design technique based on a one‐to‐one mapping method between two‐dimensional (2D) space and one‐dimensional (1D) space is presented. The 2D desired response is mapped to the 1D frequency axis after a suitable sorting of the coefficients. A min‐max or Tchebycheff approximation to the desired response is obtained in the 1D frequency domain and mapped back to the 2D frequency domain. The algorithm is suitable for multiband 2D filter design. No aliasing damage is inherent in the linear moveout filters designed using this technique because the approximation is done in the frequency‐wavenumber (f, k)‐domain.Linear moveout filters designed by using the present coefficient mapping technique achieve better pass domain approximations than the corresponding conventional least‐squares filters. Compatible reject domain approximations can be obtained from suitable mappings of the origin coefficient of the desired (f k)‐response to the 1D frequency axis. The (fk)‐responses of linear moveout filters designed by using the new technique show equi‐ripple behavior. Synthetic and real data applications show that the present technique is superior to the optimum least‐squares filters and straight stacking in recovering and enhancing the signal events with relatively high residual statics. Their outputs also show higher resolution than those of the optimum least‐squares filters.

H-sets in linear approximation

This paper studies the role H-sets play in finding the best linear Tchebycheff approximation to a given continuous function. A simple definition is given for H-sets and the algebraic theory for linear approximation is developed. We find that many of the theorems where the Haar condition is supposed can be generalized in terms of H-sets; thus a general framework for Linear Tchebycheff Approximation is made.

Moment-theory investigations of photoabsorption and dispersion profiles in atoms and ions

Moment-theory methods for the construction of photoabsorption and dispersion profiles from associated dipole spectral moments are described and applied to simple atoms and ions. A previously devised (Stieltjes) moment approach, which provides convergent histogram approximations to absorption and dispersion profiles, is refined and extended to the use of arbitrarily large numbers of spectral moments, and an improved (Tchebycheff) moment approach is introduced which gives profiles that are continuous in the photoionization region and exhibit the 8-function-like behavior associated with discrete transitions at the appropriate frequencies. Recurrence relations for the polynomials orthogonal and quasiorthogonal with respect to the distributions are employed in solving the necessary moment problems involving large numbers of spectral moments. The methods are applied in illustrative calculations of absorption and dispersion profiles in oneand two-electron atoms and ions. In the case of one-electron atomic systems the necessary polynomial recurrence coefficients are obtained in closed form from the known spectral moments, allowing the construction of distributions which reproduce the known profiles with high accuracy, employing as many as 100 spectral moments. Variational calculations using large basis sets of square-integrable functions, including the special functions required to satisfy sum rules, provide accurate spectral moments for atomic helium and the negative hydrogen ion. A simple moment-extension procedure is devised to interpolate the associated recurrence coefficients to infinite order employing their known asymptotic values. The associated Stieltjes and Tchebycheff approximations to the absorption and dispersion profiles obtained in these cases are in excellent agreement with available measurements and previous accurate calculations employing discrete and continuum ;;ave functions.

Tchebycheff extraction of the periodic vector set from the Patterson function

The image-seeking method of Buerger and the procedure of Tokonami and Hosoya appear to be capable of dealing with complex crystal structures via the Patterson function, provided that the periodic vector set is accurately determined. Unfortunately, a general and powerful method for the location of peaks in the Patterson function has not yet been developed, and it is the lack of such a method which now prevents the formulation of a general Patterson method of structure analysis. This paper presents further results in the author's attempts to formulate a general method of vector-set extraction by representing the Patterson function as a linear generalized polynomial in a system of independent interatomic functions. This approach has the advantage in that the essentially non-linear problem of vector-set extraction is reduced to an apparently simple linear problem, namely that of determining the coefficients of the approximating polynomial. In the present paper, the Tchebycheff approximation norm is employed with coefficient determination by linear-programming procedures. Since linear-programming methods are flexible and extremely powerful, this Tchebycheff vector-set-extraction procedure is much more promising than the author's earlier published methods, which were based on interpolatory approximations.

Nonlinear Tchebycheff approximation with constraints

Nonlinear Tchebycheff approximation with constraints

Discrete tchebycheff approximation for multivariate splines

Discrete tchebycheff approximation for multivariate splines

A method for solving the linear tchebycheff approximation problem when the haar condition is not satisfied

Die vorliegende Arbeit gibt eine Verallgemeinerung desRemes-Algorithmus zur linearenTschebyscheff-Approximation bei nicht erfullterHaarscher Bedingung an. Gegenuber fruheren Verfahren wird unter schwacheren voraussetzungen eine Folge von Approximationen konstruiert, die Minimalfolge des Approximationsproblems ist und sich mit weitgehend reduziertem Speicherbedarf und Rechenaufwand berechnen last.

