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 wrt
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
Gegenbauer \(\left(1-x^2\right)^{\alpha-\frac{1}{2}}\). cf Funk-Hecke formula
??? Unit ball Does this have a name?

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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