Zero distribution of complex orthogonal polynomials with respect to exponential weights

Asymptotics of Laurent Polynomials of Even Degree Orthogonal with Respect to Varying Exponential Weights

Let Λ denote the linear space over R spanned by z, k ∈Z. Define the real inner product (with varying exponential weights) 〈·, ·〉L : Λ ×ΛR→R, ( f , g) 7→ ∫ R f (s)g(s) exp(−N V(s)) ds, N∈N, where the external fieldV satisfies: (i)V is real analytic onR\ {0}; (ii) lim|x|→∞(V(x)/ ln(x2+1))=+∞; and (iii) lim|x|→0(V(x)/ ln(x−2+1))=+∞. Orthogonalisation of the (ordered) base {1, z−1, z, z−2, z2, . . ...

Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Let Λ denote the linear space over R spanned by z, k ∈Z. Define the real inner product (with varying exponential weights) 〈·, ·〉L : Λ ×ΛR→R, ( f , g) 7→ ∫ R f (s)g(s) exp(−N V(s)) ds, N∈N, where the external fieldV satisfies: (i)V is real analytic onR\ {0}; (ii) lim|x|→∞(V(x)/ ln(x2+1))=+∞; and (iii) lim|x|→0(V(x)/ ln(x−2+1))=+∞. Orthogonalisation of the (ordered) base {1, z−1, z, z−2, z2, . . ...

Orthogonal polynomials with respect to varying weights 1

In this paper we review a connection of orthogonal polynomials with respect to varying weights to weighted approximation, multipoint Pad6 approximation and to some questions of theoretical physics. (~) 1998 Elsevier Science B.V. All rights reserved.

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...

Exact Bounds for Orthogonal Polynomials Associated with Exponential Weights*