Approximation on an Arc by Polynomials with Restricted Zeros

On the Zeros of Polynomials with Restricted Coefficients

It is proved that a polynomial p of the form

Bounded Approximation by Polynomials Whose Zeros Lie on a Circleo

Some compact generalization of inequalities for polynomials with prescribed zeros

‎Let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial‎ ‎of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$‎. ‎In this paper we obtain some new results about the dependence of $|p(Rz)|$ on $|p(rz)| $ for $r^2leq rRleq k^2$‎, ‎$k^2 leq rRleq R^2$ and for $Rleq r leq k$‎. ‎Our results refine and generalize certain well-known polynomial inequalities‎.

On Orthogonal Polynomials with Regularly Distributed Zeros

In this case the points 0k,, = are sin xkn are equidistributed in Weyl's sense . A non-negative measure da for which the array xkn (da) has the distribution function fl o(t) will be called an arc-sine measure . If du(x) = w(x) dx is absolutely continuous, we apply, replacing dca by w, the notations pn(W,x), yn (w), xkn(w) and call a non-negative w(x) an arc-sine weight if da(x) = w(x) dx is an ...

On the approximation by {polynomials

As usual, p∗ is called a best approximation (b.a.) to f in (or, by elements of) IPγ,n. To give some examples, let X = Lp[0, 1] and set γ(t) = G(·, t), where G(s, t) is defined on [0, 1] × T . With G Green’s function for a k–th order ordinary linear initial value problem on (0, 1] and T = [0, 1), one has approximation by generalized splines. With G(s, t) = e and T = IR, one has approximation by ...