# Polynomial bases

January 29, 2021 — July 28, 2023

convolution
functional analysis
Hilbert space
nonparametric
signal processing

Placeholder.

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 .

### 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?

## 4 References

Dugmore, Keller, McGovern, et al. 2001. In Adapting to Climate Change.
Dugmore, Keller, and McGovern. 2007. Arctic Anthropology.
Golub, and Meurant. 2010. Matrices, Moments and Quadrature with Applications.
Ismail, and Zhang. 2017. Journal of the Egyptian Mathematical Society.
Morris. 1982. The Annals of Statistics.
———. 1983. The Annals of Statistics.
Morris, and Lock. 2009. The American Statistician.
O’Hagan. 2013. “Polynomial Chaos: A Tutorial and Critique from a Statistician’s Perspective.”
Smola, Óvári, and Williamson. 2000. In Proceedings of the 13th International Conference on Neural Information Processing Systems. NIPS’00.
Solin, and Särkkä. 2020. Statistics and Computing.
Voelker, Kajic, and Eliasmith. n.d. “Legendre Memory Units: Continuous-Time Representation in Recurrent Neural Networks.”
Withers. 2000. Statistics & Probability Letters.
Xiu, and Karniadakis. 2002. SIAM Journal on Scientific Computing.
Xu. 2001. Journal of Computational and Applied Mathematics, Numerical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials,.
———. 2004. Advances in Computational Mathematics.
Zhao, Castañeda, Salacup, et al. 2022. Science Advances.