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.     

A Bound on the Approximation Order of Surface Splines

The functions φ _m :=|.| ^2m-d if d is odd, and φ _m :=|.| ^2m-d \log|.| if d is even, are known as surface splines, and are commonly used in the interpolation or approximation of smooth functions. We show that if one's domain is the unit ball in R ^d , then the approximation order of the translates of φ _m is at most m . This is in contrast to the case when the domain is all of R ^d where it is known that the approximation order is exactly 2m . April 23, 1996. Date revised: May 5, 1997.     

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.     

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.     

Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines

Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on R is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions [CDP] is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev norms ||⋅|| _Hs([0,1]) for s∈ (-0.824926,2.5) . In particular, the multiwavelets form Riesz bases for L ₂ ([0,1]) . February 2, 1998. Date revised: February 19, 1999. Date accepted: March 5, 1999.     

The Critical Values of a Polynomial

It has been known for a long time that any real sequence y ₁ , . . . ,y _n-1 is the sequence of critical values of some real polynomial. Here we show that any complex sequence w ₁ , . . . ,w _n-1 is the sequence of critical values of some complex polynomial.     

The Remainder in Ruijsenaars’ Asymptotic Expansion of Barnes Double Gamma Function

We show that the remainder in Ruijsenaars’ asymptotic expansion of the logarithm of Barnes double gamma function gives rise to a completely monotone function. Fourier expansions of the multiple Bernoulli polynomials are also obtained. Research supported by the Carlsberg Foundation.     

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.     

Delta-semidefinite and Delta-convex Quadratic Forms in Banach Spaces

A continuous quadratic form (“quadratic form”, in short) on a Banach space X is: (a) delta-semidefinite (i.e., representable as a difference of two nonnegative quadratic forms) if and only if the corresponding symmetric linear operator factors through a Hilbert space; (b) delta-convex (i.e., representable as a difference of two continuous convex functions) if and only if T is a UMD-operator. It follows, for instance, that each quadratic form on an infinite-dimensional L _p (μ) space (1 ≤ p ≤ ∞) is: (a) delta-semidefinite iff p ≥ 2; (b) delta-convex iff p > 1. Some other related results concerning delta-convexity are proved and some open probms are stated. The first author was supported by NSF grant DMS-0555670. The second author was supported by the Russian Foundation for Basic Research, Grant 05-01-00066, and by Grant NSh-5813.2006.1. The third author was supported in part by the Ministero dell’Università e della Ricerca of Italy.     

A new characterization of euler's gamma function by a functional equation

This paper gives a new characterization of Euler's gamma function from the aspect of complex analysis. To this end the Gauss multiplication formula is used.     

The operator remainder in the Euler-Maclaurin formula

We give a formula for the remainder term in the operator-theoretic formulation of the Euler-Maclaurin formula, using Laplace transforms. There is also a related formula for the partial sums.Keywords and phrases: Euler-Maclaurin sum formula, remainder term, Laplace transform, Bernoulli polynomial.Dedicated to Professor Janos Aczél on his 60th birthday     

Approximation by boolean sums of positive linear operators III: Estimates for some numerical approximation schemes

In the present note we intröduce and investigate certain sequences of discrete positive linear operators and Boolean sum modifications of them. The mappings considered are obtained by discretizing a class of transformed convolution-type operators using Gaussian quadrature of appropriate order. For our operators and their modifications we prove pointwise Jackson-type theorems involving the first and second order moduli of smoothness, thus providing new and elegant proofs of earlier results by Timan, Telyakowskii, Gopengauz and DeVore. Due to their discrete structure, optimal order of approximation and ease of computation, the operators appear to be useful for numerical approximation. In an intermediate step we solve an old problem in Approximation Theory; its importance was only recently emphasized in a paper of Butzer.     

On Existence and Weak Stability of Matrix Refinable Functions

We consider the existence of distributional (or L ₂ ) solutions of the matrix refinement equation where P is an r×r matrix with trigonometric polynomial entries. One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional solution if and only if the matrix P (0) has an eigenvalue of the form 2 ⁿ , . A characterization of the existence of L ₂ -solutions of the above matrix refinement equation in terms of the mask is also given. A concept of L ₂ -weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions of a matrix refinement equation, we characterize this weak stability in terms of the mask. August 1, 1996. Date revised: July 28, 1997. Date accepted: August 12, 1997.     

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.     

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.     

On orders of approximation of plane curves by parametric cubic splines

The aim of the present paper is to show that the convergence rate of the parametric cubic spline approximation of a plane curve is of order four instead of order three. For the first and second derivatives, the rates are of order three and two, respectively. Finally some numerical examples are given to illustrate the predicted error behaviour.     

Iterations of linear maps over finite fields

We study the dynamics of the evolution of Ducci sequences and the Martin-Odlyzko-Wolfram cellular automaton by iterating their respective linear maps on . After a review of an algebraic characterization of cycle lengths, we deduce the relationship between the maximal cycle lengths of these two maps from a simple connection between them. For n odd, we establish a conjugacy relationship that provides a more direct identification of their dynamics. We give an alternate, geometric proof of the maximal cycle length relationship, based on this conjugacy and a symmetry property. We show that the cyclic dynamics of both maps in dimension 2n can be deduced from their periodic behavior in dimension n. This link is generalized to a larger class of maps. With restrictions shared by both maps, we obtain a formula for the number of vectors in dimension 2n belonging to a cycle of length q that expresses this number in terms of the analogous values in dimension n.     

An update on orthogonal polynomials and weighted approximation on the real line

We briefly review the state of orthogonal polynomials on (–, ), concentrating on analytic aspects, such as asymptotics and bounds on orthogonal polynomials, their zeros and their recurrence coefficients. We emphasize results rather than proofs. We also discuss applications to mean convergence of orthogonal expansions, Lagrange interpolation, Jackson-Bernstein theorems and the weighted incomplete polynomial approximation problem.     

Exponential Convergence for a System of Conuclear Space-Valued Diffusions with Mean-Field Interaction

We consider an interacting system of n diffusion processes X ⁿ _j (t): t∈[0,1] , j=1,2,. . ., n , taking values in a conuclear space Φ' . Let ζ ⁿ _t =(1/n)Σ ⁿ _j=1 δ _Xnj(t) be the empirical process. It has been proved that ζ ⁿ , as n→∞ , converges to a deterministic measure-valued process which is the unique solution of a nonlinear differential equation. In this paper we show that, under suitable conditions, ζ ⁿ converges to ζ at an exponential rate. Accepted 20 October 1997