Three-Convex Approximation by Free Knot Splines in C [0, 1]

A Jackson-type estimate is obtained for the approximation of 3 -convex functions by 3 -convex splines with free knots. The order of approximation is the same as for the Jackson-type estimate for unconstrained approximation by splines with free knots. Shape-preserving free knot spline approximation of k -convex functions, k > 3 , is also considered. January 15, 1996. Date revised: December 9, 1996.     

On the Asymptotically Optimal Rate of Approximation of Multiplier Operators from L p into L q

The asymptotic behavior of the n -widths of multiplier operators from L _p [0,1] into L _q [0,1] is studied. General upper and lower bounds for the n -widths in terms of the multipliers are established. Moreover, it is shown that these upper and lower bounds coincide for some important concrete examples. August 3, 1994. Date revised: November 15, 1996.     

A Problem of Totik on Fast Decreasing Polynomials

We solve a problem posed by V. Totik on the existence of fast-decreasing polynomials p _n of degree with p _n (0)=1 and for . For the largest c for which such polynomials exist was known. We give the solution for β > 2 . April 18, 1996.     

Gaussian Versus Optimal Integration of Analytic Functions

We consider error estimates for optimal and Gaussian quadrature formulas if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szeg?-type weight functions and ellipses as regions of analyticity. In this situation, the error constants for the Gaussian formulas are close to the obtained lower bounds, which proves the quality of the Gaussian formulas and also of the lower bounds. In the sequel, different regions of analyticity are investigated. It turns out that almost exclusively for ellipses, the Gaussian formulas are near-optimal. For classes of simply connected regions of analyticity, which are additionally symmetric to the real axis, the asymptotic of the worst ratio between the error constants of the Gaussian formulas and the optimal error constants is calculated. As a by-product, we prove explicit lower bounds for the Christoffel-function for the constant weight function and arguments outside the interval of integration. September 7, 1995. Date revised: October 25, 1996.     

On Measures Induced by Extremal Signatures in Best Real Polynomial Approximation

We consider the limit distribution of measures μ _n , that appear in extremal signatures in the best polynomial approximation of a real-valued function . Relations between structural properties of the function f and weak-star limit points of (μ _n ) _n are proved. April 4, 1996. Date revised: October 25, 1996.     

Polynomial Approximation of Conformal Maps

Let G be a finite domain, bounded by a Jordan curve Γ , and let f ₀ be a conformal map of G onto the unit disk. We are interested in the best rate of uniform convergence of polynomial approximation to f ₀ , in the case that Γ is piecewise-analytic without cusps. In particular, we consider the problem of approximating f ₀ by the Bieberbach polynomials π _n and derive results better than those in [5] and [6] for the case that the corners of Γ have interior angles of the form π/N . In the proof, the Lehman formulas for the asymptotic expansion of mapping functions near analytic corners are used. We study the question when these expansions contain logarithmic terms. December 6, 1995. Date revised: August 5, 1996.     

TURBS—Topologically Unrestricted Rational B-Splines

We present a new approach to the construction of piecewise polynomial or rational C ^k -spline surfaces of arbitrary topological structure. The basic idea is to use exclusively parametric smoothness conditions, and to solve the well-known problems at extraordinary points by admitting singular parametrizations. The smoothness of the spline surfaces is guaranteed by specifying a regular smooth reparametrization explicitly. The resulting space of topologically unrestricted rational B-splines (TURBS) is linear and possesses a natural refinement property. Compared with all known methods the construction principle of TURBS is of striking simplicity and the required polynomial bi-degree is essentially decreased from O(k ² ) to d=2k+2 . January 5, 1996. Date revised: September 5, 1996.     

Uniform Asymptotic Expansions for Meixner Polynomials

Meixner polynomials m _n (x;β,c) form a postive-definite orthogonal system on the positive real line x > 0 with respect to a distribution step function whose jumps are Unlike classical orthogonal polynomials, they do not satisfy a second-order linear differential equation. In this paper, we derive two infinite asymptotic expansions for m _n (nα;β,c) as . One holds uniformly for , and the other holds uniformly for , where a and b are two small positive quantities. Both expansions involve the parabolic cylinder function and its derivative. Our results include all five asymptotic formulas recently given by W. M. Y. Goh as special cases. April 16, 1996. Date revised: October 30, 1996.     

Hyperbolic Wavelet Approximation

We study the multivariate approximation by certain partial sums (hyperbolic wavelet sums) of wavelet bases formed by tensor products of univariate wavelets. We characterize spaces of functions which have a prescribed approximation error by hyperbolic wavelet sums in terms of a K -functional and interpolation spaces. The results parallel those for hyperbolic trigonometric cross approximation of periodic functions [DPT]. October 16, 1995. Date revised: August 28, 1996.     

