Doubly Orthogonal Concentrated Polynomials

We seek an nth degree polynomial f₀^ (n) (x) which maximizes the ratio (f) = {₈䂯 {| {f (x) |² } dx} / ₈₁ {| {f (x) |² } dx}}, \ where Iₐ and Ib are two intervals on the real line. R (f) may be interpreted as an energy ratio and f₀^ (n) (x) as the polynomial having its energy most concentrated into Iₐ at the expense of its energy in Ib Maximizing R (f) is equivalent to finding the largest eigenvalue ₀^ (n) and corresponding eigenfunction f₀^ (n) (x) of an eigenvalue problem. The other eigenfunctions, which are also polynomials of degree n, have interest because the eigenfunctionsfⱼ^ (n) (x), j = 0, , n, are orthogonal both on Iₐ and on Ib simultaneously. For small n the eigenvalue problem can be solved numerically by standard matrix methods. We give special attention to asymptotic results for n large. When Iₐ and Ib are disjoint, ₀^ (n) grows as C₁ n^ - 1 C₂ⁿ. We give C₁ and C₂ as functions of Iₐ and Ib. We also solve the problem when Iₐ is centrally positioned inside Ib, say, Iₐ = - a, a, Ib = - 1, 1, with a < 1. Then, for large n, ₀^ (n) has the behavior 1 - C₃ n^{1 / 2} C₄ⁿ and we obtain C₃ and C₄. In both these cases the eigenvalue problem can be put into differential equation form. When Iₐ and Ib are disjoint we maximize other ratios, related to R (f), to obtain maximizing polynomials which are simple expressions involving Chebyshev or Legendre polynomials. These polynomials have R (f) growing with the same exponential term C₂ⁿ as ₀^ (n) but with constant factors different from C₁.