Functional equations for exponential polynomials

Nonlinear functional equations satisfied by orthogonal polynomials

Let c be a linear functional defined by its moments c(x) = ci for i = 0, 1, . . .. We proved that the nonlinear functional equations P (t) = c(P (x)P (x + t)) and P (t) = c(P (x)P (xt)) admit polynomial solutions which are the polynomials belonging to the family of formal orthogonal polynomials with respect to a linear functional related to c. Other types of nonlinear functional equations whose...

Exponential stability for a class of functional differential equations

Generalized Exponential Euler Polynomials and Exponential Splines

Here presented is constructive generalization of exponential Euler polynomial and exponential splines based on the interrelationship between the set of concepts of Eulerian polynomials, Eulerian numbers, and Eulerian fractions and the set of concepts related to spline functions. The applications of generalized exponential Euler polynomials in series transformations and expansions are also given.

Bounds for orthogonal polynomials for exponential weights

Orthogonal polynomials pn(W ; x) for exponential weights W 2 = e−2Q on a nite or in nite interval I , have been intensively studied in recent years. We discuss e orts of the authors to extend and unify some of the theory; our deepest result is the bound |pn(W ; x)|W (x)|(x − a−n)(x − an)|6C; x∈ I with C independent of n and x. Here a±n are the Mhaskar–Rahmanov–Sa numbers for Q and Q must satisf...

Extremal Problems for Polynomials with Exponential Weights

For the extremal problem: E„r(a):= min||exp(-W«)(x-+ ■■■)\\L„ a > 0, where U (0 < r < oo) denotes the usual integral norm over R, and the minimum is taken over all monic polynomials of degree n, we describe the asymptotic form of the error E„ r(a) (as n -» oo) as well as the limiting distribution of the zeros of the corresponding extremal polynomials. The case r = 2 yields new information regar...