Showing papers in "Constructive Approximation in 1985"

Where Does the Sup Norm of a Weighted Polynomial Live? (A Generalization of Incomplete Polynomials)

Abstract: A characterization is given of the sets supporting the uniform norms of weighted polynomials (w(x))nPn(x), where Pn is any polynomial of degree at most n. The (closed) support r, of w(x) may be bounded or unbounded; of special interest is the case when w(x) has a nonempty zero set Z. The treatment of weighted polynomials consists of associating each admissible weight with a certain functional defined on subsets of r~ - Z. One main result of this paper states that there is a unique compact set (in- dependent of n and P,) maximizing this functional that contains the points where the norms of weighted polynomials are attained. The distribution of the zeros of Chebyshev polynomials corresponding to the weights (w(x))" is also studied. The main theorems give a unified method of investigating many particular examples. Applications to weighted approximation on the real line with respect to a fixed weight are included.

ConstrainedL p approximation

Abstract: In this paper, we solve a class of constrained optimization problems that lead to algorithms for the construction of convex interpolants to convex data.

Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle

Abstract: Consider a system {φ n } of polynomials orthonormal on the unit circle with respect to a measuredμ, withμ′>0 almost everywhere. Denoting byk n the leading coefficient ofφ n , a simple new proof is given for E. A. Rakhmanov's important result that lim n→∞,k n /k n+1=1; this result plays a crucial role in extending Szego's theory about polynomials orthogonal with respect to measuresdμ with logμ′∈L 1 to a wider class of orthogonal polynomials.

A Recurrence Relation for Chebyshevian B-Splines

Abstract: The fundamental recurrence relation for polynomialB-splines is generalized to ChebyshevianB-splines.

Local linear independence of the translates of a box spline

Abstract: Let Ξ=(ξ i ) l be a sequence of vectors inR m . The box splineM Ξ is defined as the distribution given by $$M_\Xi :\varphi \to \int_{[0,1]^n } \varphi \left( {\sum\limits_{i = 1}^n {\lambda (i)\xi _i } } \right)d\lambda ,\varphi \in C_c^\infty (R^m ).$$ . Suppose that Ξ contains a basis forR m . ThenM Ξ∈L ∞(R m ). Assume $$\Xi \subset V: = z^m .$$ . Consider the translatesM v :=M Ξ(·−v),v∈V. It is known that (M v ) V is linearly dependent unless (*) $$|\det Z| = 1forallbasesZ \subset \Xi$$ . This paper demonstrates that under condition (*), (M v ) V is locally linearly independent, i.e., $$\{ M_v ;\sup p M_v \cap A e ot 0\}$$ is linearly independent over any open setA.

Pointwise Estimates for Monotone Polynomial Approximation

Abstract: We prove that if f is increasing on ( - 1,1), then for each n = 1, 2 ..... there is an increasing algebraic polynomial P. of degree n such that {f(x) - P.(x){ < cw2( f, V/I - x 2/n), where w2 is the second-order modulus of smooth- ness. These results complement the classical pointwise estimates of the same type for unconstrained polynomial approximation. Using these results, we char- acterize the monotone functions in the generalized Lipschitz spaces through their approximation properties.

On the bernstein conjecture in approximation theory

Abstract: WithE 2n (|x|) denoting the error of best uniform approximation to |x| by polynomials of degree at most 2n on the interval [−1, +1], the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constantβ for which lim 2nE 2n (|x|)=β.n→∞ Moreover, by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds forβ: 0.278<β<0.286. Now, the average of these bounds is 0.282, which, as Bernstein noted as a “curious coincidence,” is very close to 1/(2√π)=0.2820947917... This observation has over the years become known as the Bernstein Conjecture: Isβ=1/(2√π)? We show here that the Bernstein conjecture isfalse. In addition, we determine rigorous upper and lower bounds forβ, and by means of the Richardson extrapolation procedure, estimateβ to approximately 50 decimal places.

Asymptotics for orthogonal polynomials defined by a recurrence relation

Abstract: Asymptotic expansions are given for orthogonal polynomials when the coefficients in the three-term recursion formula generating the orthogonal polynomials form sequences of bounded variation.

n-Widths of Sobolev spaces inL p

Abstract: LetWp(r)={f:f∈Cr−1[0, 1],f(r−1) abs.cont., ∥f(r)∥p<∞}, and setBp(r)={f:f∈Wp(r),∥f(r)∥p≤1}. We find the exact Kolmogorov, Gel'fand, linear, and Bernsteinn-widths ofBp(r) inLp for allp∈(1, ∞). For the Kolmogorovn-width we show that forn≥r there exists an optimal subspace of splines of degreer−1 withn−r fixed simple knots depending onp.

