Convex polynomial and spline approximation in C[−1, 1]

We prove that a convex functionf C[–1, 1] can be approximated by convex polynomialsp _n of degreen at the rate of ₃(f, 1/n). We show this by proving that the error in approximatingf by C² convex cubic splines withn knots is bounded by ₃(f, 1/n) and that such a spline approximant has anL third derivative which is bounded by n³₃(f, 1/n). Also we prove that iff C²[–1, 1], then it is approximable at the rate ofn ^–2 (f, 1/n) and the two estimates yield the desired result.Communicated by Ronald A. DeVore.     

Distribution of poles of diagonal rational approximants to functions of fast rational approximability

We show that the most of the time, most poles of diagonal multipoint Padé or best rational approximants to functions admitting fast rational approximation, leave the region of meromorphy. Following is a typical result: Letf be single-valued and analytic in CS, where cap(S)=0. Let {n _j } _j=1 be an increasing sequence of positive integers withn _j+1/n _j 1 asj. Then there exists an infinite sequenceL of positive integers such that asj,jL the total multiplicity of poles of any sequence of type (n _j ,n _j ) multipoint Padé or best rational approximants tof, iso(n _j ) in any compactK in whichf is meromorphic. The sequenceL is independent of the particular sequence of multipoint Padé or best approximants, and yields the same behavior for near-best approximants. If the errors of best approximation on some compact set satisfy a weak regularity condition, then we may takeL={1,2,3,}.Communicated by Edward B. Saff.     

A discrepancy theorem concerning polynomials of best approximation inL w p [−1, 1]

Letw be a suitable weight function,B _n,p denote the polynomial of best approximation to a functionf inL _w ^p [–1, 1],v _n be the measure that associates a mass of 1/(n+1) with each of then+1 zeros ofB _n+1,p–B _n,p and be the arcsine measure defined by . We estimate the rate at which the sequencev _n converges to in the weak-^* topology. In particular, our theorem applies to the zeros of monic polynomials of minimalL _w ^p norm.This author gratefully acknowledges partial support from NSA contract #A4235802 during 1992, AFSOR Grant 226113 during 1993 and The Alexander von Humboldt Foundation during both of these years.     

Moduli of smoothness andK-functionals inL p, 0<p<1

It is shown that forL _p, 0p<1, the=">K-functional betweenL _p andW _p ^r is identically zero. Useful measures that are equivalent to the moduli of smoothness are found. The equivalence results that are given are valid for 0p.Communicated by Vilmos Totik.     

The bestL 2-approximation by finite sums of functions with separable variables

Summary We consider the problem of the best approximation of a given functionh L ² (X × Y) by sums _k=1 ⁿ f _k f _k, with a prescribed numbern of products of arbitrary functionsf _k L ² (X) andg _k L ² (Y). As a co-product we develop a new proof of the Hilbert—Schmidt decomposition theorem for functions lying inL ² (X × Y).     

Uniform rational approximation of functions with first derivative in the real hardy space ReH 1

The main result proved in the paper is: iff is absolutely continuous in (–, ) andf' is in the real Hardy space ReH ¹, then for everyn1, whereR _n(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.Communicated by Ronald A. DeVore.     

Cardinal hermite interpolation with box splines II

Summary In the present work we extent the results in [RS] on CHIP, i.e. Cardinal Hermite Interpolation by the span of translates of directional derivatives of a box spline. These directional derivatives are that ones which define the type of the Hermite Interpolation. We admit here several (linearly independent) directions with multiplicities instead of one direction as in [RS]. Under the same assumptions on the smoothness of the box spline and its defining matrixT we can prove as in [RS]: CHIP has a system of fundamental solutions which are inL L ² together with its directional derivatives mentioned above. Moreover, for data sequences inl ^p ( ^d ), 1p2, there is a spline function inL ^p, 1/p+1/p=1, which solves CHIP.Research supported in part by NSERC Canada under Grant # A7687. This research was completed while this author was supported by a grant from the Deutscher Akademischer Austauschdienst     

