Polynomial bases

January 29, 2021 — July 28, 2023

functional analysis
Hilbert space
signal processing
Figure 1


You know what is a bloody great introduction to polynomial bases? Golub and Meurant (2010). They cram it into the first chapter and then do computational stuff in the subsequent chapters. It is excellent.

1 Fun things

Terry Tao on Conversions between standard polynomial bases.

Representation in terms of basis is useful for analyzing the Natural exponential family with quadratic variance function (Morris 1982).

1.1 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.

2 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 \([-1,1]\)
Gegenbauer \(\left(1-x^2\right)^{\alpha-\frac{1}{2}}\) on \([-1,1]\) cf Funk-Hecke formula; special case of Jacobi
Chebyshev \(\left(\sqrt{1-x^2}\right)^{\pm1}\) on \([-1,1]\) Special case of Gegenbauer
Charlier Poisson distribution
Meixner negative binomial distribution
Krawtchouk binomial distribution
Hahn hypergeometric distribution
??? Unit ball Does this have a name?

3 Tools

4 References

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Golub, and Meurant. 2010. Matrices, Moments and Quadrature with Applications.
Ismail, and Zhang. 2017. A Review of Multivariate Orthogonal Polynomials.” Journal of the Egyptian Mathematical Society.
Morris. 1982. Natural Exponential Families with Quadratic Variance Functions.” The Annals of Statistics.
———. 1983. Natural Exponential Families with Quadratic Variance Functions: Statistical Theory.” The Annals of Statistics.
Morris, and Lock. 2009. Unifying the Named Natural Exponential Families and Their Relatives.” The American Statistician.
O’Hagan. 2013. “Polynomial Chaos: A Tutorial and Critique from a Statistician’s Perspective.”
Smola, Óvári, and Williamson. 2000. Regularization with Dot-Product Kernels.” In Proceedings of the 13th International Conference on Neural Information Processing Systems. NIPS’00.
Solin, and Särkkä. 2020. Hilbert Space Methods for Reduced-Rank Gaussian Process Regression.” Statistics and Computing.
Voelker, Kajic, and Eliasmith. n.d. “Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks.”
Withers. 2000. A Simple Expression for the Multivariate Hermite Polynomials.” Statistics & Probability Letters.
Xiu, and Karniadakis. 2002. The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations.” SIAM Journal on Scientific Computing.
Xu. 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,.
———. 2004. Polynomial Interpolation on the Unit Sphere and on the Unit Ball.” Advances in Computational Mathematics.
Zhao, Castañeda, Salacup, et al. 2022. Prolonged Drying Trend Coincident with the Demise of Norse Settlement in Southern Greenland.” Science Advances.