Showing papers in "Constructive Approximation in 1986"
IBM1
TL;DR: In this paper, it was shown that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R Franke, which is a conjecture that was later proved in the present paper.
Abstract: Among other things, we prove that multiquadric surface interpolation is always solvable, thereby settling a conjecture of R Franke
1,476 citations
TL;DR: In this article, the authors introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x��i)=y fixmei fori e {0,1,⋯,N}.
Abstract: Let a data set {(x
i,y
i) ∈I×R;i=0,1,⋯,N} be given, whereI=[x
0,x
N]⊂R. We introduce iterated function systems whose attractorsG are graphs of continuous functionsf∶I→R, which interpolate the data according tof(x
i)=y
i fori e {0,1,⋯,N}. Results are presented on the existence, coding theory, functional equations and moment theory for such fractal interpolation functions. Applications to the approximation of naturally wiggly functions, which may show some kind of geometrical self-similarity under magnification, such as profiles of cloud tops and mountain ranges, are envisaged.
736 citations
TL;DR: In this paper, the authors studied the asymptotic behavior of polynomials with degree ≥ n satisfying the orthogonal relation and proved that ωm+n(z) is a polynomial of degreem+n+1 with all its zeros contained inV, and the path of integrationC separatesV from the setE.
Abstract: In this paper we continue our study of the asymptotic behavior of polynomialsQmn(z), m, n ∈N, of degree≤n satisfying the orthogonal relation
$$( * )\oint_c {\zeta ^l Q_{mn} (\zeta )} \frac{{f(\zeta )d\zeta }}{{\omega _{m + n} (\zeta )}} = 0,l = 0,...,n - 1,$$
(*)
and all its singularities are supposed to be contained in a set\(E \subseteq \hat C\) of capacity zero, ωm+n(z) is a polynomial of degreem+n+1 with all its zeros contained inV, and the path of integrationC separatesV from the setE. We state and prove results concerning the asymptotic magnitude of the integral in (*) forl=n,n+1,⋯.
160 citations
TL;DR: In this article, the authors focus on the analysis of numerical techniques for solving the nonlinear system and on the theoretical issues that arise when certain extensions of the convex interpolation problem are considered.
Abstract: Numerical and theoretical questions related to constrained interpolation and smoothing are treated. The prototype problem is that of finding the smoothest convex interpolant to given univariate data. Recent results have shown that this convex programming problem with infinite constraints can be recast as a finite parametric nonlinear system whose solution is closely related to the second derivative of the desired interpolating function. This paper focuses on the analysis of numerical techniques for solving the nonlinear system and on the theoretical issues that arise when certain extensions of the problem are considered. In particular, we show that two standard iteration techniques, the Jacobi and Gauss-Seidel methods, are globally convergent when applied to this problem. In addition we use the problem structure to develop an efficient implementation of Newton's method and observe consistent quadratic convergence. We also develop a theory for the existence, uniqueness, and representation of solutions to the convex interpolation problem with nonzero lower bounds on the second derivative (strict convexity). Finally, a smoothing spline analogue to the convex interpolation problem is studied with reference to the computation of convex approximations to noisy data.
81 citations
TL;DR: In this article, an axiomatic approach to vector-valued rational interpolation is presented, in which unique defined interpolants are constructed for vectorvalued data so that the components of the resulting vector-value rational interpolant share a common denominator polynomial.
Abstract: In 1963, Wynn proposed a method for rational interpolation of vector-valued quantities given on a set of distinct interpolation points. He used continued fractions, and generalized inverses for the reciprocals of vector-valued quantities. In this paper, we present an axiomatic approach to vector-valued rational interpolation. Uniquely defined interpolants are constructed for vector-valued data so that the components of the resulting vector-valued rational interpolant share a common denominator polynomial. An explicit determinantal formula is given for the denominator polynomial for the cases of (i) vector-valued rational interpolation on distinct real or complex points and (ii) vector-valued Pade approximation. We derive the connection with thee-algorithm of Wynn and Claessens, and we establish a five-term recurrence relation for the denominator polynomials.
69 citations
TL;DR: For the weights exp (−|x|λ), 0<λ≤1, the Markov-Bernstein constant turns out to be of order logn for λ = 1 and of order 1 for 0 <λ < 1 as discussed by the authors.
Abstract: For the weights exp (−|x|λ), 0<λ≤1, we prove the exact analogue of the Markov-Bernstein inequality. The Markov-Bernstein constant turns out to be of order logn for λ=1 and of order 1 for 0<λ<1. The proof is based on the solution of the problem of how fast a polynomialPn can decrease on [−1,1] ifPn (0)=1. The answer to this problem has several other consequences in different directions; among others, it leads to a general theorem about the incompleteness of the set of polynomials in weightedLp spaces.
