Tangential Markov Inequalities on Transcendental Curves

Abstract. We show that on the curves γ:=(x,e ^t(x) ) , x∈ [a,b] , where t(x) is a fixed polynomial, there holds a tangential Markov inequality of exponent four for algebraic polynomials P _N (x,y) of degree at most N in each variable x,y: ||(P _N (x,e ^t(x) ))'|| _[a,b] ≤ CN ⁴ ||P _N || _γ , and the exponent four is sharp. On the other hand, the corresponding tangential Markov factors on curves y=x ^α with irrational α grow exponentially in the degree of the polynomials.     

Exact Markov-Type Inequalities for Oscillating Perfect Splines

We prove the inequality for each perfect spline s of degree r with n-r knots and n zeros in [-1,1] . Here T _rn is the Tchebycheff perfect spline of degree r with n-r knots, normalized by the condition ||T _rn || _C[-1,1] := max  _{-1≤ x ≤ 1} |T _rn (x)| =1 . The constant ||T _rn ^(k) || _Lp[-1,1] in the above inequality is the best possible. June 8, 1999. Date revised: May 31, 2000. Date accepted: January 16, 2001.     

Uniqueness of Markov-Extremal Polynomials on Symmetric Convex Bodies

For a compact set K\subset R ^d with nonempty interior, the Markov constants M _n (K) can be defined as the maximal possible absolute value attained on K by the gradient vector of an n -degree polynomial p with maximum norm 1 on K . It is known that for convex, symmetric bodies M _n (K) = n ² /r(K) , where r(K) is the ``half-width' (i.e., the radius of the maximal inscribed ball) of the body K . We study extremal polynomials of this Markov inequality, and show that they are essentially unique if and only if K has a certain geometric property, called flatness. For example, for the unit ball B ^d (\smallbf 0, 1) we do not have uniqueness, while for the unit cube [-1,1] ^d the extremal polynomials are essentially unique. September 9, 1999. Date revised: September 28, 2000. Date accepted: November 14, 2000.     

Best Meromorphic Approximation of Markov Functions on the Unit Circle

Let E subset(-1,1) be a compact set, let μ be a positive Borel measure with support supp μ =E , and let H _p (G), 1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z colon |z|=1} . Let Δ _n,p be the error in best approximation of the Markov function frac{1}{2π i} ∈t_E frac{d μ(x)}{z-x} in the space L _p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H _p (G), Q is a polynomial of degree at most n , Qnot equiv 0 . We investigate the rate of decrease of Δ _n,p , 1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1<p≤∈fty and the measure μ with support E=[a,b] satisfies the Szegő condition ∈t_a^b frac{log(d μ/ d x)}{sqrt{(x-a)(b-x)}} dx >- ∈fty. July 27, 2000. Final version received: May 19, 2001.     

L p -Error Bounds for Hermite Interpolation and the Associated Wirtinger Inequalities

The B-spline representation for divided differences is used, for the first time, to provide L _p -bounds for the error in Hermite interpolation, and its derivatives, thereby simplifying and improving the results to be found in the extensive literature on the problem. These bounds are equivalent to certain Wirtinger inequalities. The major result is the inequality where H_Θ f is the Hermite interpolant to f at the multiset of n points Θ, and is the diameter of . This inequality significantly improves upon Beesack's inequality, on which almost all the bounds given over the last 30 years have been based. Date received: June 24, 1994 Date revised: February 4, 1996.     

Inequalities with exponential weights

Let R=(-∞,∞)$\mathbb{R}=(-\infty ,\infty )$ and let Q∈C²:R→R⁺=[0,∞)$Q\in C^2:\mathbb{R}\to {\mathbb{R}}^{+}=[0,\infty )$ be an even function. Then in this paper we consider the infinite–finite range inequality, an estimate for the Christoffel function, and the Markov–Bernstein inequality with the exponential weights _wρ(x)=|x|^ρe^-Q(x),x∈R$w_{\rho }(x)=|x{|}^{\rho }{\mathrm{e}}^{-Q(x)},x\in \mathbb{R}$.     