Conditionally positive functions andp-norm distance matrices

In [4], deep results were obtained concerning the invertibility of matrices arising from radial basis function interpolation. In particular, the Euclidean distance matrix was shown to be invertible for distinct data. In this paper, we investigate the invertibility of distance matrices generated byp-norms. In particular, we show that, for anyp(1, 2), and for distinct pointsx ¹,,x ^{n d} , wheren andd may be any positive integers, with the proviso thatn2, the matrixA ^n×n defined by

Lototsky-Schnabl operators associated with a strictly elliptic differential operator and their corresponding Feller semigroup

Given an open bounded convex subset of ^p , a strictly elliptic differential operatorL and a continuous function , and denoted withT _L the Dirichlet operator associated withL, the Lototsky-Schnabl operators associated withT _L and are investigated. In particular, conditions are established which ensure the existence of a Feller semigroup represented by limit of powers of these operators. Then the analytic expression of the infinitesimal generator is determined and some properties of the semigroup are deduced. Finally, the saturation class of Lototsky-Schnabl operators is determined.Work supported by a C.N.R. Research Grant (n. 201.19.1, November 30, 1994)     

Invariance Principles for Energy Functionals on Spheres

We study a generalised version of the g-energy functionals defined by Damelin and Grabner. We comment on invariance principles for finite energies and use these principles to obtain expansions of these latter energies in terms of cap discrepancies for a subclass of g. This allows for discrepancy estimates knowing bounds on the energy and vice versa. We are, in particular, able to carefully analyse the case when g gives a Riesz kernel g^R_s when 0<sd or a logarithmic kernel g^L₀ in the limits when 0⁺.The author is supported by the START project Y96-MAT of the Austrian Science Fund.     

Interpolating them-th power ofx at the zeros of then-th Chebyshev-polynomial yields an almost best Chebyshev-approximation

LetX={x ₁,x ₂,..., _n }I=[–1, 1] and . ForfC ₁(I) definef* byf–p _f =f*, wherep _f denotes the interpolation-polynomial off with respect toX. We state some properties of the operatorf f*. In particular, we treat the case whereX consists of the zeros of the Chebyshev polynomialT _n (x) and obtain x ^m –p _x ^m8eE _n–1(x ^m ), whereE _n–1(f) denotes the sup-norm distance fromf to the polynomials of degree less thann. Finally we state a lower estimate forE _n (f) that omits theassumptionf ⁽ⁿ⁺¹⁾>0 in a similar estimate of Meinardus.     

Rational approximation to Lipschitz and Zygmund classes

In this paper, characterizations for lim _n(R _n (f)/(n ^–1)=0 inH and for lim _n(n ^r+ R _n (f)=0 inW ^r Lip ,r1, are given, while, forZ, a generalization to a related result of Newman is established.Communicated by Ronald A. DeVore.     

Regarding thep-norms of radial basis interpolation matrices

A radial basis function approximation has the form where:R ^d R is some given (usually radially symmetric) function, (y _j ) ₁ ⁿ are real coefficients, and the centers (x _j ) ₁ ⁿ are points inR ^d . For a wide class of functions , it is known that the interpolation matrixA=((x _j –x _k )) _j,k=1 ⁿ is invertible. Further, several recent papers have provided upper bounds on ||A ^–1||₂, where the points (x _j ) ₁ ⁿ satisfy the condition ||x _j –x _k ||₂,jk, for some positive constant . In this paper we calculate similar upper bounds on ||A ^–1||₂ forp1 which apply when decays sufficiently quickly andA is symmetric and positive definite. We include an application of this analysis to a preconditioning of the interpolation matrixA _n = ((j–k)) _j,k=1 ⁿ when (x)=(x ²+c ²)^1/2, the Hardy multiquadric. In particular, we show that sup _n ||A _n ^–1 || is finite. Furthermore, we find that the bi-infinite symmetric Toeplitz matrix enjoys the remarkable property that ||E ^–1|| _p = ||E ^–1||₂ for everyp1 when is a Gaussian. Indeed, we also show that this property persists for any function which is a tensor product of even, absolutely integrable Pólya frequency functions.Communicated by Charles Micchelli.     