The attractive Coulomb potential polynomials

Abstract: TheJ matrix method in quantum mechanics developed by Heller, Reinhardt, and Yamani points to a set of orthogonal polynomials having a nonempty continuous spectrum in addition to an infinite discrete spectrum. Asymptotic methods are used to investigate the spectral properties of these polynomials. We also obtain generating functions for both numerator and denominator polynomials. The corresponding continued fraction is computed and the Stieltjes inversion formula is used to determine the distribution function.

Padé tables of entire functions of very slow and smooth growth, II

Abstract: Letf(z):=Σ j=0 ∞ a j z j , where aj≠ 0,j large enough, and for someq e C such that ¦q¦ 0, we show that form large enough, and for everyn = 1,2,3,..., $$(1 - \varepsilon )^n \leqslant \left| {{{D(m/n)} \mathord{\left/ {\vphantom {{D(m/n)} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right. \kern- ulldelimiterspace} {\left\{ {a_m^n \mathop \Pi \limits_{j - 1}^{n - 1} (1 - q_m^j )^{n - j} } \right\}}}} \right| \leqslant (1 + \varepsilon )^n .$$ We apply this to show that any sequence of Pade approximants {[m k /n k ]} 1 ∞ tof, withm k →∞ ask→ ∞, converges locally uniformly in C. In particular, the diagonal sequence {[n/n]} 1 ∞ converges throughout C. Further, under additional assumptions, we give sharper asymptotics forD(m/n).

Approximate Newton Methods and Homotopy for Stationary Operator Equations

Abstract: A quadratically convergent algorithm based on a Newton-type iteration is defined to approximate roots of operator equations in Banach spaces. Frechet derivative operator invertibility is not required; approximate right inverses are used in a neighborhood of the root. This result, which requires an initially small residual, is sufficiently robust to yield existence; it may be viewed as a generalized version of the Kantorovich theorem. A second algorithm, based on continuation via single, Euler-predictor-Newton-corrector iterates, is also presented. It has the merit of controlling the residual until the homotopy terminates, at which point the first algorithm applies. This method is capable of yielding existence of a solution curve as well. An application is given for operators described by compact perturbations of the identity.

On the convergence rates of subdivision algorithms for box spline surfaces

Abstract: Dahmen and Micchelli [8] have shown that in general the coefficients of the refined control nets of a box spline surface converge to the surface at (at least) the rate of the refinement. The purpose of this article is to show that under mild additional assumptions the convergence rate is even quadratic. Although this rate is in general best possible, we point out under what circumstances even higher rates are obtained (locally).

Divergence of diagonal Padé approximants and the asymptotic behavior of orthogonal polynomials associated with nonpositive measures

Abstract: In this paper we prove the existence of real- and complex-valued measuresμ on the interval [−1,1] with the property that the diagonal Pade approximants [n/n],n=1,2,..., to the functionf(z)=∫dμ(x)/(x−z) neither converge at any fixed pointz∈C∼[−1,1] nor converge in capacity on any open (nonempty) setS inC∼[−1,1]. This result is derived from a theorem on the asymptotic behavior of orthogonal polynomials. It will be shown that it is possible to construct measuresμ. on [−1,1] such that for any arbitrarily prescribed asymptotic behavior there exist subsequences of the associated orthogonal polynomialsQ n that have this behavior.

Convergence of bivariate cardinal interpolation

Abstract: We give necessary and sufficient conditions for the convergence of cardinal interpolation with bivariate box splines as the degree tends to infinity.

Order of approximation by electrostatic fields due to electrons

Abstract: LetD be a Jordan domain in the complex plane andA q (D) the Bers space with norm ∥ ∥ q . IfD is the unit disk, it is known that ∥S n 0∥2≥π/18, whereS n =∑ k=1 n l/(z−z nk ) withz nk ∈∂D, so that approximation in ∥ ∥ q ,q<-2, is not possible. In this paper, we give an order of estimate of ∥f−S n ∥ q for 2

On the rational approximation ofe −x on [0, ∞)

Abstract: In recent years the problem of uniform approximation ofe −x on [0, ∞) by rational functions has found wide interest. In this paper we present a method for the construction of rational approximations toe −x that seem to come arbitrarily near to the asymptotically best one. We start with a generalization of the integral form of the Pade approximant by introducing certain real parametersa j ,b i ,k andl. The corresponding error function has again an integral representation and is estimated for everyx∈[0,∞) by the Laplace method. This leads to the consideration of a finite number of new error functionsφ i·j whose maxima are determined by a set of nonlinear equations and, under some restrictions on thea j ,b i ,k, andl, are also unique. Variation of these parameters according to a special algorithm and computation of the corresponding maxima of the functionsφ i·j shows that forn→∞ the order of rational approximation ofe −x on [0, ∞) is at least 9.03−n .

