# Polynomial bases

January 29, 2021 — July 28, 2023

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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 analysing 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

- Orcuslc/OrthNet: TensorFlow, PyTorch and Numpy layers for generating Orthogonal Polynomials
- Julia’s ApproxFun.jl (see a write-up in my julia notebook.
- Chebfun is a classic MATLAB toolkit for working with functions represented by Chebyshev polynomials.

## 4 References

*Adapting to Climate Change*.

*Arctic Anthropology*.

*Matrices, Moments and Quadrature with Applications*.

*Journal of the Egyptian Mathematical Society*.

*The Annals of Statistics*.

*The Annals of Statistics*.

*The American Statistician*.

*Proceedings of the 13th International Conference on Neural Information Processing Systems*. NIPS’00.

*Statistics and Computing*.

*Statistics & Probability Letters*.

*SIAM Journal on Scientific Computing*.

*Journal of Computational and Applied Mathematics*, Numerical Analysis 2000. Vol. V: Quadrature and Orthogonal Polynomials,.

*Advances in Computational Mathematics*.

*Science Advances*.