An Adaptive Compression Algorithm in Besov Spaces

Given a function f on [0,1] and a wavelet-type expansion of f , we introduce a new algorithm providing an approximation $\tilde f of f with a prescribed number D of nonzero coefficients in its expansion. This algorithm depends only on the number of coefficients to be kept and not on any smoothness assumption on f . Nevertheless it provides the optimal rate D ^-α of approximation with respect to the L _q -norm when f belongs to some Besov space B ^α _p,∈fty whenever α>(1/p-1/q) ⁺ . These results extend to more general expansions including splines and piecewise polynomials and to multivariate functions. Moreover, this construction allows us to compute easily the metric entropy of Besov balls. June 21, 1996. Dates revised: April 9, 1998; October 14, 1998. Date accepted: October 20, 1998.     

Kergin Interpolants of Holomorphic Functions

Let D be a C-convex domain in C ⁿ . Let , and d = 0,1,2, ..., be an array of points in a compact set . Let f be holomorphic on and let K _d (f) denote the Kergin interpolating polynomial to f at A _d0 ,... , A _dd . We give conditions on the array and D such that . The conditions are, in an appropriate sense, optimal. This result generalizes classical one variable results on the convergence of Lagrange—Hermite interpolants of analytic functions. Date received: October 21, 1995. Date revised: May 1, 1996.     

Variational Principles and Sobolev-Type Estimates for Generalized Interpolation on a Riemannian Manifold

The purpose of this paper is to study certain variational principles and Sobolev-type estimates for the approximation order resulting from using strictly positive definite kernels to do generalized Hermite interpolation on a closed (i.e., no boundary), compact, connected, orientable, m -dimensional C ^∞ Riemannian manifold , with C ^∞ metric g _ij . The rate of approximation can be more fully analyzed with rates of approximation given in terms of Sobolev norms. Estimates on the rate of convergence for generalized Hermite and other distributional interpolants can be obtained in certain circumstances and, finally, the constants appearing in the approximation order inequalities are explicit. Our focus in this paper will be on approximation rates in the cases of the circle, other tori, and the 2 -sphere. April 10, 1996. Dates revised: March 26, 1997; August 26, 1997. Date accepted: September 12, 1997. Communicated by Ronald A. DeVore.     

Bounds on Projections onto Bivariate Polynomial Spline Spaces with Stable Local Bases

L _& ∞ bounds for norms of projections onto bivariate polynomial spline spaces on regular triangulations with stable local bases are established. The general results are then applied to obtain error bounds for best L ₂ - and l ₂ -approximation by splines on quasi-uniform triangulations. March 8, 2000. Date revised: November 20, 2000. Date accepted: July 9, 2001.     

On the Convergence in Capacity of Rational Approximants

Let {r _n } be a sequence of rational functions deg( r _n ≤ n) that converge rapidly in measure to an analytic function f on an open set in C ^N . We show that {r _n } converges rapidly in capacity to f on its natural domain of definition W _f (which, by a result of Goncar, is an open subset of C ^N ). In particular, for f meromorphic on C ^N and analytic near zero the sequence of Padé approximants {π _n (z, f, λ)} (as defined by Goncar) converges rapidly in capacity to f on C ^N . January 14, 1999. Date revised: October 7, 1999. Date accepted: November 1, 1999.     

Convergence of Rational Interpolants with Preassigned Poles

Let Ω be a domain in the extended complex plane such that ∞∈Ω . Further, let K= C / Ω and, for each n , let Q _n be a monic polynomial of degree n with all its zeros in K . This paper is concerned with whether (Q _n ) can be chosen so that, if f is any holomorphic function on Ω and P _n is the polynomial part of the Laurent expansion of Q _n f at ∞ , then (P _n /Q _n ) converges to f locally uniformly on Ω . It is shown that such a sequence (Q _n ) can be chosen if and only if either K has zero logarithmic capacity or Ω is regular. January 21, 1999. Date accepted: August 17, 1999.     

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.     

The approximation by q-Bernstein polynomials in the case q ↓ 1

Let B_n (f, q; x), n=1, 2, ... , 0 < q < ∞, be the q-Bernstein polynomials of a function f, B_n (f, 1; x) being the classical Bernstein polynomials. It is proved that, in general, {B_n (f, q_n; x)} with q_n ↓ 1 is not an approximating sequence for f ∈C[0, 1], in contrast to the standard case q_n ↓ 1. At the same time, there exists a sequence 0 < δ_n ↓ 0 such that the condition implies the approximation of f by {B_n (f, q_n; x)} for all f ∈C[0, 1]. Received: 15 March 2005     

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.     

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.     

