A sequence of continuous real functions ${(f_m)}_{m\in {\mathbb{N}}_0}$, with exponential decay at infinity that implies their Fourier-Laplace transforms $F_m(iz)$ are analytic on $|\Re (z)|<B$, $z\in \mathbb{C}$, generates a sequence of polynomials ${(Q_k)}_{k\in {\mathbb{N}}_0}$, that are biorthogonal to the distributional derivatives ${\mu }_m:={(-1)}^m{f}_m^{(m)}$, $m\in {\mathbb{N}}_0$. The generating function is a generalized Taylor series expansion of the Fourier-Laplace kernel $e^{xz}$ in terms of ${\hat{\mu }}_m(iz)=z^m{F}_m(iz)$, $m\in {\mathbb{N}}_0$. If ${\mu }_m=\omega P_m$, where $P_m$ are polynomials of degree m and ω is a weight function, $Q_m$ is a constant multiple of $P_m$ and they are orthogonal polynomials. In the case where $f_m$ are uniform or quasi-uniform B-splines, the corresponding biorthogonal polynomials exhibit properties reminiscent of Appell sequences, while in the case where $f_m$ are refinable functions they are eigenfunctions of the adjoint of the corresponding refinement operators. For nonuniform B-splines, a suitable choice of the knot sequence defines ${(f_m)}_{m\in {\mathbb{N}}_0}$ in which the corresponding polynomials $Q_k$ reduce to the Jacobi polynomials and the biorthogonal spline functions ${\mu }_m$ reduce to the corresponding weighted Jacobi polynomials, if there are no interior knots.

一系列连续的实函数 ${（F_{米}）}_{米\in {\mathbb{ñ}}_0}$，在无穷大处有指数衰减，这意味着它们的傅里叶-拉普拉斯变换 $F_{米}（\mathrm{一世}ž）$ 正在分析 $|\Re （ž）|<乙$， $ž\in \mathbb{C}$，生成多项式序列 ${（{问}_{ķ}）}_{ķ\in {\mathbb{ñ}}_0}$，与分布导数成直角 ${\mu }_{米}：={（-\mathrm{1个}）}^{米}{F}_{米}^{（米）}$， $米\in {\mathbb{ñ}}_0$。生成函数是傅里叶-拉普拉斯核的广义泰勒级数展开${Ë}^{Xž}$ 就......而言 ${\hat{\mu }}_{米}（\mathrm{一世}ž）={ž}^{米}{F}_{米}（\mathrm{一世}ž）$， $米\in {\mathbb{ñ}}_0$。如果${\mu }_{米}=\omega P_{米}$，在哪里 $P_{米}$是次多项式米和ω是一个权重函数，${问}_{米}$ 是...的常数倍 $P_{米}$它们是正交多项式。在这种情况下$F_{米}$是均匀的或准均匀的B样条，相应的双正交多项式表现出让人联想到Appell序列的特性，而在$F_{米}$是可细化函数，它们是相应细化算符伴随的本征函数。对于非均匀的B样条，合适的结序列选择定义${（F_{米}）}_{米\in {\mathbb{ñ}}_0}$ 其中对应的多项式 ${问}_{ķ}$ 简化为Jacobi多项式和双正交样条函数 ${\mu }_{米}$ 如果没有内部结，则简化为相应的加权Jacobi多项式。