49 citations
TL;DR: In this article, the asymptotic behavior of polynomials of degreen was shown to be asn→∞ and showed that it agrees with a previous conjecture.
Abstract: Polynomialsp1,(z),p2(z), of degreen are defined by the relation\(p_1 (z) + p_2 (z)\prod
olimits_{i = 1}^3 {(z - b_l )^{v_1 } } = O(z^{ - n - 1} ),z \to \infty \), where\(\sum
olimits_{i = 1}^3 {v_i = 0} \). We obtain the asymptotic behavior of these polynomials asn→∞ and show that it agrees with a previous conjecture.
41 citations
TL;DR: In this paper, it was shown that even ordered PC-fraction converges to Caratheodory functions and that the odd ordered denominators of positive PC-Fractions are the odd-order Szego polynomials.
Abstract: General T-fractions and M-fractions whose approximants form diagonals in two-point Pade tables are subsumed here under the study of Perron-Caratheodory continued fractions (PC-fractions) whose approximants form diagonals in weak two-point Pade tables. The correspondence of PC-fractions with pairs of formal power series is characterized in terms of Toeplitz determinants. For the subclass of positive PC-fractions, it is shown that even ordered approximants converge to Caratheodory functions. This result is used to establish sufficient conditions for the existence of a solution to the trigonometric moment problem and to provide a new starting point for the study of Szego polynomials orthogonal on the unit circle. Szego polynomials are shown to be the odd ordered denominators of positive PC-fractions. Positive PC-fractions are also related to Wiener filters used in digital signal processing [3], [25].
38 citations
TL;DR: In this article, exact asymptotics and explicit formulas are obtained for the sequences {ℰm, {pm, qm, and rm}, where rm has lead coefficient 1 is considered.
Abstract: Approximation to exp of the form
wherepm,qm, andrm are polynomials of degree at mostm andpm has lead coefficient 1 is considered. Exact asymptotics and explicit formulas are obtained for the sequences {ℰm}, {pm}, {qm}, and {rm}. It is observed that the above sequences all satisfy the simple four-term recursion:
$$\begin{array}{*{20}c} {T_{m + 3} = \frac{1}{{3m + 4}}[( - 6m - 14)z^3 T_m } \\ { + (9m + 15)(z^2 + (3m + 4)(3m + 7))T_{m + 1} + 3zT_{m + 2} ].} \\ \end{array} $$
It is also observed that these generalized Pade-type approximations can be used to asymptotically minimize expressions of the above form on the unit disk.
37 citations
TL;DR: In this paper, the problem of finding a polynomial P(x, y) whose values and the values of their derivatives at given points match given data is studied, and methods of Birkhoff interpolation are used throughout.
Abstract: This paper is devoted to bivariate interpolation. The problem is to find a polynomialP(x, y) whose values and the values of whose derivatives at given points match given data. Methods of Birkhoff interpolation are used throughout. We define interpolation matricesE, their regularity, their almost regularity, and finally the regularity of the pairE, Z for a given set of knotsZ. Many concrete examples and applications are possible.
29 citations
TL;DR: In this article, the Bernstein polynomial Bnf for ϕ−αf∈C[0,1] was derived in terms of the modulus of continuity (of second order).
Abstract: The purpose of this paper is to derive the estimate (0≤α≤2,n∈N,ϕ(x)=[x(1−x)]1/2)
$$\omega _\alpha (n^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-
ulldelimiterspace} 2}} ,f) \leqslant M_\alpha n^{ - 1} \sum\limits_{k = 1}^n {\left\| {\varphi ^{ - \alpha } (B_k f - f)} \right\|} c$$
in terms of the modulus of continuity (of second order)
$$\omega _\alpha (t,f): = \sup \{ \varphi ^{ - \alpha } (x)|\Delta _{h\varphi (x)}^ * f(x)|:x,x \pm h\varphi (x) \in [0,1],0< h \leqslant t\} $$
and the Bernstein polynomial Bnf for ϕ−αf∈C[0,1].
TL;DR: In this article, an asymptotic expansion including error bounds is given for polynomials {Pn, Qn} that are biorthogonal on the unit circle with respect to the weight function (1−eiθ)α+β(1−e−iα)α−β.