Uniqueness in the Uniform Approximation of a Function and Its Derivatives

Previous article Uniqueness in the Uniform Approximation of a Function and Its DerivativesF. J. SchuurmannF. J. Schuurmannhttps://doi.org/10.1137/0706029PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] N. I. Achieser, Theory of approximation, Translated by Charles J. Hyman, Frederick Ungar Publishing Co., New York, 1956x+307 MR0095369 0072.28403 Google Scholar[2] R. Creighton Buck, Rudolph Langer, Linear spaces and approximation theory, On numerical approximation. Proceedings of a Symposium, Madison, April 21-23, 1958, Edited by R. E. Langer. Publication no. 1 of the Mathematics Research Center, U.S. Army, the University of Wisconsin, The University of Wisconsin Press, Madison, 1959, 11–23 MR0101611 0083.28801 Google Scholar[3] E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York, 1966xii+259 MR0222517 0161.25202 Google Scholar[4] D. G. Moursund, Chebyshev approximations of a function and its derivatives, Math. Comp., 18 (1964), 382–389 MR0166529 0119.32903 CrossrefISIGoogle Scholar[5] E. Ya. Remez, General computational methods of Chebyshev approximation. The problem with linear and real parameters, United States Atomic Energy Commission, Washington, D.C., 1962 Google Scholar[6] John R. Rice, Tchebycheff approximation in several variables, Trans. Amer. Math. Soc., 109 (1963), 444–466 MR0157165 0144.05802 CrossrefGoogle Scholar[7] Lee Johnson, Unicity in approximation of a function and its derivatives, Math. Comp., 22 (1968), 873–875 MR0241856 0176.14101 CrossrefISIGoogle Scholar Previous article FiguresRelatedReferencesCited byDetails Simultaneous approximation, interpolation and Birkhoff systemsJournal of Approximation Theory, Vol. 37, No. 2 Cross Ref Hermite—Birkhoff interpolation and approximation of a function and its derivativesJournal of Approximation Theory, Vol. 30, No. 2 Cross Ref Volume 6, Issue 2| 1969SIAM Journal on Numerical Analysis History Submitted:23 January 1968Published online:14 July 2006 InformationCopyright © 1969 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0706029Article page range:pp. 305-315ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

Uniform Generalized Weight Function Polynomial Approximation with Interpolation

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

Introduction to Approximation Theory

Introduction: 1 Examples and prospectus 2 Metric spaces 3 Normed linear spaces 4 Inner-product spaces 5 Convexity 6 Existence and unicity of best approximations 7 Convex functions The Tchebycheff Solution of Inconsistent Linear Equations: 1 Introduction 2 Systems of equations with one unknown 3 Characterization of the solution 4 The special case 5 Polya's algorithm 6 The ascent algorithm 7 The descent algorithm 8 Convex programming Tchebycheff Approximation by Polynomials and Other Linear Families: 1 Introduction 2 Interpolation 3 The Weierstrass theorem 4 General linear families 5 the unicity problem 6 Discretization errors: General theory 7 Discretization: Algebraic polynomials. The inequalities of Markoff and Bernstein 8 Algorithms Least-squares Approximation and Related Topics: 1 Introduction 2 Orthogonal systems of polynomials 3 Convergence of orthogonal expansions 4 Approximation by series of Tchebycheff polynomials 5 Discrete least-squares approximation 6 The Jackson theorems Rational Approximation: 1 Introduction 2 Existence of best rational approximations 3 The characterization of best approximations 4 Unicity Continuity of best-approximation operators 5 Algorithms 6 Pade Approximation and its generalizations 7 Continued fractions Some Additional Topics: 1 The Stone approximation theorem 2 The Muntz theorem 3 The converses of the Jackson theorems 4 Polygonal approximation and bases in $C[a, b]$ 5 The Kharshiladze-Lozinski theorems 6 Approximation in the mean Notes References Index.

