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.     

An uncertainty inequality for Fourier-Dunkl series

An uncertainty inequality for the Fourier-Dunkl series, introduced by the authors in [Ó. Ciaurri, J.L. Varona, A Whittaker-Shannon-Kotel’nikov sampling theorem related to the Dunkl transform, Proc. Amer. Math. Soc. 135 (2007) 2939-2947], is proved. This result is an extension of the classical uncertainty inequality for the Fourier series.     

Identities for trigonometric B-splines with an application to curve design

In this paper we investigate some properties of trigonometric B-splines. We establish a complex integral representation for these functions, which is in certain analogy to the polynomial case, but the proof of which has to be done in a different and more complicated way. Using this integral representation, we can prove some identities concerning the evaluation of a trigonometric B-spline, its derivative and its partial derivative w.r.t. the knots. Finally we show that—in the case of equidistant knots—the trigonometric B-splines of odd order form a partition of a constant, and therefore the corresponding B-spline curve possesses the convex-hull property. This is illustrated by a numerical example.     

Monotonicity preserving representations of non-polynomial surfaces

We prove that, in contrast to the case for rational surfaces, some tensor product representations through spaces containing algebraic, trigonometric and hyperbolic polynomials are monotonicity preserving. The surface representations provided in this paper are the only known monotonicity preserving surfaces in addition to the tensor product Bézier and tensor product B-spline surfaces.     

A fractional differential inequality with application

We establish a fractional differential inequality using desingularization techniques combined with some generalizations of algebraic Bihari-type inequalities. We use this inequality to prove global existence and determine the asymptotic behavior of solutions for a family of fractional differential equations.     

On two inequalities similar to Opial's inequality in two independent variables

In the present paper we establish two new integral inequalities similar to Opial's inequality in two independent variables. The inequalities established in this paper are similar to the analogues of Calvert's generalizations of Opial's inequality, in two independent variables and contains in the special case the analogue of Opial's inequality given by G. S. Yang in two independent variables.     

Nikol'skij-type and maximal inequalities for generalized trigonometric polynomials

We consider some Nikol'skij-type inequalities, thus inequalities between different metrics of a function, for almost periodic trigonometric polynomials. Some basic methods of probability theory are applied to prove the existence of the distribution function for an almost periodic function in the sense of Besicovitch. Finally, the Maximal function of Hardy and Littlewood is considered and maximal inequalities on Besicovitch spaces are proved. Received: 23 July 1998 / Revised version: 8 March 1999     

Kolmogorov's problem on the class of multiply monotone functions

In this paper we give necessary and sufficient conditions for the system of positive numbers _Mk1,_Mk2,…,_Mkd$M_{k_1},M_{k_2},\dots ,M_{k_d}$, 0≤_k1<…<_kd≤r$0\le k_1<\dots <k_d\le r$, to guarantee the existence of an r  -monotone function defined on the negative half-line _R−${\mathbb{R}}_{-}$ and such that _{‖x(ki)‖∞}=_Mki${\Vert x^{(k_i)}\Vert }_{\infty }=M_{k_i}$, i=1,2,…,d$i=1,2,\dots ,d$. We also discuss some applications of the obtained results and connections with other problems.     

On the convergence of hybrid polynomial approximation to higher derivatives of rational curves

In this paper, we derive the bounds on the magnitude of l  th (l=2,3)$(l=2,3)$ order derivatives of rational Bézier curves, estimate the error, in the _L∞$L_{\infty }$ norm sense, for the hybrid polynomial approximation of the l  th (l=1,2,3)$(l=1,2,3)$ order derivatives of rational Bézier curves. We then prove that when the hybrid polynomial approximation converges to a given rational Bézier curve, the l  th (l=1,2,3)$(l=1,2,3)$ derivatives of the hybrid polynomial approximation curve also uniformly converge to the corresponding derivatives of the rational curve. These results are useful for designing simpler algorithms for computing tangent vector, curvature vector and torsion vector of rational Bézier curves.     

Collections of paths and rays in the plane which fix its topology

A collection $\text{F}$ of proper maps into a locally compact Hausdorff space X fixes the topology of X if the only locally compact Hausdorff topology on X which makes each element of $\text{F}$ continuous and proper is the given topology. In I²=[-1, 1]×[-1, 1], neither the collection of analytic paths nor the collection of regular twice differentiable paths fixes the topology. However, in I², both the collection of C^∞ arcs and the collection of regular C¹ arcs fix the topology. In $\text{I}{}^2\text{=[?1,1]×[?1,1], the collection of polynomial rays together with any collection of paths does not fix the topology. However, in}\text{R}{}^2$, the collection of regular injective entire rays together with either the collection of C^∞ arcs or the collection of regular C¹ arcs fixes the topology.     

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {r_n(z)}_n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {r_n(z)}_n∈N in the complement CP¹?D_f, where D_f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {r_n(z)}_n∈N. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.     

Almost Hermitian trigonometric interpolation on three equidistant nodes

Summary We find necessary and sufficient conditions for the regularity (or unique solvability) of a trigonometric interpolation problem on three equidistant nodes in [0, 2) where the data are almost Hermitian. We shall show that in many cases the conditions only depend upon the number of even and odd derivatives prescribed in the problem. In general such a simplification seems to be difficult.Dedicated to the memory of Alexander M. Ostrowski on occasion of the 100th anniversary of his birth     

An error analysis for absolute and relative approximation

We consider the vectorial algorithm for finding best polynomial approximationsp P _n to a given functionf C[a, b], with respect to the norm · _s , defined byp – f _s =w ₁ (p – f)+w ₂ (p – f) A bound for the modulus of continuity of the best vectorial approximation operator is given, and using the floating point calculus of J. H. Wilkinson, a bound for the rounding error in the algorithm is derived. For givenf, these estimates provide an indication of the conditioning of the problem, an estimate of the obtainable accuracy, and a practical method for terminating the iteration.This paper was supported in part by the Canadian NCR A-8108, FCAC 74-09 and G.E.T.M.A.Part of this research was done during the first-named author's visit to theB! Chair of Applied Mathematics, University of Athens, Spring term, 1975.     

Convergence and summability of Gabor expansions at the Nyquist density

It is well known that Gabor expansions generated by a lattice of Nyquist density are numerically unstable, in the sense that they do not constitute frame decompositions. In this paper, we clarify exactly how bad such Gabor expansions are, we make it clear precisely where the edge is between enough and too little, and we find a remedy for their shortcomings in terms of a certain summability method. This is done through an investigation of somewhat more general sequences of points in the time-frequency plane than lattices (all of Nyquist density), which in a sense yields information about the uncertainty principle on a finer scale than allowed by traditional density considerations. An important role is played by certain Hilbert scales of function spaces, most notably by what we call the Schwartz scale and the Bargmann scale, and the intrinsically interesting fact that the Bargmann transform provides a bounded invertible mapping between these two scales. This permits us to turn the problems into interpolation problems in spaces of entire functions, which we are able to treat.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

A constructive method related to generalizations of the uniform boundedness principle and its applications

The present paper suggests a general constructive method, which improves and generalizes a result of Dickmeis, Nessel, and van Wickeren. In Section 3, three applications will be discussed.     

On certain integral inequalities related to Opial's inequality

The aim of the present paper is to establish some new integral inequalities involving three functions and their derivatives which in the special cases yield the well known Opial inequality and some of its generalizations.