On the Divergence Phenomenon in Hermite–Fejér Interpolation

Generalizing results of L. Brutman and I. Gopengauz (1999, Constr. Approx.15, 611–617), we show that for any nonconstant entire function f and any interpolation scheme on [−1, 1], the associated Hermite–Fejér interpolating polynomials diverge on any infinite subset of \[−1, 1]. Moreover, it turns out that even for the locally uniform convergence on the open interval ]−1, 1[ it is necessary that the interpolation scheme converges to the arcsine distribution.     

On the Positivity of the Fundamental Polynomials for Generalized Hermite–Fejér Interpolation on the Chebyshev Nodes

It is shown that the fundamental polynomials for (0, 1, …, 2m+1) Hermite–Fejér interpolation on the zeros of the Chebyshev polynomials of the first kind are non-negative for −1x1, thereby generalising a well-known property of the original Hermite–Fejér interpolation method. As an application of the result, Korovkin's 10theorem on monotone operators is used to present a new proof that the (0, 1, …, 2m+1) Hermite–Fejér interpolation polynomials offC[−1, 1], based onnChebyshev nodes, converge uniformly tofasn→∞.     

On Generalized Hermite–Fejér Interpolation of Lagrange Type on the Chebyshev Nodes

For fC[−1, 1], let H_m, n(f, x) denote the (0, 1, …,anbsp;m) Hermite–Fejér (HF) interpolation polynomial of f based on the Chebyshev nodes. That is, H_m, n(f, x) is the polynomial of least degree which interpolates f(x) and has its first m derivatives vanish at each of the zeros of the nth Chebyshev polynomial of the first kind. In this paper a precise pointwise estimate for the approximation error |H_2m, n(f, x)−f(x)| is developed, and an equiconvergence result for Lagrange and (0, 1, …, 2m) HF interpolation on the Chebyshev nodes is obtained. This equiconvergence result is then used to show that a rational interpolatory process, obtained by combining the divergent Lagrange and (0, 1, …, 2m) HF interpolation methods on the Chebyshev nodes, is convergent for all fC[−1, 1].     

Convergence of Hermite–Fejér type interpolation of higher order on an arbitrary system of nodes

A criterion of convergence for general Hermite–Fejér type interpolation of higher order on an arbitrary system of nodes is given.     

Uniform or Mean Convergence of Hermite–Fejér Interpolation of Higher Order for Freud Weights

In this paper we show the uniform or mean convergence of Hermite–Fejér interpolation polynomials of higher order based at the zeros of orthonormal polynomials with the typical Freud weight.     

On Divergence of Hermite—Fejér Interpolation to f(z)=z in the Complex Plane

We show that for a broad class of interpolatory matrices on [-1,1] the sequence of polynomials induced by Hermite—Fejér interpolation to f(z)=z diverges everywhere in the complex plane outside the interval of interpolation [-1,1] . This result is in striking contrast to the behavior of the Lagrange interpolating polynomials. June 15, 1998. Date accepted: January 26, 1999.     

L∞ convergence of interpolation and associated product integration for exponential weights

We investigate convergence in a weighted L_∞ norm of Hermite–Fejér, Hermite, and Grünwald interpolations at zeros of orthogonal polynomials with respect to exponential weights such as Freud, Erd s, and exponential weight on (−1,1). Convergence of product integration rules induced by the various approximation processes is deduced. We also give more precise weight conditions for convergence of interpolations with respect to above three types of weights, respectively.     

B-splines and Hermite–Padé approximants to the exponential function

This paper is the continuation of a work initiated in [P. Sablonnière, An algorithm for the computation of Hermite–Padé approximations to the exponential function: divided differences and Hermite–Padé forms. Numer. Algorithms 33 (2003) 443–452] about the computation of Hermite–Padé forms (HPF) and associated Hermite–Padé approximants (HPA) to the exponential function. We present an alternative algorithm for their computation, based on the representation of HPF in terms of integral remainders with B-splines as Peano kernels. Using the good properties of discrete B-splines, this algorithm gives rise to a great variety of representations of HPF of higher orders in terms of HPF of lower orders, and in particular of classical Padé forms. We give some examples illustrating this algorithm, in particular, another way of constructing quadratic HPF already described by different authors. Finally, we briefly study a family of cubic HPF.     

Quadratic Hermite–Padé polynomials associated with the exponential function

The asymptotic behavior of quadratic Hermite–Padé polynomials associated with the exponential function is studied for n→∞. These polynomials are defined by the relation

(*)

p_n(z)+q_n(z)e^z+r_n(z)e^2z=O(z³ⁿ⁺²) as z→0,

where O(·) denotes Landau's symbol. In the investigation analytic expressions are proved for the asymptotics of the polynomials, for the asymptotics of the remainder term in (*), and also for the arcs on which the zeros of the polynomials and of the remainder term cluster if the independent variable z is rescaled in an appropriate way. The asymptotic expressions are defined with the help of an algebraic function of third degree and its associated Riemann surface. Among other possible applications, the results form the basis for the investigation of the convergence of quadratic Hermite–Padé approximants, which will be done in a follow-up paper.     

Noyaux potentiels associés à une filtration

We study from the point of view of potential theory some operators V which are “integrals of martingales” and noteworthy the formula (I + V)⁻¹ = I − N where N is a submarkovian kernel. We give an explicit expression of N when the filtration is finite and get the general case with an usual approximation procedure. Some links are made with the matrix theory (ultrametric and Stieltjes matrices) and the graph theory (flows and capacities) when the space is finite.