Chebyshev and $l^1 $-Solutions of Linear Equations Using Least Squares Solutions

Previous article Next article Chebyshev and $l^1 $-Solutions of Linear Equations Using Least Squares SolutionsC. S. Duris and V. P. SreedharanC. S. Duris and V. P. Sreedharanhttps://doi.org/10.1137/0705040PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] E. Ya. Remez, General computation methods for Čebyšev approximation. Problems with real parameters entering linearly, Izdat. Akad. Nauk Ukrainsk. SSR. Kiev, 1957, 454–, A translation is available through the Office of Technical Services, Department of Commerce, Washington, D. C. MR0088788 Google Scholar[2] Alston S. Householder, The theory of matrices in numerical analysis, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964xi+257 MR0175290 0161.12101 Google Scholar[3] H. S. Wilf, A. Ralston and , Herbert S. Wilf, Matrix inversion by the method of rank annihilationMathematical methods for digital computers, Wiley, New York, 1960, 73–77 MR0117911 Google Scholar[4] Eduard L. Stiefel, An introduction to numerical mathematics, Translated by Werner C. Rheinboldt and Cornelie J. Rheinboldt, Academic Press, New York, 1963x+286 MR0181077 0154.16702 Google Scholar[5] D. Jackson, The Theory of Approximation, American Mathematical Society, Providence, 1930 56.0936.01 Google Scholar[6] M. S. Bartlett, An inverse matrix adjustment arising in discriminant analysis, Ann. Math. Statistics, 22 (1951), 107–111 MR0040068 0042.38203 CrossrefISIGoogle Scholar[7] C. De La Vallée Poussin, Sur la méthode de l'approximation minimum, Ann. Soc. Sci. Bruxelles, 35 (1911), 1–16, Seconde Partie, Mémoires 42.0255.02 Google Scholar[8] David Moursund, Chebyshev solution of $n+1$ linear equations in n unknowns, J. Assoc. Comput. Mach., 12 (1965), 383–387 MR0182139 0142.11505 CrossrefISIGoogle Scholar[9] E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York, 1966xii+259 MR0222517 0161.25202 Google Scholar[10] John R. Rice, The approximation of functions. Vol. I: Linear theory, Addison-Wesley Publishing Co., Reading, Mass.-London, 1964xi+203 MR0166520 0114.27001 Google Scholar[11] Allen A. Goldstein and , Ward Cheney, A finite algorithm for the solution of consistent linear equations and inequalities and for the Tchebycheff approximation of inconsistent linear equations, Pacific J. Math., 8 (1958), 415–427 MR0101505 0084.01902 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Theoretical Upperbound of the Spurious-Free Dynamic Range in Direct Digital Frequency Synthesizers Realized by Polynomial Interpolation MethodsIEEE Transactions on Circuits and Systems I: Regular Papers, Vol. 54, No. 10 Cross Ref Minimum ℓ1, ℓ2, and ℓ∞ Norm Approximate Solutions to an Overdetermined System of Linear EquationsDigital Signal Processing, Vol. 12, No. 4 Cross Ref An algorithm for estimating the parameters in multiple linear regression model with linear constraintsComputers & Industrial Engineering, Vol. 28, No. 4 Cross Ref Minimization technique for a convex function with application to multiple regression model27 June 2007 | Optimization, Vol. 19, No. 2 Cross Ref THE CHEBYSHEV ADJUSTMENT OF A GEODETIC LEVELLING NETWORK19 July 2013 | Survey Review, Vol. 28, No. 220 Cross Ref The Chebyshev solution of the linear matrix equationAX+YB=CNumerische Mathematik, Vol. 46, No. 3 Cross Ref On a particular case of the inconsistent linear matrix equation AX+YB=CLinear Algebra and its Applications, Vol. 66 Cross Ref Discrete Chebyshev Approximation with Linear ConstraintsMichael Brannigan14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 22, No. 1AbstractPDF (1337 KB)The strict Chebyshev solution of overdetermined systems of linear equations with rank deficient matrixNumerische Mathematik, Vol. 40, No. 3 Cross Ref Computational methods of linear algebraJournal of Soviet Mathematics, Vol. 15, No. 5 Cross Ref Least absolute values estimation: an introduction27 June 2007 | Communications in Statistics - Simulation and Computation, Vol. 6, No. 4 Cross Ref An overdetermined linear systemJournal of Approximation Theory, Vol. 18, No. 3 Cross Ref Annotated Bibliography on Generalized Inverses and Applications Cross Ref Chebyshev solution of overdetermined systems of linear equationsBIT, Vol. 15, No. 2 Cross Ref A new algorithm for the Chebyshev solution of overdetermined linear systems1 January 1974 | Mathematics of Computation, Vol. 28, No. 125 Cross Ref A Finite Step Algorithm for Determining the “Strict” Chebyshev Solution to $Ax=b$C. S. Duris and M. G. Temple14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 10, No. 4AbstractPDF (787 KB)Least squares algorithms for finding solutions of overdetermined linear equations which minimize error in an abstract normNumerische Mathematik, Vol. 17, No. 5 Cross Ref Note on the fitting of non-equispaced two-dimensional dataGeoexploration, Vol. 8, No. 1 Cross Ref Solutions of overdetermined linear equations which minimize error in an abstract normNumerische Mathematik, Vol. 13, No. 2 Cross Ref Volume 5, Issue 3| 1968SIAM Journal on Numerical Analysis History Submitted:14 November 1966Accepted:06 February 1968Published online:14 July 2006 InformationCopyright © 1968 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0705040Article page range:pp. 491-505ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

A characterization of best Tchebycheff approximations in function spaces

A characterization of best Tchebycheff approximations in function spaces