# Random Reminders#

The nth Legendre polynomial, $$\boldsymbol{L}_{n}$$, is orthogonal to every polynomial with degrees less than n i.e.

• $$\boldsymbol{L}_{n} \perp \boldsymbol{P}_{i}, \ \forall i\in [0..n-1]$$
• ex: $$\boldsymbol{L}_{n} \perp x^{3}$$

$$\boldsymbol{L}_{n}$$ has n real roots and they are all $$\in [-1,1]$$

Harmonic functions => $$\Delta u(x) = 0$$

Homogenous function => $$f : \reals^{n} \rightarrow \reals^{n}, \ f(\lambda \mathbf{v})=\lambda^{k} f(\mathbf{v})$$ where $$k,\lambda \in \reals$$

General form of Newton’s divided-difference polynomial interpolation:

\begin{aligned} f_{n}(x)=& f\left(x_{0}\right)+\left(x-x_{0}\right) f\left[x_{1}, x_{0}\right]+\left(x-x_{0}\right)\left(x-x_{1}\right) f\left[x_{2}, x_{1}, x_{0}\right] \\ &+\cdots+\left(x-x_{0}\right)\left(x-x_{1}\right) \cdots\left(x-x_{n-1}\right) f\left[x_{n}, x_{n-1}, \ldots, x_{0}\right] \end{aligned}

Lagrange interpolating polynomial scheme is just a reformulation of Newton scheme that avoids computation of divided differences

$f_{n}(x)=\sum_{i=0}^{n} L_{i}(x) f\left(x_{i}\right)\newline L_{i}(x)=\prod_{j=0 \atop j \neq i}^{n} \frac{x-x_{j}}{x_{i}-x_{j}}$

• allows for accurately approximating functions where $$f(x) \in P_{2n-1}$$ with only n coefficients

Regression Schemes (Linear or nonlinear)

• Curves do not necessarily go through sample points so error at said points might be large
• Round-off error becomes pronounced for higher order versions and ill-conditioned matrices are a problem
• Orthogonal polynomials do not necessarily suffer from this

Interpolation Schemes (splines, lagriangina/newtonian, etc)

• Curves must go through sample points so error at said points is small
• Not ill conditioned

Thin plate splines

• construction is based on choosing a function that minimizes anintegral that represents the bending energy of a surface
• the idea of thin-plate splines is to choose a functionf(x) that exactly interpolates the datapoints (xi,yi), say,yi=f(xi), and that minimizes the bending energy $$E[f]=\int_{\mathbf{R}^{n}}\left|D^{2} f\right|^{2} d X$$
• Can also choose function that doesn’t exactly interpolate all control points by using smoothing parameter for regularization $$E[f]=\sum_{i=1}^{m}\left|f\left(\mathbf{x}_{i}\right)-y_{i}\right|^{2}+\lambda \int_{\mathbb{R}^{n}}\left|D^{2} f\right|^{2} d X$$

Spherical Basis Spliens:

• Gross reduction summary: bsplines with slerp instead of lerp between control points

Laplacian => Avg of neighbors at a point - point value

• Maximal smoothness/mean curvature is zero
• Poisson equation = $$\Delta u$$ = 0
• Think of boundary condition being a wire and a soap film covering the wire. That’s a $$\Delta u(x,y) = 0$$
• Another interpretation is equilibriam state. Think of temperature
• Another interpretation is that there are no bumps or local minimas in that surface

# Spherical Splines#

Last update: November 14, 2019