Generalized Hyperinterpolation on the Sphere and the Newman—Shapiro Operators

Hyperinterpolation on the sphere, as introduced by Sloan in 1995, is a constructive approximation method which is favorable in comparison with interpolation, but still lacking in pointwise convergence in the uniform norm. For this reason we combine the idea of hyperinterpolation and of summation in a concept of generalized hyperinterpolation. This is no longer projectory, but convergent if a matrix method A is used which satisfies some assumptions. Especially we study A partial sums which are defined by some singular integral used by Newman and Shapiro in 1964 to derive a Jackson-type inequality on the sphere. We could prove in 1999 that this inequality is realized even by the corresponding discrete operators, which are generalized hyperinterpolation operators. In view of this result the Newman—Shapiro operators themselves gain new attention. Actually, in their case, A furnishes second-order approximation, which is best possible for positive operators. As an application we discuss a method for tomography, which reconstructs smooth and nonsmooth components at their adequate rate of convergence. However, it is an open question how second-order results can be obtained in the discrete case, this means in generalized hyperinterpolation itself, if results of this kind are possible at all. March 9, 2000. Date revised: October 2, 2000. Date accepted: March 8, 2001.     

Markov—Bernstein-Type Inequality for Trigonometric Polynomials with Respect to Doubling Weights on [-ω,ω]

   Abstract. Various important, weighted polynomial inequalities, such as Bernstein, Marcinkiewicz, Nikolskii, Schur, Remez, etc., have been proved recently by Giuseppe Mastroianni and Vilmos Totik under minimal assumptions on the weights. In most cases this minimal assumption is the doubling condition. Here, based on a recently proved Bernstein-type inequality by D. S. Lubinsky, we establish Markov—Bernstein-type inequalities for trigonometric polynomials with respect to doubling weights on [-ω,ω] . Namely, we show the theorem below. Theorem Let p ∈ [1,∞) and ω ∈ (0, 1/2] . Suppose W is a weight function on [-ω,ω] such that W(ω cos t) is a doubling weight. Then there is a constant C depending only on p and the doubling constant L so that

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.     

Asymptotic Uniqueness of Best Rational Approximants of Given Degree to Markov Functions in L 2 of the Circle

We improve over a sufficient condition given in [8] for uniqueness of a nondegenerate critical point in best rational approximation of prescribed degree over the conjugate-symmetric Hardy space of the complement of the disk. The improved condition connects to error estimates in AAK approximation, and is necessary and sufficient when the function to be approximated is of Markov type. For Markov functions whose defining measure satisfies the Szego condition, we combine what precedes with sharp asymptotics in multipoint Padé approximation from [43], [40] in order to prove uniqueness of a critical point when the degree of the approximant goes large. This lends perspective to the uniqueness issue for more general classes of functions defined through Cauchy integrals.     

On the Power of Standard Information for Weighted Approximation

We study weighted approximation of multivariate functions for classes of standard and linear information in the worst case and average case settings. Under natural assumptions, we show a relation between n th minimal errors for these two classes of information. This relation enables us to infer convergence and error bounds for standard information, as well as the equivalence of tractability and strong tractability for the two classes. April 11, 2001. Final version received: May 29, 2001.     

More on Comonotone Polynomial Approximation

The main achievement of this paper is that we show, what was to us, a surprising conclusion, namely, twice continuously differentiable functions in (0,1) (with some regular behavior at the endpoints) which change monotonicity at least once in the interval, are approximable better by comonotone polynomials, than are such functions that are merely monotone. We obtain Jackson-type estimates for the comonotone polynomial approximation of such functions that are impossible to achieve for monotone approximation. July 7, 1998. Date revised: May 5, 1999. Date accepted: July 23, 1999.     

On Rational Interpolation to |x|

We consider Newman-type rational interpolation to |x| induced by arbitrary sets of interpolation nodes, and we show that under mild restrictions on the location of the interpolation nodes, the corresponding sequence of rational interpolants converges to |x|. Date received: August 18, 1995. Date revised: January 10, 1996.     

