We develop a functional calculus for d-tuples of non-commuting elements in a Banach algebra. The functions we apply are free analytic functions, that is nc functions that are bounded on certain polynomial polyhedra.

In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By virtue Faber polynomial expansions, estimation n?th(n?3) Taylor–Maclaurin coefficients an obtained. Furthermore, bounds value first two such established.

Let Ln [f ] denote the Lagrange interpolation polynomial to a function f at the zeros of a polynomial Pn with distinct real zeros. We show that f − Ln [f ] = −PnHe [ H [f ] Pn ] , where H denotes the Hilbert transform, and He is an extension of it. We use this to prove convergence of Lagrange interpolation for certain functions analytic in (−1, 1) that are not assumed analytic in any ellipse wi...

Méray has given f the following illustration to show that polynomials formed from a given function by interpolation do not necessarily converge to that function. Interpolate to the f unction ƒ (s) = 1/z by means of the polynomials pn(z) of respective degreest w = l, 2, 3, • • • , required to coincide with ƒ (z) in the (n + l)th roots of unity; this condition defines the polynomials pn(z) unique...

We construct a sequence of globally defined polynomial valued operators, using linear combinations of either the coefficients in the orthogonal polynomial expansion of the target function with respect to a very general mass distribution μ or the values of the function at suitable points on [−1, 1], so that the degree of approximation by these operators globally is commensurate with the degree o...

A function is called arc-analytic if it is real analytic on each real analytic arc. In real analytic geometry there are many examples of arc-analytic functions that are not real analytic. Arc analytic functions appear while studying the arc-symmetric sets and the blow-analytic equivalence. In this paper we show that the non-analyticity locus of an arc-analytic function is arc-symmetric. We disc...

Just as the Bernstein basis and Be Âzier points give us tools to manipulate polynomials, the Poisson basis and control points provide a new framework for investigating analytic functions. Using the Poisson representation, two different subdivision algorithms for analytic functions can be derived as extensions of the Be Âzier polynomial subdivision algorithm. The ®rst procedure generates an appr...

Abstract We consider approximating analytic functions on the interval $$[-1,1]$$ [ - 1 , ] from their values at a set of $$m+1$$ m + equispaced nodes. A result Platte, Trefethen & K...