Some properties of biorthogonal polynomials and their application to Padé approximations

Transformations of biorthogonal polynomials under certain transformations of biorthogonalizable sequences are studied. The obtained result is used to construct Padé approximants of orders [N?1/N],N ε ?, for the functions $$\tilde f(z) = \sum\limits_{m = 0}^M {\alpha _m } \frac{{f(z) - T_{m - 1} [f;z]}}{{z^m }},$$ wheref(z) is a function with known Padé approximants of the indicated orders,T _j [f;z] are Taylor polynomials of degreej for the functionf(z), and α _{m, M} = $\overline {1,M} $ are constants.     

An application of Szegő polynomials to the computation of certain weighted integrals on the real line

In this paper, a new approach in the estimation of weighted integrals of periodic functions on unbounded intervals of the real line is presented by considering an associated weight function on the unit circle and making use of both Szegő and interpolatory type quadrature formulas. Upper bounds for the estimation of the error are considered along with some examples and applications related to the Rogers-Szegő polynomials, the evaluation of the Weierstrass operator, the Poisson kernel and certain strong Stieltjes weight functions. Several numerical experiments are finally carried out.     

Approximation of analytic functions by polynomials “close” to polynomials of best approximation

The rate of approximation of analytic functions at interior points of compact sets with connected complement by polynomials close to polynomials of best approximation is investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 208–214, February, 1992.     

Confluent and related mappings on arc-like continua—an application to homogeneity

It is shown that the pseudo-arc is the only nondegenerate arc-like continuum which is homogeneous with respect to the class of confluent (monotone, open) mappings. Further a concept of a pseudo-end point of an arc-like continuum is introduced and is applied to obtain another characterization of the pseudo-arc in terms of homogeneity with respect to a class of mappings.     

Orthogonal polynomials—centroid of their zeroes

Let c _n,k (k=1,...,n) the n zeroes of the monic orthogonal polynomials family P _n (x). The centroid of these zeroes: $s_n=\frac1n \sum\limits^n_{k=1}c_{n,k}$ controls globally the distribution of the zeroes, and it is relatively easy to obtain information on s _n , like bounds, inequalities, parameters dependence, ..., from the links between s _n , the coefficients of the expansion of P _n (x), and the coefficients β _n , γ _n in the basic recurrence relation satisfied by P _n (x). After a review of basic properties of the centroid on polynomials, this work gives some results on the centroid of a large class of orthogonal polynomials.     

On zeros of polynomials of best approximation to holomorphic andC ∞ functions

LetE be a compact subset of the complex planeC such that Leja's extremal functionL _E forE is continuous. If almost all zeros of the polynomials of best approximation to a functionfC(E) are outside the setE _R ={zC:L _E (z<R)}, for someR>1, thenf is extendible to a holomorphic function inE _R . If the zeros ofn-th, polynomial of best approximation tof are outside and the sequence {R _n ^–n } rapidly decreases to zero thenf can be extended to aC function on 075-4}.     

Asymptotic analysis of Emden–Fowler type equation with an application to power flow models

Emden–Fowler type equations are nonlinear differential equations that appear in many fields such as mathematical physics, astrophysics and chemistry. In this paper, we perform an asymptotic analysis of a specific Emden–Fowler type equation that emerges in a queuing theory context as an approximation of voltages under a well-known power flow model. Thus, we place Emden–Fowler type equations in the context of electrical engineering. We derive properties of the continuous solution of this specific Emden–Fowler type equation and study the asymptotic behavior of its discrete analog. We conclude that the discrete analog has the same asymptotic behavior as the classical continuous Emden–Fowler type equation that we consider.     

A parametric solution for simple stress—strength model of failure with an application

Estimation of P(R ⩽ S) is considered for the simple stress—strength model of failure. Using the Pareto and Power distributions together with their combined form a useful parametric solution is obtained and is illustrated numerically. It is shown that these models are also applicable when only the tails of distributions for R and S are considered. An application to the failure study concerning the fractures is also included