Polynomial bases


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.

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

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.


This list is extracted from a few places including Xiu and Karniadakis (2002).

FamilyOrthogonal wrt
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
CharlierPoisson distribution
Meixnernegative binomial distribution
Krawtchoukbinomial distribution
Hahnhypergeometric distribution
???Unit ballDoes this have a name?



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