Doubly Orthogonal Concentrated Polynomials
Doubly Orthogonal Concentrated Polynomials
复制标题
双正交集中多项式
DOI：
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发表时间：
1977
期刊：
影响因子：
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通讯作者：
D. Slepian
中科院分区：
文献类型：
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作者：
E. Gilbert;D. Slepian
我们寻找一个n次多项式$f_0^{(N)}(X)$，它使比率[R(F)={{intLimits_{i_a}{Left|{f(X)}]最大化
Ight|^2}dx}/{intLimits_{i_b}{Left|{f(X)}
Ight|^2}dx}，]其中$i_a$和$i_b$是实数行上的两个间隔。$R(F)$可解释为一个能量比，而$f_0^{(N)}(X)$可解释为能量最集中于$i_a$的多项式，而在$i_b中，使$R(F)$最大化等价于求一个特征值问题的最大本征值$lambda_0^{(N)}$及相应的特征函数$f_0^{(N)}(X)$。其他特征函数也是n次多项式，因为特征函数$f_j^{(N)}(X)$，$j=0，cdots，n$同时在$i_a$和$i_b$上是正交的，对于较小的n，特征值问题可以用标准矩阵方法数值求解。我们特别注意关于n大数的渐近结果。当$i_a$和$i_b$不相交时，$lambda_0^{(N)}$增长为$C_1 n^{-1}C_2^n…
We seek an nth degree polynomial $f_0^{(n)} (x)$ which maximizes the ratio [R(f) = {{intlimits_{I_a } {left| {f(x)}
ight|^2 } dx} / {intlimits_{I_b } {left| {f(x)}
ight|^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...