Résumé

On étudie, du point de vue de la théorie du potentiel, des opérateurs V du type “intégrales de martingale”, et notamment la formule (I + V)⁻¹ = I − N où N est un noyau sous-markovien. On donne une expression explicite de N dans le cas d'une filtration finie, et on traite le cas général par un procédé d'approximation usuel. On fait le lien avec la théorie des matrices (matrices ultramétriques et de Stieltjes) et la théorie des graphes (flots et capacités) quand l'espace est fini.     

Near-circularity of the error curve in complex Chebyshev approximation

Let f(z) be analytic on the unit disk, and let p^*(z) be the best (Chebyshev) polynomial approximation to f(z) on the disk of degree at most n. It is observed that in typical problems the “error curve,” the image of the unit circle under (f − p^*)(z), often approximates to a startling degree a perfect circle with winding number n + 1. This phenomenon is approached by consideration of related problems whose error curves are exactly circular, making use of a classical theorem of Carathéodory and Fejér. This leads to a technique for calculating approximations in one step that are roughly as close to best as the best approximation error curve is close to circular, and hence to strong theorems on near-circularity as the radius of the domain shrinks to 0 or as n increases to ∞. As a computational example, very tight bounds are given for approximation of e^z on the unit disk. The generality of the near-circularity phenomenon (more general domains, rational approximation) is discussed.     

The weak type inequality for the maximal operator of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series

The main aim of this paper is to prove that the maximal operator σ^* of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series is bounded from the Hardy space H_2/3 to the space weak-L_2/3.     

Padé approximation of Stieltjes series

Using Nuttall's compact formula for the [n, n − 1] Pad'e approximant, the authors show that there is a natural connection between the Padé approximants of a series of Stieltjes and orthogonal polynomials. In particular, we obtain the precise error formulas. The [n, n − 1] Padé approximant in this case is just a Gaussian quadrature of the Stieltjes integral. Hence, analysis of the error is now possible and under very mild conditions it is shown that the [n, n + j], j − 1, Padé approximants converge to the Stieltjes integral.     

On Extremal Permutations AvoidingωN = NN − 1 . . . 1

Dedicated to the memory of Marcel–Paul Schützenberger Cet article présente une étude des permutations qui évitent le motif de la permutation maximaleω_N = NN − 1 . . . 1. Après avoir donné les définitions classiques, nous montrons que l’ensemble de ces permutations est un idéal pour l’ordre de Bruhat faible et faisons l’étude de ses éléments maximaux. Nous exhibons alors un algorithme pour calculer ces éléments et nous montrons que ceux-ci peuvent être obtenus à partir d’un automate. Nous terminons en donnant des estimations asymptotiques de leur nombre. This paper presents a study of permutations avoiding the patternω_N = NN − 1 . . . 1. After recalling the basic definitions, we prove that this set of permutations is an ideal for the weak Bruhat order and begin the study of its maximal elements. We then present an algorithm that generates these elements and find out that they can be obtained from an automaton. Finally, we give some asymptotics about their number.     

Ahlfors-régularité des quasi-minima de Mumford–Shah

Let be an open set. We consider on Ω the competitors (U,K) for the reduced Mumford–Shah functional, that is to say the Mumford–Shah functional in which the -norm of U term is removed, where K is a closed subset of Ω and U is a function on ΩK with gradient in  . The main result of this paper is the following: there exists a constant c for which, whenever (U,K) is a quasi-minimizer for the reduced Mumford–Shah functional and B(x,r) is a ball centered on K and contained in Ω with bounded radius, the -measure of is bounded above by cr^N−1 and bounded below by c⁻¹r^N−1.     

Matrix Mean Series in Terms of Boundary Orthogonal Systems and Functions in the Classes H and E

We give conditions for orthonormal systems on the boundary of a plane Jordan domain which are necessary and sufficient for an arbitrary series in terms of this orthonormal system to be the Fourier series of some function in H^∞(G) resp. E^P(G)(1<p<∞). Our results contain a classical criterion of Fejér for the boundedness of a holomorphic function in the unit disk.     

Multivariate Interpolation Using Carlitz's Inversion Formulas

Multivariate rational exponential Lagrange interpolation formulas, Hermite interpolation formulas, and Hermite–Fejér interpolation formulas of the Newton type are established by using Carlitz's inversion formulas. The recurrence relation for constructing Lagrange interpolation is also given. In addition, by setting q1 in the obtained formulas, we obtain the corresponding polynomial interpolation formulas with combinatorial form.     

A new proof of some polynomial inequalities related to pseudo-splines

Pseudo-splines of type I were introduced in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003) 1–46] and [Selenick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2000) 163–181] and type II were introduced in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104]. Both types of pseudo-splines provide a rich family of refinable functions with B-splines, interpolatory refinable functions and refinable functions with orthonormal shifts as special examples. In [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104], Dong and Shen gave a regularity analysis of pseudo-splines of both types. The key to regularity analysis is Proposition 3.2 in [B. Dong, Z. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007) 78–104], which also appeared in [A. Cohen, J.P. Conze, Régularité des bases d'ondelettes et mesures ergodiques, Rev. Mat. Iberoamericana 8 (1992) 351–365] and [I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia, 1992] for the case l=N−1. In this note, we will give a new insight into this proposition.     

The Divergence of Lagrange Interpolation for |x| at Equidistant Nodes

In 1918 S. N. Bernstein published the surprising result that the sequence of Lagrange interpolation polynomials to |x| at equally spaced nodes in [−1, 1] diverges everywhere, except at zero and the end-points. In the present paper, we prove that the sequence of Lagrange interpolation polynomials corresponding to |x|^α (0<α1) on equidistant nodes in [−1, 1] diverges everywhere in the interval except at zero and the end-points.