Inequalities for generalized nonnegative polynomials

We define generalized polynomials as products of polynomials raised to positive real powers. The generalized degree can be defined in a natural way. We prove Markov-, Bernstein-, and Remez-type inequalities inL ^p (0p) and Nikolskii-type inequalities for such generalized polynomials. Our results extend the corresponding inequalities for ordinary polynomials.Communicated by George G. Lorentz.     

The Gibbs phenomenon for bestL 1-trigonometric polynomial approximation

The classical Gibbs phenomenon for the Fourier sections (bestL ₂-trigonometric polynomial approximants) of a jump function asserts that, near the jump, these sections overshoot the function by an asymptotically constant factorg (theL ₂-Gibbs constant). In this paper we show that, for a class of one-jump discontinuous functions, a similar phenomenon holds for the trigonometric polynomials of bestL ₁-approximation. We determine theL ₁-Gibbs constant , which is substantially smaller thang. Furthermore, we prove that uniform convergence of bestL ₁-approximants takes place on intervals that avoid the jump. In the analysis we obtain some strong uniqueness theorems for bestL ₁-approximants.Communicated by Vladimir N. Temlyakov.     

On the convergence of multivariable Lagrange interpolants

LetA _dj be a triangular array in a compact setXC ⁿ . Forf analytic in a neighborhood ofX, letL _d (f) denote the Lagrange interpolant tof at staged of the array. In the caseX is locally regular, we construct a continuous function satisfying the complex Monge-Ampère equation onC ⁿ –X, such that iff is analytic onR forR>1 then, for someB>0, we have L _d (f)–f _x B exp(–d logR. In particular, since1 onX, iff is analytic on1, then lim _d L _d (f)–f _x =0.Communicated by Edward B. Saff.     

EfficientL 2 approximation by splines

LetS _N ^k (t) be the linear space ofk-th order splines on [0, 1] having the simple knotst _i determined from a fixed functiont by the rulet _i=t(i/N). In this paper we introduce sequences of operators {Q _N } _N =1 fromC ^k [0, 1] toS _N ^k (t) which are computationally simple and which, asN, give essentially the best possible approximations tof and its firstk–1 derivatives, in the norm ofL ₂[0, 1]. Precisely, we show thatN ^k–1((f–Q _N f) ⁱ –dist₂(f ⁽¹⁾,S _N ^k–1 (t)))0 fori=0, 1, ...,k–1. Several numerical examples are given.The research of this author was partially supported by the National Science Foundation under Grant MCS-77-02464The research of this author was partially supported by the U.S. Army Reesearch Office under Grant No. DAHC04-75-G-0816     

Besov spaces andK-functionals inL p, 0<p<1*

We investigate Besov spaces and their connection with trigonometric polynomial approximation inL _p[−π,π], algebraic polynomial approximation inL _p[−1,1], algebraic polynomial approximation inL _p(S), and entire function of exponential type approximation inL _p(R), and characterizeK-functionals for certain pairs of function spaces including (L _p[−π,π],B _s ^a(L _p[−π,π])), (L _p(R),_s ^a(L_p(R))), , and , where 0<s≤∞, 0<p<1,S is a simple polytope and 0<α<r. This project is supported by the National Science Foundation of China.     

Gaussian Estimates of Order α and L p -Spectral Independence of Generators of C 0-Semigroups

We prove L^p-spectral independence for generators of C₀-semigroups estimated by the positive C₀-semigroup . In the preliminary process of the proof, we obtain the asymptotic expansion formula for the integral kernel of the C₀-semigroup .