Polynomial bases



Placeholder.

Fun things

Terry Tao on Conversions between standard polynomial bases.

Well known facts

Xiu and Karniadakis (2002) mention the following “Well known facts”:

All orthogonal polynomials \(\left\{Q_{n}(x)\right\}\) satisfy a three-term recurrence relation \[ -x Q_{n}(x)=A_{n} Q_{n+1}(x)-\left(A_{n}+C_{n}\right) Q_{n}(x)+C_{n} Q_{n-1}(x), \quad n \geq 1 \] where \(A_{n}, C_{n} \neq 0\) and \(C_{n} / A_{n-1}>0 .\) Together with \(Q_{-1}(x)=0\) and \(Q_{0}(x)=1,\) all \(Q_{n}(x)\) can be determined by the recurrence relation.

It is well known that continuous orthogonal polynomials satisfy the second-order differential equation \[ s(x) y^{\prime \prime}+\tau(x) y^{\prime}+\lambda y=0 \] where \(s(x)\) and \(\tau(x)\) are polynomials of at most second and first degree, respectively, and \[ \lambda=\lambda_{n}=-n \tau^{\prime}-\frac{1}{2} n(n-1) s^{\prime \prime} \] are the eigenvalues of the differential equation; the orthogonal polynomials \(y(x)=\) \(y_{n}(x)\) are the eigenfunctions.

Zoo

This list is extracted from a few places including Xiu and Karniadakis (2002).

Family Orthogonal in measure
Monomial n/a
Bernstein n/a
Legendre \(\operatorname{Unif}([-1,1])\)
Hermite \(\mathcal{N}(0,1)\)
Laguerre \(x^{\alpha}\exp -x, \, x>0\)
Jacobi \((1-x)^{\alpha }(1+x)^{\beta }\) on \([-1,1]\)
Charlier Poisson distribution
Meixner negative binomial distribution
Krawtchouk binomial distribution
Hahn hypergeometric distribution
??? Unit ball

References

Ismail, Mourad E. H., and Ruiming Zhang. 2017. “A Review of Multivariate Orthogonal Polynomials.” Journal of the Egyptian Mathematical Society 25 (2): 91–110. https://doi.org/10.1016/j.joems.2016.11.001.
O’Hagan, Anthony. 2013. “Polynomial Chaos: A Tutorial and Critique from a Statistician’s Perspective,” 20.
Smola, Alex J., Zoltán L. Óvári, and Robert C. Williamson. 2000. “Regularization with Dot-Product Kernels.” In Proceedings of the 13th International Conference on Neural Information Processing Systems, 290–96. NIPS’00. Cambridge, MA, USA: MIT Press. https://openreview.net/forum?id=ryXbEvbdWS.
Voelker, Aaron R, Ivana Kajic, and Chris Eliasmith. n.d. “Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks,” 10.
Withers, C. S. 2000. “A Simple Expression for the Multivariate Hermite Polynomials.” Statistics & Probability Letters 47 (2): 165–69. https://doi.org/10.1016/S0167-7152(99)00153-4.
Xiu, Dongbin, and George Em Karniadakis. 2002. “The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations.” SIAM Journal on Scientific Computing 24 (2): 619–44. https://doi.org/10.1137/S1064827501387826.
Xu, Yuan. 2001. “Orthogonal Polynomials and Cubature Formulae on Balls, Simplices, and Spheres.” Journal of Computational and Applied Mathematics, Numerical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials, 127 (1): 349–68. https://doi.org/10.1016/S0377-0427(00)00504-5.
———. 2004. “Polynomial Interpolation on the Unit Sphere and on the Unit Ball.” Advances in Computational Mathematics 20 (1): 247–60. https://doi.org/10.1023/A:1025851005416.

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