论文标题
迈向通用吉布斯常数
Towards a Universal Gibbs Constant
论文作者
论文摘要
在本文中，我们建立在\ cite {kaber}的工作的基础上，其中表明gegenbauer多项式（GP）的单参数家族在跳跃不连续性时表现出Gibbs现象。我们表明，一般性拉瓜多项式（GLP）的单参数家族也表现出Gibbs现象。在许多差异中，一个主要的是，GLP在$ \ r $的非紧凑子集上是正交的，而GP则是$ [ - 1,1] $的正交。我们的策略遵循\ cite {kaber}的策略，我们使用完全基本的方法来得出我们的结果。作为特殊情况，我们表明，Hermite多项式也具有Gibbs现象。我们以一个数字示例结束，该示例表现出与GLP的特殊值的收敛速率和猜想的身份。
In this paper we build on the work of \cite{kaber} where it was shown that the one-parameter family of Gegenbauer Polynomials (GP) exhibit a Gibbs Phenomenon at a jump discontinuity. We show that the one-parameter family of Generalized Laguerre Polynomials (GLP) also exhibit a Gibbs Phenomenon. Among many differences, a major one is that the GLP are orthogonal on a non-compact subset of $\R$, while the GP are orthogonal on $[-1,1]$. Our strategy follows that of \cite{kaber} and we use entirely elementary methods to arrive at our result. As a special case we show that the Hermite Polynomials also possess a Gibbs Phenomenon. We conclude with a numerical example exhibiting the rate of convergence to the Gibbs constant and a conjectured identity for special values of the GLP.
