Interpolation by Polynomials in z and z-1 on an Annulus

A polynomial of degree n in z^–1 and n–1 in z isdefined by an interpolation projection from the space of functionsanalytic in the annulus r|z|R and continuous on its boundary.The points of interpolation are chosen to coincide with then roots of zⁿ=Rⁿeⁱⁿ (0<<2/n) and the n roots of zⁿ=rⁿ.The behaviour of the corresponding Lebesgue function is studied,and an estimate for the operator norm is obtained. The resultsof the present paper give a partial affirmative answer to twoconjectures suggested earlier by Mason on the basis of numericalcomputations.     

On interpolation by radial polynomials

A lemma of Micchelli's, concerning radial polynomials and weighted sums of point evaluations, is shown to hold for arbitrary linear functionals, as is Schaback's more recent extension of this lemma and Schaback's result concerning interpolation by radial polynomials. Schaback's interpolant is explored. Happy 60th and beyond, Charlie! Mathematics subject classifications (2000) 41A05, 41A6.     

Near-minimax Interpolation by a Polynomial in z and z-1 on a Circular Annulus

A polynomial of degree n in z^–1 and n–1 in z isdefined by an interpolation projection from the space A(N_p) of functions f analytic in the circular annulusp^–1 < <p and continuous on itsboundaries = p^–1, p. The points ofinterpolation are chosen to be spaced at equal angles aroundthe two boundaries, with arguments on the inner boundary midwaybetween those on the outer boundary. By calculating the Lebesgueconstants numerically, is found to be close to a minimax approximation for all p 1and all degrees n in the range 1 n 15. In the limiting casesp = 1 and, it is proved that is asymptotic to 2^–1 log n. More specifically and , where _nis the Lebesgueconstant of Gronwall for equally spaced interpolation on a circleby a polynomial of degree n. It is also demonstrated that is not in general monotonic in p, and that is not everywhere differentiable in p.     

On Birkhoff interpolation by polynomials in several variables

Chui and Lai (1987) have discussed a kind of multivariate polynomial interpolation problem defined on the straight line type node configuration C (SLTNCC). In this paper, we define general Birkhoff interpolation problems for the SLTNCC, and show, under some restrictions, that these interpolation problems are unisolvent. Also we give some generalizations of the SLTNCC.     

On a conjecture of Fernando,Hou and Lappano concerning permutation polynomials over finite fields

Let ${\mathbb{F}}_q$ be the finite field with q elements and let $p=\text{char}{\mathbb{F}}_q$. It was conjectured that for integers $e\ge 2$ and $1\le a\le pe-2$, the polynomial $X^{q-2}+X^{q^2-2}+\cdots +X^{q^a-2}$ is a permutation polynomial of ${\mathbb{F}}_{q^e}$ if and only if (i) $a=2$ and $q=2$, or (ii) $a=1$ and $\text{gcd}(q-2,q^e-1)=1$. In the present paper we confirm this conjecture.     

Approximation of periodic functions by interpolation polynomials in L1

Asymptotically precise estimates are obtained for the deviation, in the L₁-norm, of interpolation polynomials with equally-spaced nodes from certain classes of functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 781–786, June, 1990.     

On a conjecture by Ghahramani-Lau and related problems concerning topological centres

Let A be a Banach algebra, and consider A^** equipped with the first Arens product. We establish a general criterion which ensures that A is left strongly Arens irregular, i.e., the first topological centre of A^** is reduced to A itself. Using this criterion, we prove that for a very large class of locally compact groups, Ghahramani-Lau's conjecture (cf. [Lau 94] and [Gha-Lau 95]) stating the left strong Arens irregularity of the measure algebra M(G), holds true. (Our methods obviously yield as well the right strong Arens irregularity in the situation considered.)Furthermore, the same condition used above implies that every linear left A^**-module homomorphism on A^* is automatically bounded and w^*-continuous. We finally show that our criterion also yields a partial answer to a question raised by Lau-Ülger (Trans. Amer. Math. Soc. 348 (3) (1996) 1191) on the topological centre of the algebra (A^*⊙A)^*, for A having a right approximate identity bounded by 1.     

On the conjecture on APN functions and absolute irreducibility of polynomials

An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field \(\mathbb {F}\) is called exceptional APN, if it is also APN on infinitely many extensions of \(\mathbb {F}\). In this article we consider the most studied case of \(\mathbb {F}=\mathbb {F}_{2^n}\). A conjecture of Janwa–Wilson and McGuire–Janwa–Wilson (1993/1996), settled in 2011, was that the only monomial exceptional APN functions are the monomials \(x^n\), where \(n=2^k+1\) or \(n={2^{2k}-2^k+1} \) (the Gold or the Kasami exponents, respectively). A subsequent conjecture states that any exceptional APN function is one of the monomials just described. One of our results is that all functions of the form \(f(x)=x^{2^k+1}+h(x)\) (for any odd degree h(x), with a mild condition in few cases), are not exceptional APN, extending substantially several recent results towards the resolution of the stated conjecture. We also show absolute irreducibility of a class of multivariate polynomials over finite fields (by repeated hyperplane sections, linear transformations, and reductions) and discuss their applications.     

On Kemnitz’ conjecture concerning lattice-points in the plane

In 1961, Erdős, Ginzburg and Ziv proved a remarkable theorem stating that each set of 2n−1 integers contains a subset of size n, the sum of whose elements is divisible by n. We will prove a similar result for pairs of integers, i.e. planar lattice-points, usually referred to as Kemnitz’ conjecture. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—11B50.     

On the van der Waerden conjecture and zeros of polynomials

Let

$\text{F(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{A}{}_{\mathrm{n}}\text{≠ 0,}$

and

$\text{G(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{B}{}_{\mathrm{n}}\text{≠ 0,}$

be polynomials with real zeros satisfying A_n?1 = B_n?1 = 0, and let

$\text{H(x) =}\text{∑}\text{k=o}\text{n-2}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}\text{.}$

Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H(x) are obtained. These results are related to classical results on composite polynomials.     

On the approximation of the continuous functions by some Hermite-Fejer interpolation polynomials

The object of this note is to improve Some wellknown results, which are related with the approximation problems of the continuous functions by Hermite-Fejér interpolation which based on the zeros of Chebyshev polynomials of the first or second kind.     

On a conjecture of Fink and Jacobson concerning k-domination and k-dependence

A set D of vertices of a graph is k-dependent if every vertex of D is joined to at most k?1 vertices in D. Let β_k(G) be the maximum order of a k-dependent set in G. A set D of vertices of G is k-dominating if every vertex not in D is joined to at least k vertices of D. Let γ_k(G) be the minimum order of a k-dominating set in G. Here we prove the following conjecture of Fink and Jacobson: for any simple graph G and any positive integer k, γ_k(G) ≤ β_k(G).