Two 1935 questions of Mazur about polynomials in Banach spaces: counter-examples

We give various constructions of which the most significant is a continuous scalar-valued 2-polynomial W on the separable Hilbert space l ₂ and an unbounded set R ? l ₂ such that (i) W is bounded on an ?-neighbourhood of R; (ii) W is unbounded on ½R; (iii) W does not factor through any (continuous or not) 1-polynomial mapping from l ₂ to a normed space and sending R to a bounded set. This answers in the negative two 1935 questions asked by Mazur (Problems 55 and 75 in the Scottish Book). The construction is valid both over the real and complex fields. (In finite dimensions, the questions were answered in the positive by Auerbach soon after being asked.)     

ON THE SATURATION OF Lpw-APPROXIMATION BY (0-q'-q) TYPE HERMITE-FEJ(E)R INTERPOLATING POLYNOMIALS

The "o" saturation theorem and the degree of Lpw approximation by (0- q' -q) type Hermite-Fejer interpolating polynomials for mean convergence are obtained.     

On the Difference of Orthonormal Polynomials

We establish an estimate on the difference of orthonormal polynomials for a general class of exponential weights.     

Some integral mean estimates for polynomials

In this paper we establish Lq inequalities for polynomials, which in particular yields interesting generalizations of some Zygmund-type inequalities.     

CONVERGENCE ARTE FOR INTERATES OF q-BERNSTEIN POLYNOMIALS

Recently, q-Bernstein polynomials have been intensively investigated by a number of authors. Their results show that for q ≠ 1, q-Bernstein polynomials possess of many interesting properties. In this paper, the convergence rate for iterates of both q-Bernstein when n →∞ and convergence rate of Bn(f,q;x) when f ∈ Cn-1[0, 1], q →∞ are also presented.     

Approximation by rational functions with polynomials of positive coefficients as the denominators

The present paper proves that if f(x) ∈ C[0,1], changes its sign exactly l times at 0 ＜ y1 ＜ y2 ＜y1＜1 in(0,1),then there exists a pn (x)пn( ),such that |f(x)-p(x)/pn(x)|≤ Cωφ(f,n-1/2),where ρ(x) is defined by ρ(x)=l∏i=1(x - yi), if f(x) ≥ 0 for x ∈ (yl, 1),-1∏i=1(x-yi), iff(x) ＜ 0 for x ∈ (y1,1).which improves and generalizes the result of [7].     

Bounds for the zeros of polynomials

Let P(z)=∑↓j=0↑n ajx^j be a polynomial of degree n. In this paper we prove a more general result which interalia improves upon the bounds of a class of polynomials. We also prove a result which includes some extensions and generalizations of Enestrǒm-Kakeya theorem.     

Jacobi polynomials used to invert the laplace transform

Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Jacobi polynomials. The main contribution of our work is the development of a new and very effective method to determine the coefficients in the finite series expansion that approximation f(t) in terms of Jacobi polynomials. Some numerical examples are illustrated.     

A note on Sobolev orthogonality for Laguerre matrix polynomials

Let {Ln(A,λ)(x)}n≥0 be the sequence of monic Laguerre matrix polynomials defined on [0, ∞) by Ln(A,λ)(x)=n!/(-λ)n∑nk=0(-λ)κ/k!(n-1)! (A I)n[(A I)k]-1 xk,where A ∈ Cr×r. It is known that {Ln(A,λ)(x)}n≥0 is orthogonal with respect to a matrix moment functional when A satisfies the spectral condition that Re(z) ＞ - 1 for every z ∈σ(A).In this note we show that forA such that σ(A) does not contain negative integers, the Laguerre matrix polynomials Ln(A,λ) (x) are orthogonal with respect to a non-diagonal SobolevLaguerre matrix moment functional, which extends two cases: the above matrix case and the known scalar case.