Abstract: An asymptotic expansion including error bounds is given for polynomials {Pn, Qn} that are biorthogonal on the unit circle with respect to the weight function (1−eiθ)α+β(1−e−iθ)α−β. The asymptotic parameter isn; the expansion is uniform with respect toz in compact subsets ofC{0}. The pointz=1 is an interesting point, where the asymptotic behavior of the polynomials strongly changes. The approximants in the expansions are confluent hyper-geometric functions. The polynomials are special cases of the Gauss hyper-geometric functions. In fact, with the results of the paper it follows how (in a uniform way) the confluent hypergeometric function is obtained as the limit of the hypergeometric function2F1(a, b; c; z/b), asb→±∞,z≠b, withz=0 as “transition” point in the uniform expansion.
TL;DR: In this article, several fast numerical schemes based on solving an initial-value problem for the Cauchy-Riemann equations are analyzed, which are well-suited for generating level curves and stream lines of conformal mappings.
Abstract: The problem of computing the analytic continuation of a holomorphic function known on a circle is considered. Several fast numerical schemes based on solving an initial-value problem for the Cauchy-Riemann equations are analyzed. To avoid instability problems, some of the schemes consist of two parts: one for integrating the Cauchy-Riemann equations, and one for smoothing the function values so obtained. We show that with appropriate integration and smoothing methods, the stability and accuracy of such schemes is sufficient for many applications. The schemes are well-suited for generating level curves and stream lines of conformal mappings. Computed examples are presented. We also indicate how the schemes can be used to generate near-orthogonal boundary-fitted grids with given mesh sizes along the boundary.
TL;DR: In this paper, the existence of n-fold antiderivatives of a holomorphic function φ is shown to hold for all n ∈ N0 and z ∈ Ω.
Abstract: Let Ω ⊂C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ
(−n)} ofn-fold antiderivatives (i.e., we haveφ
(0)(z)∶=φ(z) andφ
(−n)(z)=dφ
(−n−1)(z)/dz for alln ∈ N0 and z ∈ Ω) such that the following properties hold:
TL;DR: Aq-integral representation of Rogers' q-ultraspherical polynomials is obtained by using Sears' summation formula for balanced non-terminating series as discussed by the authors, which is then used to give a simple derivation of the Gasper-Rahman formula for the Poisson kernel.
Abstract: Aq-integral representation of Rogers'q-ultraspherical polynomialsC
n
(x;β∥q) is obtained by using Sears' summation formula for balanced non-terminating3
φ
2 series. It is then used to give a simple derivation of the Gasper-Rahman formula for the Poisson kernel ofC
n
(x;β∥q). As another application it is shown how this representation can be directly used to give an asymptotic expansion of theq-ultraspherical polynomials.
TL;DR: In this article, it was shown that for each carrier of positive capacity there is a measurev which is carrier related to μ, such that the equilibrium measure of the carrierB is the weak limit of the sequence {v n(v)} = 1/∞potion.
Abstract: Letμ be a positive unit Borel measure with infinite support on the interval [−1, 1]. LetP
n(x, μ) denote the monic orthogonal polynomial of degreen associated withμ, and letv
n(μ) denote the unit measure with mass 1/n at each zero ofP
n(x, μ). A carrier is a Borel subset of the support ofμ having unitμ-measure, and a measurev is carrier related toμ when it has the same carriers asμ. We demonstrate that for each carrierB of positive capacity there is a measurev, which is carrier related toμ, such that the equilibrium measure of the carrierB is the weak limit of the sequence {v
n(v)}
=1/∞
.
TL;DR: In this article, the authors determined exactly the nonnegative quantity for any complex polynomial of degree at mostn (n≥ 1) and any complex number of degree n (n ≥ 1) for which the three-dimensional surface, generated by the points (Reμ, Imμ, Sn(μ)) has the interesting shape of a volcano.
Abstract: WithPn denoting the set of complex polynomials of degree at mostn (n≥1), define, for any complex numberμ, the subset
$$P_n (\mu ): = \{ p_n (z) \in P_n :p_n (0) = 1 and p_n (1) = \mu \} .$$
In this paper, we determine exactly the nonnegative quantity
$$S_n (\mu ): = \mathop {\sup \{ \min |p_n (z)|\} }\limits_{p_n \in P_n (\mu )|z| \leqslant 1} ,$$
as a function ofn andμ. For fixedn≥2, the three-dimensional surface, generated by the points (Reμ, Imμ,Sn(μ)) for all complex numbersμ, has the interesting shape of a volcano.
TL;DR: The class of continuous functions possessing n−α(1/p <α≤1) order of approximation by Bernstein polynomials in Lp[0, 1] is characterized in this paper.