Weighted Maximum Over Minimum Modulus of Polynomials, Applied to Ray Sequences of Padé Approximants

Let a≥ 0 , ɛ >0 . We use potential theory to obtain a sharp lower bound for the linear Lebesgue measure of the set Here P is an arbitrary polynomial of degree ≤ n . We then apply this to diagonal and ray Padé sequences for functions analytic (or meromorphic) in the unit ball. For example, we show that the diagonal \left{ [n/n]\right} _n=1 ^∞ sequence provides good approximation on almost one-eighth of the circles centre 0 , and the \left{ [2n/n]\right} _n=1 ^∞ sequence on almost one-quarter of such circles. July 18, 2000. Date revised: . Date accepted: April 19, 2001.     

Asymptotics of Differentiated Bernstein Polynomials

It is shown that the m th-order derivative of the n th-order Bernstein polynomial of a function f satisfying a certain Lipschitz condition, can be written for n\rightarrow +∈fty as a singular integral of Gauss—Weierstrass type, m times differentiated (in a certain sense) under the integral sign. The theorem is applied to yield an overdifferentiation formula, involving p times differentiated Bernstein polynomials of functions that are not C ^p . December 1, 1998. Dates revised: July 22, 1999 and January 11, 2000. Date accepted: February 1, 2000.     

On Stable Local Bases for Bivariate Polynomial Spline Spaces

Stable locally supported bases are constructed for the spaces \cal S _d ^r (\triangle) of polynomial splines of degree d≥ 3r+2 and smoothness r defined on triangulations \triangle , as well as for various superspline subspaces. In addition, we show that for r≥ 1 , in general, it is impossible to construct bases which are simultaneously stable and locally linearly independent. February 2, 2000. Date revised: November 27, 2000. Date accepted: March 7, 2001.     

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.     

Greedy Algorithm and m -Term Trigonometric Approximation

We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form , where is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients of function f . We compare the efficiency of this method with the best m -term trigonometric approximation both for individual functions and for some function classes. It turns out that the operator G _m provides the optimal (in the sense of order) error of m -term trigonometric approximation in the L _p -norm for many classes. September 23, 1996. Date revised: February 3, 1997.     

Minimally supported error representations and approximation by the constants

Summary. Distribution theory is used to construct minimally supported Peano kernel type representations for linear functionals such as the error in multivariate Hermite interpolation. The simplest case is that of representing the error in approximation to f by the constant polynomial f(a) in terms of integrals of the first order derivatives of f. This is discussed in detail. Here it is shown that suprisingly there exist many representations which are not minimally supported, and involve the integration of first order derivatives over multidimensional regions. The distance of smooth functions from the constants in the uniform norm is estimated using our representations for the error. Received June 30, 1997 / Revised version received April 6, 1999 / Published online February 17, 2000     

Approximation by p -Faber polynomials in the weighted Smirnov class Ep( G,ω ) and the Bieberbach polynomials

Let G\subset C be a finite domain with a regular Jordan boundary L . In this work, the approximation properties of a p -Faber polynomial series of functions in the weighted Smirnov class E ^p (G,ω) are studied and the rate of polynomial approximation, for f∈ E ^p ( G,ω) by the weighted integral modulus of continuity, is estimated. Some application of this result to the uniform convergence of the Bieberbach polynomials π _n in a closed domain \overline G with a smooth boundary L is given. February 25, 1999. Date revised: October 20, 1999. Date accepted: May 26, 2000.     

On the Best Approximation by Ridge Functions in the Uniform Norm

We consider the best approximation of some function classes by the manifold M _n consisting of sums of n arbitrary ridge functions. It is proved that the deviation of the Sobolev class W _p ^r,d from the manifold M _n in the space L _q for any 2≤ q≤ p≤∈fty behaves asymptotically as n ^-r/(d-1) . In particular, we obtain this asymptotic estimate for the uniform norm p=q=∈fty . January 10, 2000. Date revised: March 1, 2001. Date accepted: March 12, 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

On the Lebesgue Function of Weighted Lagrange Interpolation. I. (Freud-Type Weights)

For a wide class of Freud-type weights of form w = exp(-Q) we investigate the behavior of the corresponding weighted Lebesgue function λ _n (w,X,x) , where X = { x _kn } (-∞,∞) is an interpolatory matrix. We prove that for arbitrary X (-∞,∞) and ɛ > 0 , fixed, λ _n (w, X, x) ≥ c ɛ log n, x ∈ [-a _n , a _n ]\H _n , n ≥ 1, where a _n is the MRS number and |H _n | ≤ 2 ɛ a _n . The result corresponds to the behavior of the ``ordinary' Lebesgue function in [-1,1] . Other exponential weights are considered in our subsequent paper. October 28, 1996. Date revised: April 7, 1997. Date accepted: March 18, 1998.