Upper and Lower Bounds on the Dimension of Superspline Spaces

Upper and lower bounds are provided on the dimension of bivariate polynomial superspline spaces which are defined by enforcing smoothness conditions across the interior edges of the underlying triangulation. The results generalize known bounds for classical spline spaces. As an example of the usefulness of such bounds, we show how they can be applied to analyze a new macroelement.     

Shape-Preserving Widths of Weighted Sobolev-Type Classes of Positive, Monotone, and Convex Functions on a Finite Interval

   Abstract. Let I be a finite interval, r∈ N and ρ(t)= dist {t, I} , t∈ I . Denote by Δ ^s ₊ L _q the subset of all functions y∈ L _q such that the s -difference Δ ^s _τ y(t) is nonnegative on I ,

Note on the Approximation of Powers of the Distance in Two-Dimensional Domains

Although Newman's trick has been mainly applied to the approximation of univariate functions, it is also appropriate for the approximation of multivariate functions that are encountered in connection with Green's functions for elliptic differential equations. The asymptotics of the real-valued function on a ball in 2-space coincides with that for an approximation problem in the complex plane. The note contains an open problem. May 17, 1999. Date revised: October 20, 1999. Date accepted: March 17, 2000.     

Compactly Supported, Piecewise Affine Scaling Functions on Triangulations

The methods of using intertwining MRAs to find orthogonal scaling functions have previously been applied to one-dimensional MRAs and are here extended to two-dimensional bases. Two examples are constructed from MRAs consisting of continuous, compactly supported, piecewise affine functions of two variables. The resulting scaling functions can be conveniently restricted to compact domains. November 23, 1997. Date accepted: January 22, 1999.     

Asymptotic Distribution of Nodes for Near-Optimal Polynomial Interpolation on Certain Curves in R 2

Let E\subset \Bbb R ^s be compact and let d _n ^E denote the dimension of the space of polynomials of degree at most n in s variables restricted to E . We introduce the notion of an asymptotic interpolation measure (AIM). Such a measure, if it exists , describes the asymptotic behavior of any scheme τ _n ={ \bf x _k,n } _k=1 ^dnE , n=1,2,\ldots , of nodes for multivariate polynomial interpolation for which the norms of the corresponding interpolation operators do not grow geometrically large with n . We demonstrate the existence of AIMs for the finite union of compact subsets of certain algebraic curves in R ² . It turns out that the theory of logarithmic potentials with external fields plays a useful role in the investigation. Furthermore, for the sets mentioned above, we give a computationally simple construction for ``good' interpolation schemes. November 9, 2000. Date revised: August 4, 2001. Date accepted: September 14, 2001.     

On Markoff's Inequality

For compact subsets of the real line the Markoff factors increase at least as fast as n ² . In this paper it is shown that there are sets of measure zero with Markoff factors of order n ² . We shall also show that this cannot happen for compact sets of logarithmic capacity zero. In connection with Markoff's inequality for polynomials of two variables we show a set E

Piecewise Monotone Pointwise Approximation

Let r, k, s be three integers such that , or We prove the following: Proposition. Let Y:={y _i } _i=1 ^s be a fixed collection of distinct points y _i ∈ (-1,1) and Π (x):= (x-y ₁ ). ... .(x-y _s ). Let I:=[-1,1]. If f ∈ C ^(r) (I) and f'(x)Π(x) ≥ 0, x ∈ I, then for each integer n ≥ k+r-1 there is an algebraic polynomial P _n =P _n (x) of degree ≤ n such that P _n '(x) Π (x) ≥ 0 and $$ \vert f(x)-P_n(x) \vert \le B\left(\frac{1}{n^2}+\frac{1}{n}\sqrt{1-x^2}\right)^r \omega_k \left(f^{(r)};\frac{1}{n^2}+\frac{1}{n}\sqrt{1-x^2}\right) \legno{(1)}$$ for all x∈ I, where ω _k (f ^(r) ;t) is the modulus of smoothness of the k -th order of the function f ^(r) and B is a constant depending only on r , k , and Y. If s=1, the constant B does not depend on Y except in the case (r=1, k=3). In addition it is shown that (1) does not hold for r=1, k>3. March 20, 1995. Dates revised: March 11, 1996; December 20, 1996; and August 7, 1997.