On L p Extremal Polynomials

A Turán-type inequality for L _p extremal polynomials is given and mean convergence of Lagrange interpolation based on the zeros of L _p extremal polynomials is investigated. November 8, 1994. Date revised: January 23, 1997.     

Hardy Approximation to L p Functions on Subsets of the Circle with 1< =p< = infinity

We consider approximation of L ^p functions by Hardy functions on subsets of the circle for . After some preliminaries on the possibility of such an approximation which are connected to recovery problems of the Carleman type, we prove existence and uniqueness of the solution to a generalized extremal problem involving norm constraints on the complementary subset. December 6, 1995. Date revised: August 26, 1996.     

Extremal Polynomials with Preassigned Zeros and Rational Approximants

Let p _n be the n th orthonormal polynomial with respect to a positive finite measure μ supported by Δ=[-1,1] . It is well known that, uniformly on compact subsets of C/Δ , and, for a large class of measures μ , where g _Ω (z) is Green's function of with pole at infinity. It is also well known that these limit relations give convergence of the diagonal Padé approximants of the Markov function to f on Ω with a certain geometric speed measured by g _Ω (z) . We prove corresponding results when we restrict the freedom of p _n by preassigning some of the zeros. This means that the Padé approximants are replaced by Padé-type approximants where some of the poles are preassigned. We also replace Δ by general compact subsets of C. July 12, 1995. Date revised: October 1, 1996.     

Multipoint Padé-Type Approximants. Exact Rate of Convergence

We study the rate with which sequences of interpolating rational functions, whose poles are partially fixed, approximate Markov-type analytic functions. Applications to interpolating quadratures are given. January 25, 1996. Date revised: December 26, 1996.     

Plancherel—Rotach Asymptotics for the Charlier Polynomials

We derive an asymptotic approximation of Plancherel—Rotach type for the Charlier polynomials on the positive real line. July 26, 1993. Date revised: December 2, 1996.     

Critical Points and Error Rank in Best H 2 Matrix Rational Approximation of Fixed McMillan Degree

This paper deals with best rational approximation of prescribed McMillan degree to matrix-valued functions in the real Hardy space of the complement of the unit disk endowed with the Frobenius L ₂ -norm. We describe the topological structure of the set of approximants in terms of inner-unstable factorizations. This allows us to establish a two-sided tangential interpolation equation for the critical points of the criterion, and to prove that the rank of the error F-H is at most k-n when F is rational of degree k , and H is critical of degree n . In the particular case where k=n , it follows that H=F is the unique critical point, and this entails a local uniqueness result when approximating near-rational functions. January 23, 1996. Date revised: September 16, 1996.     

Finite-dimensional approximation of neutral systems with v₀-stable root chains

An approximate finite-dimensional representation is constructedfor a linear, time-invariant, neutral delay system whoseroot chains are v₀-stable.     

Semigroups of linear operators on p-Fréchet spaces, 0 < p < 1

We develop the beginning of a theory of semigroups of linear operators on p-Fréchet spaces, 0 < p < 1 (which are non-locally convex F-spaces), and give some applications.     

A Simple Proof of the Theorem Concerning Optimality in a One-Dimensional Ergodic Control Problem

We give a simple proof of the theorem concerning optimality in a one-dimensional ergodic control problem. We characterize the optimal control in the class of all Markov controls. Our proof is probabilistic and does not need to solve the corresponding Bellman equation. This simplifies the proof. Accepted 24 March 1998     

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.     

Berezin Forms on Line Bundles over Complex Hyperbolic Spaces

We consider complex hyperbolic spaces where and , line bundles , over them and representations of in smooth sections of (the representation is induced by a character of ). We define a Berezin form $, associated with , and give an explicit decomposition of this form into invariant Hermitian (sesqui-linear) forms for irreducible representations of the group for all and . It is the main result of the paper. Besides it, we give the Plancherel formula for . As it turns out, this formula is, en essence, one of the particular cases of the Plancherel formula for the quasiregular representation for rank one semisimple symmetric spaces, see [20], it can be obtained from the quasiregular Plancherel formula for hyperbolic spaces (complex, quaternion, octonion) by analytic continuation in the dimension of the root subspaces. The decomposition of the Berezin form allows us to define and study the Berezin transform, - in particular, to find out an explicit expression of this transform in terms of the Laplacian. Using that, we establish the correspondence principle (an asymptotic expansion as ). At last, considering , we observe an interpolation in the spirit of Neretin between Plancherel formulae for and for the similar representation for a compact form of the space . Submitted: July 12, 2001?Revised: February 12, 2002