Doubly Orthogonal Concentrated Polynomials

We seek an nth degree polynomial f₀⁽ⁿ⁾ (x) which maximizes the ratio \[R(f) = {{∫I_a {| {f(x)} |^2 } dx} / {∫I_b {| {f(x)} |^2 } dx}},\] 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 λ 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, λ 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, λ 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 λ ₀⁽ⁿ⁾ but with constant factors different from C₁.