Doubly Orthogonal Concentrated Polynomials
E. N. Gilbert, David Slepian · SIAM Journal on Mathematical Analysis · 1977
We seek an nth degree polynomial $f_0^{(n)} (x)$ which maximizes the ratio \[R(f) = {{\int\limits_{I_a } {\left| {f(x)} \right|^2 } dx} / {\int\limits_{I_b } {\left| {f(x)} \right|^2 } dx}},\] where $I_a $ and $I_b $ are two intervals on the real line. $R(f)$ may be interpreted as an energy ratio and $f_0^{(n)} (x)$ as the polynomial having its energy most concentrated into $I_a $ at the expense of its energy in $I_b $ Maximizing $R(f)$ is equivalent to finding the largest eigenvalue $\lambda _0^{(n)} $ and corresponding eigenfunction $f_0^{(n)} (x)$ of an eigenvalue problem. The other eigenfunctions, which are also polynomials of degree n, have interest because the eigenfunctions$f_j^{(n)} (x)$, $j = 0, \cdots ,n$, are orthogonal both on $I_a $ and on $I_b $ 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 $I_a $ and $I_b $ are disjoint, $\lambda _0^{(n)} $ grows as $C_1 n^{ - 1} C_2^n $. We give $C_1 $ and $C_2 $ as functions of $I_a $ and $I_b $. We also solve the problem when $I_a $ is centrally positioned inside $I_b $, say, $I_a = [ - a,a]$, $I_b = [ - 1,1]$, with $a < 1$. Then, for large n, $\lambda _0^{(n)} $ has the behavior $1 - C_3 n^{{1 / 2}} C_4^n $ and we obtain $C_3 $ and $C_4 $. In both these cases the eigenvalue problem can be put into differential equation form. When $I_a $ and $I_b $ 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 $C_2^n $ as $\lambda _0^{(n)} $ but with constant factors different from $C_1 $.