Orthogonal polynomial

Orthogonal polynomials of distinct degrees are orthogonal under a positive weighted inner product.

The real Zernike polynomials form an orthogonal polynomial basis on the unit Euclidean disk, with radial factor $R_n^m(\rho )$ and angular factor $\cos m\varphi$ or $\sin m\varphi$. Here $0\le m\le n$ and $n-m$ is even. They decompose wavefront errors into modes useful for optical correction. Normalizations differ, so coefficients should specify the chosen convention.

The unnormalized rotationally symmetric mode is $Z_4^0(\rho )=6{\rho }^4-6{\rho }^2+1$. It describes balanced primary spherical aberration. The lower-degree terms make it orthogonal to constant piston and quadratic defocus under the disk area measure: direct integration gives ${\int }_0^1{Z}_4^0\rho d\rho =0$ and ${\int }_0^1{Z}_4^0(2{\rho }^2-1)\rho d\rho =0$.

A monic orthogonal polynomial has leading coefficient one. For a fixed positive weight, there is a unique monic orthogonal polynomial of each degree.

For monic orthogonal polynomials,where

The probabilists' Hermite polynomials areThey are orthogonal for the standard Gaussian weight and satisfy ${\mathrm{He}}_{n+1}=x{\mathrm{He}}_n-n{\mathrm{He}}_{n-1}$.

The space-time Probabilists' Hermite polynomial $H_n(x,t)=t^{n/2}{\mathrm{He}}_n(x/\sqrt{t})$ extends polynomially to $t=0$ and solves ${\partial }_t{H}_n+\frac{1}{2}{\partial }_x^2{H}_n=0$. Therefore $H_n(B_t,t)$ is a martingale for a standard Brownian motion, by the Itô formula and Gaussian moments.

A degree-n orthogonal polynomial for a positive interval weight has n simple zeros in the interval interior.

A Jacobi matrix is a symmetric tridiagonal matrix whose characteristic polynomials obey the orthogonal-polynomial three-term recurrence.