Zeros of Fekete Polynomials

Zero Distribution of Sequences of Classical Orthogonal Polynomials

In this paper, we study the zero distribution of sequences of Jacobi, Laguerre, and Hermite polynomials. Our approach is based on identifying these orthogonal polynomials with certain Fekete polynomials defined below, and using monotonicity properties of the zeros of the polynomials. Let E ⊂ R be a closed set that consists of finitely many intervals. Let w : E→ [0,∞) be a weight function such t...

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.

A comparison of interpolation grids over the triangle or the tetrahedron

A simple strategy for constructing a sequence of increasingly refined interpolation grids over the triangle or the tetrahedron is discussed with the goal of achieving uniform convergence and ensuring high interpolation accuracy. The interpolation nodes are generated based on a one-dimensional master grid comprised of the zeros of the Lobatto, Legendre, Chebyshev, and second-kind Chebyshev polyn...

Domain of attraction of normal law and zeros of random polynomials

Let$ P_{n}(x)= sum_{i=0}^{n} A_{i}x^{i}$ be a random algebraicpolynomial, where $A_{0},A_{1}, cdots $ is a sequence of independent random variables belong to the domain of attraction of the normal law. Thus $A_j$'s for $j=0,1cdots $ possesses the characteristic functions $exp {-frac{1}{2}t^{2}H_{j}(t)}$, where $H_j(t)$'s are complex slowlyvarying functions.Under the assumption that there exist ...

On the average number of sharp crossings of certain Gaussian random polynomials