A characterization of best complex rational approximants in a fundamental case

Abstract: Necessary conditions for a given complex rational approximantR= p/q, degp

Asymptotic centers and best compact approximation of operators intoC(K)

Abstract: The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX*. IfK is just one convergent sequence, the condition is that everyω*-convergent sequence inX* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω2) and from l1 intoL1 without best compact approximation. We also construct spacesX1,X2, isomorphic to a Hilbert space, and operatorsT1,∶X1→C(ω2),T2∶l1→X2 without best compact approximations.

Generalized monosplines and optimal approximation

Abstract: Generalized monosplines of least norm are shown to exist and to determine optimal approximation processes such as numerical integration, interpolation and best approximating spaces. This extends various classical results related to monosplines and perfect splines, which are particular cases of generalized monosplines. The analysis here also provides for a unified treatment of the two classical classes of monosplines and perfect splines of least norm, and of their extremal properties.

Sections of the Taylor expansion of exp(g(z)) whereg(z) is entire and admissible in the sense of Hayman

Abstract: The author considers $$f(z) = \exp (g(z)) = \sum\limits_{j = 0}^\infty {a_j z^j ,}$$ , whereg(z) is a real, entire, transcendental function admissible in the sense of W. K. Hayman [(1956): Reine Angew. Math.,196:67-95]. The aim of the paper is to study, asm→+∞, the distribution of the zeros of the partial sums $$s_m (z) = \sum\limits_{j = 0}^m {a_j z^j .}$$ The results are stated in terms of Hayman's auxiliary functions Ifr>0 is large enough, botha(r) andb(r) are positive,a(r) is strictly increasing, and $$a(r) \to + \infty ,b(r) \to + \infty (r \to + \infty ).$$ Define the sequence (R m ) (m>m 0) by the relationsa(R m )=m. From the following proposition, typical of those stated in the paper, it is easy to deduce accurate information regarding those zeros ofs m (z) that lie near the positive axis: Letζ be an auxiliary complex variable; then asm→+∞, and forR=R m , the functions $$\left\{ {1 + \zeta \left( {\frac{2}{{b(R)}}} \right)^{1/2} } \right\}^{ - m} \{ f(R)\} ^{ - 1} s_m \left( {R\left( {1 + \zeta \left( {\frac{2}{{b(R)}}} \right)^{1/2} } \right)} \right)$$ tend to $$\frac{1}{2}e^{\zeta ^2 } \left( {1 - \frac{2}{{\sqrt \pi }}\int_0^\zeta {e^{ - \sigma ^2 } d\sigma } } \right)$$ uniformly on every compact subset of theζ-plane. There are similar, equally precise, results covering those zeros ofs m (z) that lie near any rayte iϕ(0

Trigonometric Polynomials with Many Real Zeros

Abstract: This is a contribution to the theory of “incomplete trigonometric polynomials”Tn, but mainly for the case when their zeros are not concentrated at just one point, but are distributed in some intervalI whose length is not too large. We begin with the simple theorem that if ∥Tn∥ ≤ 1 and ifTn has ≥θn, 0 ∥∞<-1, thenf vanishes on a set of the circleT whose measure is controlled by lim sup (Nn/n), whereNn is the number of zeros ofTn onT. In turn, this has further applications to series of polynomials, to norms of Lagrange operators, and to Hardy classes.

On projections inL 1 andL

Abstract: Let (X n ) be an increasing sequence ofn-dimensional subspaces inL∞. LetP n be a sequence of projections fromL1 orLt8 ontoX n , written in the integral form(p n f)(t)=∫K n (s,t)f(s)ds. We prove that if ∥K n −Kn−1∥∞<0(logn), then sup∥p n ∥=∞. This theorem extends some results of Olevskii [3] and Kwapien and Szarek [2].