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.
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? |
Natural exponential family with quadratic variance function have infinite divisibility in all cases except binomial (Morris 1982).
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 this
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