Abstract: The class of all continuous functions possessing n−α(1/p<α≤1) order of approximation by Bernstein polynomials inLp[0, 1] is characterized.
TL;DR: In this article, the uniqueness of monosplines and perfect splines of least-L��p-norm is analyzed in the framework of generalized mono-pline and total positivity.
Abstract: The uniqueness of monosplines and perfect splines of leastL
p-norm is treated in the framework of generalized monosplines and total positivity. The analysis is based on the invariance properties of the degree of a certain mapping and on a new composition result for totally positive kernels. For theL
p-case 1
TL;DR: In this paper, a nontrivial function having the properties described in the title is given explicitly, and the properties of the function are described in terms of a nonparametric function.
Abstract: A nontrivial function having the properties described in the title is given explicitly.
TL;DR: This paper presents a technique for estimating the error incurred in replacing a matrix exponential by a rational approximation by introducing another approximation, of superior order, whose aposteriori evaluation is cheap.
Abstract: Several methods for the numerical solution of stiff ordinary differential equations require approximation of an exponential of a matrix. In the present paper we present a technique for estimating the error incurred in replacing a matrix exponential by a rational approximation. This estimation is done by introducing another approximation, of superior order, whose aposteriori evaluation is cheap. Properties of the new approximation pertaining to both its stability and the behavior of the error for matrices with negative eigenvalues are analyzed.
TL;DR: In this paper, the supremum of (max∥z∥=1∥p′n (z)∥)/ (max ∥z ∥= 1∥pn(z) ∥) taken over all polynomialspn of degree at mostn having a zero on the unit circle is shown.
Abstract: Let us denote byΛn, 1 the supremum of (max∥z∥=1∥p′n (z)∥)/ (max∥z∥=1∥pn (z)∥) taken over all polynomialspn of degree at mostn having a zero on the unit circle {z ∈ C∶∥z∥=1}. We show that Λn.1=n-(π2/16)(1/n)+O(1/n2.
TL;DR: In this article, the authors considered the space W(n, σ; F) of functions of the form σF(αjx) Pj(x) where there arem(n) terms in the sum.
Abstract: Fix a positive integern and σ>0. ForF continuous and positive on [0, ∞), we consider the spaceW(n, σ; F) of functions of the form σF(αjx) Pj(x) where there arem(≤n) terms in the sum; thePj's are polynomials of total degree not exceedingn — m; and 0≤αj≤αj+1-α, j=1, 2,⋯, m-1. Under certain conditions onF (primarily that it increase rapidly enough to ∞ asx goes to ∞),W(n, σ; F) is an existence space forC[0,1].
TL;DR: A lower bound on the error of an arbitrary algorithm is established by assuming that the information about the function is given by its Taylor coefficients, and an algorithm whose error is asymptotically at most twice the lower bound is presented.
Abstract: This paper deals with the problem of approximate evaluation of a certain class of analytic functions. The choice of this class is motivated by the problem of the summation of moment sequences. By assuming that the information about the function is given by its Taylor coefficients, we are able to establish a lower bound on the error of an arbitrary algorithm. We present also an algorithm whose error is asymptotically at most twice the lower bound, thereby showing that our estimate is asymptotically sharp.
TL;DR: In this paper, it was shown that there exists exactly one value of γ in each of min(n+1,m) interpolation intervals such that the uniform error over R− is at a local minimum.
Abstract: LetRn/m(z∶γ)=Pn(z∶γ)/(1−γz) m be a rational approximation to exp (z),z ∈C, of ordern for all real positiveγ. In this paper we show there exists exactly one value ofγ in each of min(n+1,m) interpolation intervals such that the uniform error overR− is at a local minimum.
TL;DR: In this article, it was shown that the error in best approximation to f in the supremum norm by rational functions of type (m, n) with real and complex coefficients, respectively, is Ω(n−1−1≥0======\/\/\/\/\/\/√ 1−2\/\/\/\/1−2\/\/\/1 −1,1] = \tfrac{1}{2}.
Abstract: Iff∈C[−1, 1] is real-valued, letE R mn (f) andE C mn (f) be the errors in best approximation tof in the supremum norm by rational functions of type (m, n) with real and complex coefficients, respectively. We show that form≥n−1≥0
$$\gamma _{mn} = \inf \{ {{E_{mn}^C (f)} \mathord{\left/ {\vphantom {{E_{mn}^C (f)} {E_{mn}^R (f)}}} \right. \kern-
ulldelimiterspace} {E_{mn}^R (f)}}:f \in C[ - 1,1]\} = \tfrac{1}{2}.$$