Let ${\mathcal{P}}_n$ be the vector space of polynomials of degree at most $n$, equipped with the inner product $〈f,g〉={\int }_{-1}^{1}f\left(x\right)g\left(x\right)dx$. Let $D:{\mathcal{P}}_n⟶{\mathcal{P}}_{n-1}$ be the differentiation operator and $I:{\mathcal{P}}_{n-1}⟶{\mathcal{P}}_n$ be the integration operator. In this paper the generalized (Moore–Penrose) inverses of $D$ and $I$ are computed. The answer for $D$ is straightforward, though perhaps somewhat unexpected. The answer for $I$ requires more work, and involves Legendre polynomials in an unexpected way.

中文翻译：

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微分和积分的广义逆

让 ${\mathcal{磷}}_n$ 最多为次数多项式的向量空间 $n$, 配备内积 $〈F,G〉={\int }_{-1}^{1}F\left(X\right)G\left(X\right)dX$. 让$D：{\mathcal{磷}}_n⟶{\mathcal{磷}}_{n-1}$ 是微分算子和 $\mathrm{一世}：{\mathcal{磷}}_{n-1}⟶{\mathcal{磷}}_n$成为整合运营商。在本文中，广义（Moore-Penrose）逆$D$ 和 $\mathrm{一世}$被计算。答案为$D$很简单，虽然可能有点出乎意料。答案为$\mathrm{一世}$ 需要更多的工作，并以意想不到的方式涉及勒让德多项式。